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KCSE REVISION MATHEMATICS (QUESTIONS & ANSWERS)

MATHEMATICS I

PART I

SECTION I (50 MARKS)

  1. Evaluate without mathematical tables leaving your answer in standard form

0.01712 X 3

855 X 0.531                                                                                                                  (2 Mks)

  1. Six men take 14 days working 8 hours a day to pack 2240 parcels. How many more men working

5 hours a day will be required to pack 2500 parcels in 2 days                                                      (3 Mks)

 

  1. M                                  In quadrilateral OABC, OA = 4i – 3j. OC = 2i + 7j

AB = 3OC. cm: mB = 2:3. Find in terms of  i and j

C                                                           vector Om                                           (3 Mks)

 

 

 

 

 

O                                                A

 

  1. By matrix method, solve the equations

5x + 5y = 1

4y + 3x = 5                                                                                                                         (3 Mks)

 

 

  1. In the given circle centre O, ÐABC = 1260.

Calculate ÐOAC                                           (3 Mks)

 

A                                     C

 

 

 

B

 

  1. Solve the equation

2(3x – 1)2 9 (3x – 1) + 7 = 0                                                                                               (4 Mks)

  1. Maina, Kamau and Omondi share Shs.180 such that for every one shilling Maina gets, Kamau gets 50

Cts and for every two shillings Kamau gets, Omondi gets three shillings. By how much does Maina’s

share exceed Omondi’s                                                                                                         (3 Mks)

  1. Expand (2 + 1/2x)6 to the third term. Use your expression to evaluate 2.46 correct to 3 s.f (3 Mks)
  2. The probability of failing an examination is 0.35 at any attempt. Find the probability that

(i)   You will fail in two attempts                                                                                  (1 Mk)

(ii)   In three attempts, you will at least fail once                                                                       (3 Mks)

  1. Line y = mx + c makes an angle of 1350 with the x axis and cuts the y axis at y = 5. Calculate the

equation of the line                                                                                                             (2 Mks)

  1. During a rainfall of 25mm, how many litres collect on 2 hectares? (3 Mks)
  2. Solve the equation a 3a – 7 = a – 2 (3 Mks)

3       5          6

  1. The sum of the first 13 terms of an arithmetic progression is 13 and the sum of the first 5 terms is

–25. Find the sum of the first 21 terms                                                                                (5 Mks)

  1. The curved surface of a core is made from the shaded sector on the circle. Calculate the height of

the cone.                                                                                                                            (4 Mks)

 

 

 

 

 

O

20cm      1250                   20 cm

 

 

 

 

 

 

  1. Simplify (wx – xy – wz + yz) (w + z) (3 Mks)

z2 – w2

  1. The bearing of Q from P is North and they are 4 km apart. R is on a bearing of 030 from P and on

a bearing of 055 from Q. Calculate the distance between P and R.                                        (3 Mks)

 

SECTION II (50 MARKS)

  1. In the given circle centre O, ÐQTP = 460, ÐRQT = 740 and ÐURT = 390

 

 

U                                                   T                                P

 

 

Q

S          390

      Calculate                                                                                    R

(a)  ÐRST                                                        (1 Mk)

(b)  ÐSUT                                                       (3 Mks)

(c)  Obtuse angle ROT                                    (2 Mks)

(d)  ÐPST                                                        (2 Mks)

  1. The exchange rate on March 17th 2000, was as follows: –

1 US$ = Kshs.74.75

1 French Franc (Fr) = Kshs.11.04

      A Kenyan tourist had Kshs.350,000 and decided to proceed to America

(a)  How much in dollars did he receive from his Kshs.350,000 in 4 s.f?                               (2 Mks)

(b) The tourist spend  ¼  of the amount in America and proceeded to France where he spend Fr

16,200. Calculate his balance in French Francs to 4 s.f                                                   (3 Mks)

(c) When he flies back to Kenya, the exchange rate for 1 Fr = Kshs.12.80. How much more in

Kshs. does he receive for his balance than he would have got the day he left?                 (3 Mks)

  1. On the provided grid, draw the graph of y = 5 + 2x – 3x2 in the domain -2 £ x £ 3               (4 Mks)

(a) Draw a line through points (0,2) and (1,0) and extend it to intersect with curve y = 5 + 2x – 3 x 2

read the values of x where the curve intersects with the line                                         (2 Mks)

(b)  Find the equation whose solution is the values of x in (a) above                                     (2 Mks)

  1. (a) Using a ruler and compass only, construct triangle PQR in which PQ = 3.5 cm, QR = 7 cm

and angle PQR = 300                                                                                                     (2 Mks)

(b)  Construct a circle passing through points P, Q and R                                                     (2 Mks)

(c)  Calculate the difference between area of the circle formed and triangle PQR                   (4 Mks)

  1. The given Region below (unshaded R) is defined by a set of inequalities. Determine the inequalities (8 Mks)

Y

 

4

 

 

 

2                   R              (3,3)

  

 

X

-3                           5

 

 

 

 

 

 

 

  1. The table below shows the mass of 60 women working in hotels

 

Mass (Kg) 60 – 64 65 – 69 70 – 74 75 – 79 80 – 84 85 – 89
No. of women 8 14 18 15 3 2

 

(a)   State (i)   The modal class                                                                                             (1 Mk)

(ii)  The median class                                                                                           (1 Mk)

(b)   Estimate the mean mark                                                                                                           (4 Mks)

(c)   Draw a histogram for the data                                                                                       (2 Mks)

  1. XY, YZ and XZ are tangents to the circle centre O

at points A, B, C respectively. XY = 10 cm,

YZ = 8 cm and XZ = 12 cm.                                                                                         (2 MKS)

Z

 

 

C

 

 

 

 

..                    B

X

 

A                    Y

 

 

(a)  Calculate, length XA                                                                                                    (2 Mks)

(b)  The shaded area                                                                                                                  (6 Mks)

  1. Maina bought a car at Kshs.650,000. The value depreciated annually at 15%

(a)  After how long to the nearest 1 decimal place will the value of the car be Kshs.130,000        (4 Mks)

(b)  Calculate the rate of depreciation to the nearest one decimal place which would make the value of

the  car be half of its original value in 5 years                                                              (4 Mks)

 

MATHEMATICS I

PART II

SECTION 1 (50 MARKS)

 

 

  1. Simplify 32a10   -2/5 ÷  9b4      11/2

b15             4a6                                                                                                 (2 Mks)

 

  1. Use logarithm tables to evaluate

Ö0.375 cos 75

tan 85.6                                                                                                       (4 Mks)

  1. The marked price of a shirt is Shs.600. If the shopkeeper gives a discount of 20% off the marked price, he makes a loss of 4%. What was the cost of the shirt? (3 Mks)
  2. The surface area (A) of a closed cylinder is given by A = 2pr2 + 2prh where r is radius and h is height of the cylinder. Make r the subject. (4 mks)
  3. In the circle centre O, chords AB and CD intersect at X. XD = 5 cm

      XC = 1/4 r where r is radius. AX:XO = 1:2 Calculate radius of the circle.                             (3 mks)

 

A             5cm       D

 

 

C                O

 

B

 

 

  1. Simplify     2       –        1                                                                                             (3 mks)

5 – 2Ö3     5 + 2Ö3

 

 

  1. P is partly constant and partly varies as q2. When q = 2, P = 6 and when q = 3, P = 16. Find q when P = 64                               (4 mks)
  2. The figure on the side is a tent of uniform cross-section A                           F

ABC. AC = 8m, BC = 8m, BD = 10m   and (ACB = 1200.                  8m

If a scout needs 2.5 m3 of air, how many scouts can fit                      120o C                     E

in the tent.                                                                                                            8m                   (4 mks)

B                              D

10m

  1. The length of a rectangle is given as 8 cm and its width given as 5 cm. Calculate its maximum % error in its perimeter                (3 mks)
  2. ABCD is a rectangle with AB = 6 cm, BC = 4 cm AE = DH = 4 cm BF = CG = 12 cm. Draw a

labelled net of the figure and show the dimensions of the net

  1. Expand (1 + 2x)6 to the 3rd term hence evaluate (1.04)6 (4 mks)
  2. The eye of a scout is 1.5m above a horizontal ground. He observes the top of a flag post at an

angle of elevation of 200. After walking 10m towards the bottom of the flag post, the top is observed at angle of elevation of 400. Calculate the height of the flag post                                  (4 mks)

  1. A bottle of juice contains 405ml while a similar one contains 960ml. If the base area of the

larger Container is 120 cm2. Calculate base area of the smaller container.                             (3 mks)

  1. It takes a 900m long train 2 minutes to completely overtake an 1100m long train travelling at

30km per hour. Calculate the speed of the overtaking train                                                  (3 mks)

  1. Okoth traveled 22 km in 23/4 hours. Part of the journey was at 16 km/h and the rest at 5 km/h.

Determine the distance at the faster speed                                                                           (3 mks)

  1. P and Q are points on AB such that AP:PB = 2:7 and AQ:QB = 5:4 If AB = 12 cm, find PQ

(2 Mks)

SECTION B (50 MARKS)

 

  1. The income tax in 1995 was collected as follows:

      Income in Kshs. p.a                rate of tax %

1 – 39,600                               10

39,601 – 79,200                               15

79,201 – 118,800                             25

118,801 – 158,400                           35

158,401 – 198,000                           45

      Mutua earns a salary of Kshs.8,000. He is housed by the employer and therefore 15% is added to his salary to arrive at its taxable income. He gets a tax relief of Shs.400 and pay Shs.130 service charge. Calculate his net income                                                                                    (8 Mks)

  1. The probability Kioko solves correctly the first sum in a quiz is 2/5 Solving the second correct

is 3/5 if the first is correct and it is 4/5 if the first was wrong. The chance of the third correct is

2/5 if the second was correct and it is 1/5 if the second was wrong. Find the probability that

(a)  All the three are correct                                                                                    (2 Mks)

(b)  Two out of three are correct                                                                              (3 Mks)

(c)  At least two are correct                                                                                     (3 Mks)

  1. A businessman bought pens at Shs.440. The following day he bought 3 pens at Shs.54. This

purchase reduced his average cost per pen by Sh.1.50. Calculate the number of pens bought earlier and the difference in cost of the total purchase at the two prices                                      (8 mks)

 

 

 

 

  1. In D OAB, OA = a, OB = b

OPAQ is a parallelogram.

      ON:NB = 5:-2, AP:PB = 1:3

Determine in terms of a and b vectors

(a)  OP                                                                                                                   (2 Mks)

(b)  PQ                                                                                                                   (2 Mks)

(c)  QN                                                                                                                   (2 Mks)

(d)  PN                                                                                                                   (2 mks)

 

  1. A cylindrical tank connected to a cylindrical pipe of diameter 3.5cm has water flowing at 150

cm per second. If the water flows for 10 hours a day

(a)  Calculate the volume in M3 added in 2 days                                                                   (4 ms)

(b) If the tank has a height of 8 m and it takes 15 days to fill the tank, calculate the base radius

of the tank                                                                                                                     (4 mks)

  1. A joint harambee was held for two schools that share a sponsor. School A needed Shs.15 million while

School B needed 24 million to complete their projects. The sponsor raised Shs.16.9 million while other

guest raised Shs.13.5 million.

(a) If it was decided that the sponsor’s money be shared according to the needs of the school

with the rest equally, how much does each school get                                               (5 mks)

(b) If the sponsor’s money was shared according to the schools needs while the rest was in the  ratio of

students, how much does each school get if school A has 780 students and school B 220

students                                                                                                                        (3 mks)

  1. Voltage V and resistance E of an electric current are said to be related by a law of the form

V = KEn where k and n are constants. The table below shows values of V and E

      V

0.35 0.49 0.72 0.98 1.11
E 0.45 0.61 0.89 1.17 1.35

      By drawing a suitable linear graph, determine values of k and n hence V when E = 0.75(8mks)

  1. The vertices of triangle P,Q,R are P(-3,1), Q (-1,-2), R (-2,-4)

(a)  Draw triangle PQR and its image PIQIRI of PQR under translation T =    3    on the provided grid                                                                                                                4                        (2 Mks)

(b)  Under transformation matrix m =    4  3  , PIQIRI is mapped on to PIIQIIRII. Find the

co-ordinates of PIIQIIRII and plot it   1  2    on the given grid                                          (4 Mks)

(c)  If area of D PIQIRI is 3.5 cm2, find area of the images PIIQIIRII                                        (2 Mks)

 

MATHEMATICS I

PART 1

MARKING SCHEME

 

  1. 171 X 171 X 3 X 10-5 M1

                                  855 X 531

= 2 X 10-6                                                                                     A1

  2

 

  1. No. of men = 6 X 14 X 8 X 2500 M1

                                  2 X 5 X 2240

= 75                                                                            A1

Extra men        = 75 – 6 = 69                                                                B1

 3

  1. OM = 2i + 7j + 2/5 (4i – 3j + 6i + 21j – 2i – 7j) M1

= 2i + 7j + 2/5 (8i + 11j)                                                           M1

= 26 i + 57 j

5       5                                                                               A1

  3

 

 

 

 

 

  1. 2 5       x         =      1

3  4       y                   5                                                                                    M1

 

x          -1/7   5/7       1

y    =     3/7   -2/7      5                                                                M1

 

x    =  3

y       -1

 

x, 3, y = -1                                                                                A1

 3

 

  1. Reflex ÐAOC = 126 x 2 = 2520 B1

Obtuse ÐAOC = 360 – 252 = 1080                                                               B1

= 1/2 (180 – 108)0

= 360                                                                                B1

 3

  1. 18x2 – 39x + 18 = 0

6x2 – 13x + 6 = 0                                                                                         B1Ö equation

6x2 – 9x – 4x + 6 = 0

3x(2x – 3) (3x – 2) = 0                                                                                  M1

x = 2/3  or                                                                                  A1

x =1 ½                                                                                      B1

4

 

  1. M :  K  :  O  =  4 : 2 : 3                                                                              B1Ö ratio

      Maina’s  = 4/9 X 180

= 80/-                                                                                     B1Ö Omondi’s

      Omondi’s = 60/-                                                                                          and Maina’s

      Difference = Shs.20/-                                                                                   B1 difference

3

  1. (2 + 1/2x)6 = 26 + 6(25) (1/2x + 15 (24) (1/2 x)2 M1

= 64 + 96x + 60x2                                                                     A1

2.46      = (2 + 1/2 (0.8))6

= 64 + 96 (0.8) + 60 (0.64)                                                        M1

= 179.2

@179 to 3 s.f                                                                             A1

 4

  1. P (FF) = 7/20 X 7/20

= 49/100                                                                                                            B1

P (at least one fail) = 1 – P (FI FI FI)

= 1- 13/20   3                                                      M1

= 1 – 2197                                                       M1

8000

= 5803

                                                     8000                                                                        A1

 4

 

  1. grad = term 135

= -1                                                                                                            B1

y  = mx + c

y  = -x + 5                                                                                          B1

 2

 

  1. Volume = 2 x 10,000 x 10,000 x 25 M1Ö x section area

1000                 10                                                            M1Ö conv. to litres

= 500,000 Lts                                                               A1

 3

 

  1. 10a – 6(3a – 7) = 5(a -2) M1

10a – 18a + 42 = 5a – 10

– 13a    = -52                                                                                        M1

a        = 4                                                                                           A1

 3

  1. 2a + 12d = 2

2a + 4d = -10                                                                                              M1

8d   = 12

d   = 11/2                                                                                                   A1

a   = -8                                                                                                     B1

S21  = 21/2 (-16 + 20 X 3/2)                                                                           M1

= 147                                                                                             A1

 5

 

  1. 2 p r = 120 x p x 40 M1

360

r = 6.667 cm                                                                                         A1

h =  Ö 400 – 44.44                                                                                 M1

= 18.86 cm                                                                                          A1

 4

  1. = (w (x – z) – y (x – z)) (w + z) M1Ö factor

(z – w) (z + w)

= (w – y) (x – z) (w + z)                                                             M1Ö grouping

(z – w) (z + w)

= (w – y) (x – z)

z – w                                                                                         A1

 3

 

R

250                                                                                B1Ö sketch

  1. 550

Q  125                                            PR = 4 sin 125                                              M1

Sin 25

A1

30

P                                                                                                          3

  1. (a) <RST = 1800 – 740  = 1060                                                              B1

(b) < RTQ = 900– 740           = 160                                                                B1

< PTR = 460 + 160         = 620                                                                B1

< SUT = 620 – 390         = 230                                                                B1

(c)  Reflex ÐRQT = 180 – 2 x 16

= 180 – 32 = 1480                                                                      B1

Obtuse ROT = 360 – 148 = 2120                                                                   B1

(d)  < PTS = 46 + 180 – 129 = 970                                                                      B1

< PST = 180 – (97 + 39) = 440                                                                      B1

8

(a)  Kshs.350,000 = $ 350,000                                                                           M1

74.75

= $ 4682                                                                                   A1

(b) Balance             = 3/4 x 4682

= $ 3511.5                                                                          B1

$3511.5      = Fr 3511.5 x 74.75                                                                   M1

11.04

= Fr 23780                                                                    A1

Expenditure      = Fr 16 200

Balance            = Fr 7580

(c) Value on arrival = Kshs.7580 X 12.80

= Kshs.97,024

Value on departure        = Kshs.7580 X 11.04                                                              B1 bothÖ

= Kshs.83 683.2

Difference                      = Kshs.97,024 – 83683.2                                         M1

= Kshs.13,340.80                                                   A1

 8

X -2 -1 0 1 2 3
Y -11 0 5 4 -3 -16

B1Övalues

 

y

S1Ö scale

8 —                                                                P1Ö plotting

6 —                                                                C1 Ö curve

4 —

2

 

-2 —    1            2          3                                x

-4 —

-6 —

-8 —                                                        y=2x=2

-10 —

-12 —

-14 —                                                                            x   =-0.53 + 0.1  BI

-16 —                                                                           Nx = 1.87+ 0.1

 

y = 5+2x-3x2 =2-2x                  MI for equation

3x2-4x-4x-3=0                   AI equation

8

x     = -0.53 ± 0.1                                                                     B1

mx   = 1.87 ± 0.1

 

 

y = 5 + 2x – 3x2 = 2 – 2x                               M1 Ö for equation

\ 3×2 – 4x – 3 = 0                                                         MA1 Ö equation

 8

 

 

 

 

 

 

 

 

 

20.

 

 

 

 

B1 Ö 300

 

R                                                                                                      B1 Ö 2 ^ PQ, QR

B1 Ö 2 ^ bisectors

B1 Ö circle

 

 

9                         Q

 

 

Radius = 4.2 ± 0.1                                                                                 B1Ö radius

Area of circle = 22/7 x 4.22

= 55.44 ± 3 cm2

Area of D PQR = 1/2 x 3.5 x 7.5 sin 30                                                    M1Ö D and circle

= 6.5625 cm2

Difference               = 55.44 – 6.5625                                                                 M1Ö sub

= 48.88 cm2                                                                       A1

 8

  1. Line (i) y/2 + x/5 = 1

5y + 2x = 10                                                                             B1Öequation

5y + 2x = 10                                                                             B1Ö inequality

      Line (ii)      y/4 + x/-3 = 1

3y = 4x + 12                                                                 B1Ö equation

3y < 4x + 12 or 3y – 4x < 12                                          B1Ö inequality

      Line (iii)     grad = -1/3 y inter = 4

3y + x = 12 or 3y = -x + 12                                            B1Ö equation

3y + x < 12                                                       B1Ö inequality

      Line (iv)      y – 3 = -3

x – 3      2

2y + 3x = 15                                                                 B1Ö equation

\         2y + 3x £ 15                                                                 B1Ö equation

  8

CLASS

F x Fx Cf
60 – 64

65 – 69

70 – 79

75 – 79

80 – 84

85 – 89

8

14

18

15

3

2

62

67

72

77

82

87

 496

938

1296

1155

246

174

8

28

40

55

58

60

  Sf = 60       Sfx 3809  

 

B1Ö x column

B1Ö f column

 

 

 

 

(a)  (i)  Modal class   = 70 – 74                                                                    B1Ö model class

(ii) Median class = 70 – 74                                                                    B1Ö median

 

(b)              Mean =  3809

                                         60                                                                           M1

= 63.48                                                                         A1

 

S1Ö scale

B1 Ö blocks

59.5 – 64.5

64.5 – 69.5 e.t.c.

 8

(c)

 

Histogram

 

 

 

20  —

 

 

15  —

 

 

10 – –

 

 

5  —

 

 

 

 

55    60        65        70        75        80        85        90

 

  1. (a) XA = a, YA = 10 – a, YB = 10 – a, CZ = 10 – a = ZB

YZ = 10 – a + 12 – a = 8                                                                         M1

2a = 14

a = 7 cm                                                                                 A1

Cos X = 100 + 144 – 64

240                                                                               M1Ö any angle of the D

= 0.75

X = 41.410

     1/2 X = 20.700                                                                                     A1Ö 1/2 of the angle

 

r = OA = 7tan 20.7                                                                                   B1 Ö radius

= 2.645 cm

Shaded area = 1/2 X 10 X 12 sin 41.41 – 22/7 X 2.6452                                    M1 Ö D & circle

= 39.69 – 21.99

= 17.7 cm2                                                                    A1Ö

 8

 

 

 

 

 

 

 

  1. (a) 650,000 (0.85)n = 130,000                                                         M1Ö formula

1.15n    = 0.2

n    = log 0.2                                                         M1Ö

log 0.85

1.3010

1.9294

= – 0.6990                                                        M1

– 0.0706

= 9.9 years                                                       A1

(b)  650,000 (1 – r/100) 5 = 325,000                                                                 M1

(1 – r/100) 5 = 0.5

1 – r/100     = 0.5 1/5                                                                   M1

= 0.8706

r/100 = 0.1294                                                                 A1

r    = 12.9 %                                                               B1

 8

MATHEMATICS I

PART II

MARKING SCHEME

 

SECTION I (50 MARKS)

 

 

  1. = b15      2/5    X    4a6   3/2

32a10                9b4                                                                   M1Ö reciprocal

 

 

=          2a5                                                                                           A1

27                                                                                            2

 

      No.             Log.                

0.375          1.5740 +

cos 75         1.4130

2.9870 _

tan 85.6      1.1138

3.8732 =  4 + 1.8732

2                  2

2.9366

0.0864

 

  1. S. Price =  80   X 600

100

= Shs.480                                                                         B1

Cost Price = x

96x       = 480                                                                            M1

100

x  =   Shs.500                                                                      A1

 3

  1. r2 + hr = A/2p                                                 M1

r2 + hr + (h/2)2 = A/2A + h/4                                                                            M1

(r + h/2)2 =  Ö 2A + h2

4p                                                                                        M1

r    = -h/2 ±   Ö2A + h2                                                                            A1

4p                                                                               4

 

  1. (12/3r) (1/3 r) = (1/4 r) (5) M1

4r2 – qr = 0

r(4r – q) = 0                                                                                                 M1

r = 0

or   r  = 2.25                                                                                         A1

 3

 

  1. = 2 (5 + 2Ö3) – 1 (5 – 2Ö3) M1

(5 – 2Ö3) (5 + 2Ö3)

= 10 + 4Ö3 – 5 +2Ö3                                                                                                M1

13

= 5 + 6Ö3                                                                                                    A1

13                                                                                                      3

  1. P = Kq2 + c

6 = 4k + c

16 = 9k + c                                                                                     M1 Ö subtraction

5k = 10

k = 2

c = -2                                                                                                         A1 Ö k and c

      P = 64     2q2 = 66

q  = Ö33

= ± 5.745                                                                                A1

 4

  1. Volume = 1/2 X 8 X 8 sin 120 X 10 M1 Ö area of x-section

      No. of scouts = 32 sin 60 X 10                                                         M1 Ö volume

2.5                                                                               M1

= 110.8

= 110                                                                                        A1

 3

 

  1. Max. error = 2(8.5 + 5.5) – 2(7.5 + 4.5)

2

= 2                                                                                           B1

% error = 2/26 X 100                                                                                 M1

= 7.692%                                                                                  A1

G                                                                          3

 

 

  1. B1 Ö net

 

H             D                             G                       H                                  B1 Ö dimen. FE must be 10cm

 

4cm                                                                                  4cm

 

B1 Ö labelling

E 4cm  A                        12cm      F     10cm    E                                      3

4cm                12cm

E

F

  1. (1 + 2x)6 = 1 + 6(2x) + 15 (2x)2 M1

= 1 + 12x + 60x2                                                                       A1

(1.04)6 = (1 + 2(0.02))6

= 1 + 12 (0.02) + 60(0.02)2                                                        M1

= 1.264                                                                                     A1

 4

 

 

 

 

  1. BT = 10 cm                              B1

CT = 10 sin 40                          M1

= 6.428 m                                 A1

A1 10cm    B                  C                                h = 6.428 + 1.5

1-5                                  = 7.928                                  B1

  4

 

 

  1. A.S.F = 405 2/3  =  27  2/3   =   9                                                                  B1

960           64            10

smaller area = 29  X 120                                                                        M1

164

= 67.5 cm2                                                                                A1

  3

 

  1. Relative speed = (x – 30)km/h B1

2 km     =          2 hrs

(x – 30)km/h      60                                                                             M1

2x – 60 = 120

x = 90 km/h                                                                              A1

  3

  1. 16 Km/h 5 Km/hr

x Km                          (22 – x) Km

x + 22 – x   = 11

16        5           4                                                                                    M1

5x + 352 – 16x = 220                                                                 M1Ö x-multiplication

11x  = 132

x  = 12 km                                                                  A1

  3

 

  1. AP = 2/9 x 12 = 22/3 cm B1 Ö both AP & AQ

      AQ = 5/9 x 12 = 62/3 cm

\ PQ = 62/3 – 22/3 = 4 cm                                                                            B1 Ö C.A.O

  2

 

  1. Taxable income = 115/100 x 8000 M1

= Shs.9200 p. m

= Shs.110,400 p.a                                                                   A1

Tax dues = 10/100 x 39600 + 15/100 x 39600 + 25/100 x 31200                 M1 Ö first 2 slabs

= 3960 + 5940 + 7800                                                               M1 Ö last slab

= Shs.17,700 p.a

= 1475 p.m                                                                               A1

net tax = 1475 – 400

= Shs.1075                                                                             B1 Ö net tax

Total deductions = 1075 + 130

= Shs.1205

net income = 8000 – 1205                                                                      M1

= Shs.6795                                                                   A1

  8

 

 

 

 

 

(a)  P (all correct) = 2/3 x 3/5 x 2/5                                                                  M1

= 12/125                                                              A1

(b)  P (2 correct) = 2/5 x 3/5 x 3/5 + 2/5 x 2/5 X 1/5 + 3/5 x 4/5 x 2/5

                                                                                                                        M1

= 18/125 + 4/125 + 24/125                                         M1

= 46/125                                                              A1

(c) P (at least 2 correct)

= P(2 correct or 3 correct)

= 46/125 + 12/125                                                                           M1

= 46 + 12                                                                                  M1

125

=  58

                                         125                                                                          A1

  8

  1. Old price/pen = 440

x

New price/pen = 494                                                                 B1Öboth expressions

x + 3

440494   = 1.50

x      x + 3                                                                               M1 Ö expression

440(x + 3) – 494x = 1.5x2 + 4.5x                                                M1Ö x-multiplication

x2 + 39x – 880 = 0                                                                     A1 Ö solvable quad. Eqn

x2 + 55x – 16x – 880 = 0                                                 M1 Ö factors or equivalent

(x – 16) (x + 55) = 0

x = -55

or x = 16                                                                                   A1 Ö both values

\ x = 16

difference in purchase = 19 X 1.50                                                        M1

= Shs.28.50                                                           A1

  8

  1. (a) OP = a + 1/4 (b – a) M1

= 3/4 a + 1/4 b                                                                            A1

(b)  PQ = PO + OQ

= –3/4 a – 1/4 b + 1/4 (a – b)                                                          M1

= –1/2 a – 1/2 b                                                                           A1

(c)  QN = QO + ON

= 1/4 (b – a) + 5/3 b                                                              M1

= 23/12 b – 1/4 a                                                                    A1

(d)  PN = PB + BN

= 3/4 (b – a) + 2/3 b                                                               M1

= 17/12 b – 3/4 a                                                                     A1

  8

  1. (a) Volume in 2 days = 22 x 3.5 x 3.5 x 150 x 20 x 3600 M1 Ö area of x-section

7       2        2           1,000,000                                 M1 Ö volume in cm3

= 103.95 m3                                                                  M1 Ö volume in m3

(b)  22 X r2 x 8 = 103.95 x 15   x 7                                                               M1

7                               2

 

r2 = 103.95 x 15 x 7                                                                   M1

                                  2 x 2 2x 8

= 31.01                                                                                     M1

r = 5.568 m                                                                               A1

  8

  1. (a) Ration of needs for A:B = 5:8

A’s share = 5/13 x 16.9 + 1/2 x 13.5                                                          M1

= 13.25 Million                                                                         A1

B’s share = (13.5 + 16.9) – 13.25                                                                        M1

= 13.25                                                                                     M1

  • A’s share 5/13 x 16.9 + 39/50 x 13.5

6.5 + 10.53

= 17.03 m                                                                                 A1

B’s share = 30.4 – 17.03                                                                         M1

= 13.37 Million                                                                         A1

  8

  1. Log V = n Log E = log k
Log V -0.46 -0.13 -0.14 -0.01 0.05
Log E -0.35 -0.21 -0.05 0.07 0.13

B1Ö log V all points

B1Ö log E all points

S1 Ö scale

P1Ö plotting

Log V = n log E + log K                                    L1 Ö line

                                                Log K = 0.08

K = 1.2 ± 0.01                                                  B1 Ö K

N = 0.06/0.06                                                        B1 Ö n

= 1 ± 0.1

\ v = 1.2E                                                       B1Ö v

when E = 0.75, V = 0.9 ± 0.1                            8

  1. (a) T 3 PQR ® PIQIRI

4    PI (0,5), QI (2,2) RI (1,0)

PI QI RI       PII  QII  RII

(b)  4  3    0   2   1   =   15    14   4                                                              M1 Ö

1  2     5   2   0        10     6    1                                                             A1 Ö

 

PII (15,10), QII (14,6), RII (4,1)                                                               B1Ö

(c)  Area s.f = det M

= 5

 

area of PII QII RII = 5 (area PIQIRI)

= 5 X 3.5                                                          M1Ö

= 16.5 cm2                                                        A1

  8

 

 

 

MATHEMATICS 2

PART I

 

SECTION A: 

 

  1. Use logarithm tables to evaluate                      (4 mks)

 

0.0368 x 43.92

361.8

 

  1. Solve for x by completing the square                           (3mks)

2x2  – 5x + 1 = 0

 

  1. Shs. 6000 is deposited at compound interest rate of 13%. The same amount is deposited at 15% simple interest. Find which amount is more and by how much after 2 years in the bank       (3mks)

 

  1. The cost of 3 plates and 4 cups is Shs. 380. 4 plates and 5 cups cost Shs. 110 more than this. Find the cost of each item.                                                                                                        (3mks)

 

  1. A glass of juice of 200 ml content is such that the ratio of undiluted juice to water is 1: 7 Find how many diluted glasses can be made from a container with 3 litres undiluted juice       (3mks)

 

  1. Find the value of θ within θ  < θ < 360if  Cos (2 θ + 120) =  γ3                                                     (3mks)

2

 

  1. A quantity P varies inversely as Q2 Given that P = 4 When Q = 2.  , write the equation joining P and  Q

hence find P when Q = 4                                         a                                                                      (3mks)

 

  1. A rectangle measures 3.6 cm by 2.8 cm. Find the percentage error in calculating its perimeter.                                                                                                                                                 (3mks)

 

  1. Evaluate:          11/6   x  ¾  –  11/12                                                                                              (3mks)

½  of 5/6

 

  1. A metal rod, cylindrical in shape has a radius of 4 cm and length of 14 cm. It is melted down and recast into small cubes of 2 cm length. Find how many such cubes are obtained          ( 3mks)

 

  1. A regular octagon has sides of 8 cm. Calculate its area to 3 s.f.             (4mks)

 

  1. Find the values of x and  y if                                                                                                       ( 2 mks)

3          x          1   =     2

2          1          -1         y

 

  1. An equation of a circle is given by x2 + y2 – 6x + 8y – 11 = 0       (3mks)

Find its centre and radius

 

 

 

 

 

 

  1. In the figure given AB is parallel to DE. Find the value of x and y

 

 

 

 

 

 

 

 

  1. A line pass through A (4,3) and B(8,13). Find                                                  (6 mks)

(i)  Gradient of the line

(ii)  The magnitude of AB

(iii) The equation of the perpendicular bisector of AB.

 

  1. A train is moving towards a town with a velocity of 10 m/s. It gains speed and the velocity becomes 34 m/s after 10 minutes . Find its acceleration (2mks)

 

 

SECTION B:

 

  1. Construct without using a protractor the triangle ABC so that BC=10cm, angle ABC = 600 and

BCA = 450

  1. On the diagram , measure length of AC
  2. Draw the circumference of triangle ABC
  3. Construct the locus of a set of points which are equidistant from A and B.
  4. Hence mark a point P such that APB = 450 and AP = PB
  5. Mark a point Q such that angle AQB = 450 and AB = AQ

 

  1. (a) A quadrilateral ABCD has vertices A(0,2) , B(4,0) , C(6,4) and D(2,3). This is given a

transformation by the matrix   -2  0  to obtain its image AI B I CI DI. under a second transformation

0 – 2

which has a rotation centre (0,0) through –900 , the image AII  BII  CII  DII  of AI  BI  CI  DI  is

obtained.    Plot the three figures on a cartesian plane                                                         (6mks)

(b)  Find  the  matrix of  transformation  that  maps  the  triangle  ABC  where A (2,2)   B (3,4)   C (5,2)

onto  A B C   where  A( 6,10)  B  (10,19 )  C ( 12, 13).                                                    ( 2mks)

 

 

 

 

19.

 

 

 

 

 

 

In the triangle OAB, OA = 3a , OB = 4b and OC = 5/3 OA.  M divides OB in the ratio 5:3

  1. Express AB and MC in terms of a and b
  2. By writing MN in two ways, find the ratio in which N divides
  3. AB
  4. MC

 

 

 

 

 

 

  1. In the figure below, SP = 13.2 cm, PQ = 12 cm, angle PSR = 80O and angle PQR = 900. S and Q are the centres       (8mks)

 

Calculate:

The area of the intersection of the two circles

The area of the quadrilateral  S P Q R

The area of the shaded region

 

 

 

 

 

 

 

 

 

 

 

  1. In an experiment the two quantities x and y were observed and results tabled as below
X 0 4 8 12 16 20
Y 1.0 0.64 0.5 0.42 0.34 0.28

 

  1. By  plotting  1/y  against x, confirm that y is related to x by an equation of the form

 

Y =      q

 

 

P + x

where p and q are constants.                                                                             (3mks)

 

(b)  Use your graph to determine p and q                                                                                   (3mks)

 

(c )  Estimate the value of   (i) y when x = 14

(ii) x when y = 0.46                                                             (2mks)

 

  1. A racing cyclist completes the uphill section of a mountain course of 75 km at an average speed of v km/hr. He then returns downhill along the same route at an average speed of (v + 20) km/hr. Given that the difference between the times is one hour, form and solve an equation in v.

Hence

  1. Find the times taken to complete the uphill and downhill sections of the course.
  2. Calculate the cyclists average speed over the 150km.

 

  1. In the diagram below, X is the point of intersection of the chords AC and BD of a circle. AX = 8 cm, XC = 4cm and XD = 6 cm
  2. Find the length of XB as a fraction
  3. Show that XAD is similar to XBC
  4. Given that the area of AXD = 6cm2, find the area of BXC
  5. Find the value of the ratio

Area of       AXB

Area of        DXC

 

 

 

 

 

 

 

  1. A town B is 55 km on a bearing of 0500. A third town C lies 75km due south of B. Given that D lies on a bearing of 2550 from C and 1700 from A, make an accurate scale drawing to show the positions of the four towns.                                                                                           (3mks)

(scale 1cm rep 10 km)

From this find,

(a) The distance of AD and DC in km                                                                     (2mks)

(b) The distance and bearing of B from D                                                               (2mks)

(c)  The bearing of  C from A                                                                                 (1mk)

 

MATHEMATICS 2

PART 1

MARKING SCHEME             (100MKS)

 

 

  1. No. Log

=   3.6502

0.3681              2.5660

0.3682              1.6427 +                                -4  =  1.6502      = 2.8251

0.2087              Logs                            2

361.8                2.5585              + – v   ans  (4)         6.6850 x 10 -2

3.6502                                         = 0.06685

 

  1. 2 x2 – 5x + 1 = 0

x2 5 x + ½ = 0

2

x25 x   = ½

2

x – 5x  +     5 2    =  ½   +     5    (m)

2         4                        4

 

= x –  5    = ½ +      25    =  17                    (3)

4                   16        16

 

= x – 5/4  =  17/16   =    1.0625

x – 5/4    ±  1.031

X1 = -1.031 = 1.25 = 0.2192

X2 = 1.031  + 1.25  = 1.281

 

  1. A1 = P(1 + R/100)2 = 6000  x  113/100 x 113/100 = Sh. 7661.40

 

A2 = P + PRT/100         =   6000 + 15 X 2 = 6000 + 1800

100

=   Shs. 7800

 

Amount by simple interest is more by Shs.  (7800 – 7661. 40)

Shs. 138.60

  1. Let a plate be p and a cup c.

3p + 4c = 380  x 5             15p + 20c  = 1900

4p + 5c  = 490  x 4       16p + 20c  = 1960 

-p      -60                (m)

 

 

 

 

 

p = Shs 60

 

3(60) + 4 c = 380

4c = 380 –180 = 2000                (3)

c=   Shs. 50

Plate = Shs. 60 ,            Cup = Shs. 50            (A both)

 

  1. Ratio of juice to water = 1          :           7

In 1 glass = 1/8 x 200 = Sh 25

3 litres = 300 ml (undiluted concentrate)           (3)

No. of glasses =v    3000  =  120 glasses

25

 

  1. Cos (2 θ + 120) = 3/2 = 0.866

Cos 30 , 330, 390, 690, 750 ….

            2 θ + 120                = 330

2 θ = 210          ,     = 1050                                                                                        (3)

2 θ = 390 – 120   = 2700          θ2 1350

2 θ =  690 – 120  = 5700  ,       θ3 2850       (for 4 ans)

θ4= 315o    ( for >2)

2 θ =  750 – 120   = 6300 ,

 

  1. P =          k                      4  =  K/4           (substitution)

Q2                         9

K = 4 X 4         =            16

9                           9

P =  16   v         when Q = 4

9Q2

 

P =         16        =   1/9              (A)                 (3)

9x4x4

 

  1. The perimeter = (3.6 + 2.8 ) x 2 = 12.8 cm

Max perimeter = (3.65 + 2.85) x 2 = 23 cm    Expressions

% error =   13 –12.8     x  100    m         =     0.2        x     100  (3)

12.8                                     12.8

= 1.5620%        (A)

 

  1.      1 1/6 x ¾  – 11/12   = (7/6 x ¾ )  -11/12         =  7/8 – 11/12   =   21-22  

½  of 5/6                       ½ of 5/6                        5/12              5/12

= -1/24    = -1  x 12    =  -1

5/12        24   5          10       (3)

 

  1. Volume of rod = П r2h = 22/7 x 4  x 14 = 704cm3                (m)

                    Volume of each cube = 2x2x2 = 8 cm3                         A

 

No. of cubes = 704 /8  = 88 cm3   A

 

 

 

 

 

 

 

 

 

  1. < AOB = 360 = 450

                          8

Tan 67.5 =  h

4

h = 4 x 2.414                A

=  9.650cm

Area of 1 triangle = ½ x 8 x 9.656 x 8 cm = 38.628 x 8   vm

Octagon area  =  38.628 x 8      m

=  309.0 cm2        (A)

 

  1. 3   2        -1             2

=

2              1          -1           y

 

3 – x = 2       (1)       x = 1                          (2)

2 –  1 = y                 y = 1  (A)

 

  1. x 2 + y2 – 6x + 8y – 11 = 0

x2 – 6x + (-3)2 + y2 + 8y + (4)2 = 11 + (-3)2 + (4)2         (completing the square)

(x – 3)2 + (y+4)2 = 11 + 9 + 16 = 36

(x – 3)2 + (y + 4)2 = 62                                                                                          

Centre is  (3, -4)

Radius       = 6 units           As                                            (3)

 

 

14.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figs A C B  and D C E are similar

AB       =          AC       =  and   AB       =          BC

                         DE                    DC                 DE                   CE

 

10    =  6 + x

3          6

= 10   =  15 + y,     m

3            y                                                    60 = 18 + 3x

10y  = 15 + 3y                                                   3x = 42

7y = 15                                                                x = 14

 

y = 15/7              (A)                                                                             (3)

A (4 , 3)           B(8,13)

 

  1. (i) gdt          = change in y    = 13-3 = 10     =  5

change in x       8-4       4          2

 

(ii)      Mag  AB  =  8     -4           4                                                    =

13 -3         10

Length =   Ö42 + 10   = Ö116 = 10.77 units

(iii)   Mid point  = 4 +8  ,    3 + 3

2             2

=  (6, 8)    (mid point)                                                (5 mks)

gdt of perpendicular to AB = -ve rec. of 5/2

-2/5

Eqn is  y = -2/5 x + c

8 = -2/5  x 6 + c    =  40  = -12  +  5c

= c = 52/5

 

y = -2/5 x + 52/5        (A)

 

 

  1. Acceleration = Change in velocity

Time

= (34 – 10) m/s                  = 24 m/s

60 x 10                                600

 

= 0.04m/s2-                                (2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

17.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Triangle                                    (8)

AC = 9cm

Circumference Centre

Circle

Perpendicular bisector of AB

P

Q

 

 

 

  1. (b) a b   2       3          5        6          10        12

c          d   2       4          2        10        19        13

 

2a +2b = 6  x 2       = 49 + 4b = 12

3a + 4b = 10             3a + 4b = 10

a     = 2              4 + 2b = b

 

2c + 2d = 10×2 = 4c + 4d = 20                2 b = 2  b = 1

3c + 4d  = 19        3c + 4d  = 19

c           = 1

2 (1)  + 2d = 10

2d = 8                           Matrix is           2          1      (A)

d = 4                                                  1          4

 

 

 

 

 

OC = 5/3 (31) = 5A

 

19.

 

 

(a)  = AO + OB                         MC = MO + OC

= -3a  = 4b                         = -5/8 (4b) + 5

= 5A – 5/2 b

 

(b) MN = 5 Mc     = 3(5a – 5/2 b)

= 5 s a – 5/2 s b

 

MN = BN + BN

=  3/8 (4 b) + (1 – t) (-BA)

=  3/8 (4 b) + (1 – t)(3a – 4 b)

=  3/2  b + 3 ta –4b + 4tb

= (3-3t) a (4t – 5/2)b

 

MN = MN

= 5 s a – 5/2  sb = (3-3t)a +   (4t – 5/2 )b

=  5 a =  3 – 3t       = 5s + 3t =3

= -5/2 s = 4t –5/2  v     5s + 8t = 5 

-5t = -2            t = 2/5

5 s   = 3 – 3(2/5)

= 3 – 6/5 = 9/5

= 3 – 6/5 = 9/5

s = 9/25

 

(i)    AN :     NB = 2 : 3

 

(ii)   MN :    9   :  16

 

 

 

 

 

 

 

 

20.

 

 

θ x pr2

360

 

  1. Area of sector SPR =  80/360 x 13.2 x 13.2 x 3.142

=  121.6

Area of triangle SPR ½ x 13.2  x 13.2 x sin 80

= 85.8 cm2

(m of area of ) A (at least one)

(m of area)  A(at least one)

Area of segment = 121.6 – 85.8

= 35.8 cm2

Area of sector QPR = 90/360 x 3.142 x 12 x12

 

Area of  PQR = ½ x 12 x 12 = 722

                    Area of segment = 113.1 – 72

= 41.1cm2

Area of intersection = (35.8 + 41.1) = 76.9 cm2

 

b).  Area of quadrilateral  = Area of   PQR + SPR

=  85.8 + 72 = 157.8cm2

Area of shaded region  =  Area of Quadrilateral – Area of sector SPR

=  157.8 – 121.6

=  36.2 cm2

 

 

  1. y = q                   p + x = q                       1  =  x + p

p + x                          y                      y      q    q

 

Gradient  = 1/q   at (0, 0.95)  (8,2.0)  (8,2.0)  gradient   =  2.0 – 0.95  =  1.05

8                 8

1          =  0.1312

q

=  1      =  7.619

0.1312

q =  7.62.

 

y(1/y)  Intercept   p    =  0.95     P   =  0.95

q                7.62

 

p = 7.62 x 095  =  7.27

at x =  14,  y = 2.7

at  y = 0.46,  1/y  =  2.174

x  =  9.6.

 

 

 

 

 

 

  1. a) Distance  =  75km   uphill speed  =  vkm/h

uphill Time  =  75/v hrs

Downhill speed  = ( + 20)  km/h

Downhill Time    =        75         hrs.

                                             v + 20

Takes larger uphill

75  –  75             =  1

v         v+20

75 (v+20) – 75v            = 1

v(v + 20)                    1

75v + 1500 – 75v  =  v(v + 20)  =  v2 + 20v.

v2 + 20v  – 1500  =  0

v  =  – 20 +  202 – 4(1)  (-1500)

2(1)

v  =  –20 +  400 + 6000  = –20 + v6400

2                        2.

V1     =  –20  +  80      =  30km/hr

2

V2    =   – 20 – 80      X   impossible

2

speed uphill      =  30 km /hr,  T = 75  time =  2 ½ hrs

30

speed downhill =  50 km /hr  Time = 75      Time =  2 ½ hr

50

Average speed   =  Total  distance         =  150km          =  37.5 km/ hr

                                                Total time                      4hrs

 

X 0 4 8 12 16 20
Y 1.0 0.64 0.5 0.42 0.34 0.28
1/y 1.0 1.56 2.0 2.38 2.94 3.57

 

 

  1. A                 B

 

 

 

 

D                      C

 

A x X x C  =  BX .  XD

8 x 4           =  6BX

BX       =  8 x 142          =   16  

6                     3

X AD   =  XBC

XA       =  8    =  24      =  3

XB        16        16          2

XD      =    6      =    3

XC               4              2

 

<   AXD   =   BXC            (vertically opposite  <s))

                                                    SAS holds  :  they are similar.

LSF  =   3/2    ASF  =  (3/2)2  =  9/4

Area  A x A  =  6cm2    Area  B x C  =  6 x 9       =  27   =  13.5cm2

4

 

24.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. a) AD =   50km

DC   =   35km

BD  = 90km

Bearing is 020

Bearing is 134o                                                                                                       (8mks)

 

 

MATHEMATICS  I

PART II

 

SECTION (52 MARKS)

 

  1. Without using tables, simplify

1.43 x 0.091 x 5.04

2.86 x 2.8 x 11.7                                                                                             (3mks)

 

  1. Make x the subject of the formula if

y = a/x  +  bx                                                                                                    (3mks)

 

  1. Give the combined solution for the range of x values satisfying the inequality

2x + 1<  10 – x  <   6x – 1                                                                                 (3mks)

 

  1. A man is employed at a KShs. 4000 salary and a 10% annual increment. Find the total amount of money received in the first five years                                                                   (4mks)

 

  1. A town A is 56 km from B on a bearing 0620.  A third town C is 64 km from B on the bearing of 140o.  Find

(i) The distance of A to C                                                                                        (2mks)

(ii) The bearing of A from C                                                                                          (3mks)

 

  1. Expand (x + y)6 hence evaluate (1.02) to 3d.p.                                                         (3mks)

 

  1. Rationalise the denominator in                                                                               (2mks)

 

Ö 3

1 – v3

 

 

 

  1. The table below shows daily sales of sodas in a canteen for 10 days.

 

 Day 1 2 3 4 5 6 7 8 9 10
No. of 52 41 43 48 40 38 36 40 44 45

 

Calculate the 4 day moving averages for the data                                                     (3mks)

 

  1. Find the image of the line y = 3x = 4 under the transformation whose matrix is.

3mks

2           1

-1         2

 

  1. Three points are such that A (4 , 8), B(8,7), C (16, 5). Show that the three points are collinear                                                                                                                                          (3mks)
  2. Write down the inverse of the matrix 2 – 3 hence solve for x and y if

4     3

2x  – 3y = 7

4x + 3y +5                                                                                                        (3mks)

 

  1. Use the table reciprocals to evaluate to 3 s.f. 3mks

1/7  +  3/12  +  7/0.103

 

 

 

 

 

 

 

Given that O is the centre of the circle and OA is parallel to CB, and that angle

ABC =   1070,  find

(i) Angles AOC,                (ii) OCB               (iii) OAB                                                 (3mks)

  1. Two points A and B are 1000m apart on level ground, a fixed distance from the foot of a hill. If the angles of elevation of the hill top from A and B are 60o and 30o respectively, find the height of the hill                                                                                           (4 mks)
  2. Two matatus on a dual carriageway are moving towards a bus stop and are on level 5 km from the stop. One is travelling 20 km/hr faster than the other, and arrives 30 seconds earlier. Calculate their speeds.       (5mks)
  3. If log x = a and log y = b, express in terms of a and b

Log  x 3 

VY                                                                                                             (2mks)

 

SECTION B:

 

  1. The table below gives the performance of students in a test in percentage score.
Marks 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79
No. of Students  

2

 

4

 

7

 

19

 

26

 

15

 

12

 

5

 

Using an assumed mean of 44.5, calculate

  1. The mean
  2. The standard deviation
  3. Find the median mark

 

 

 

  1. Draw the graph of y = 2x2 – x – 4 for the range of x -3  = x  =  3.  From your  graph

State the minimum co-ordinates

  1. Solve the equations
  2. 2x2 – x – 4 = 0
  3. 2x2 – 3x – 4 = 0

 

 

 

 

 

 

  1. Two concentric circles are such that the larger one has a radius of 6cm and the smaller one radius of 4 cm. Find the probability that an item dropped lands on the shaded region           4mks

 

  1. Two unbiased dice are thrown. Find the probability of obtaining (4mks)
  2. A product of 6
  3. A sum of 8

iii. The same number showing                                                                             (4mks)

 

 

 

 

 

 

 

 

 

 

Two pulley wheels centers A and B are joined by a rubber band C D E F G H C round them.  Given that larger wheel has radius of 12 cm and AB = 20 cm, CD and GF are tangents  common to  both  wheels and that CBA = 60o), Find

  1. BD (Length)
  2. CD

iii.  Arc length CHG and DEF, hence find the length of the rubber.

 

  1. V A B C D is a right pyramid with a square base A B C D of side 5 cm. Each of its four triangular

faces is inclined at 750 to the base. Calculate

  1. The perpendicular height of the pyramid
  2. The length of the slant edge VA
  3. The angle between edge VA and base A B C D
  4. The area of the face VAB

 

  1. Plot the graphs of y = sin xo and y = cos 2xo on the same axes for –180 £ x £180o.

Use your graphs to solve the equation 2 sin x = cos  2x

 

  1. The depth of the water in a rectangular swimming pool increases uniformly from 1M at the shallow

end to 3.5m at the deep end.  The pool is  25m long  and  12m  wide. Calculate the volume of the pool

in cubic meters.

The pool is emptied by a cylindrical pipe of internal radius 9cm. The water flows through the pipe at speed of 3 metres per second.  Calculate the number of litres emptied from the pool in two minutes to the nearest 10 litres.          (Take II = 3.142)

 

 

 

  1. A rectangle A B C D is such that A and C lie on the line y = 3x. The images of B and D under a

reflection in the line y = x are B1 (-1, -3) and D1 (1,3) respectively.

  1. Draw on a cartesian plane, the line y = x  and mark points B1 and D1
  2. Mark the points B and D before reflection
  3. Draw the line y = 3x hence mark the points A and C to complete and draw the rectangle ABCD.

State its co-ordinates, and these of A1 and C1.

  1. Find the image of D under a rotation, through – 900, Center the origin.

 

 

MATHEMATICS I

PART II

MARKING SCHEME.

  1. 1.43 X 0.091 X 5.04100000        91 X 504           =        7/103

                        2.86 X 2.8 X 11.7             105             2 x 28 x 117 x 103

                                                                                                                                                                                    (3)

                                                                                                                         = 0.007            (A)

  1. y = a/x + bx yx = a + bx2

Either

bx2 – yx + a = 0

 

x =     y   ±   v y2 –  4ab

2b                                                         (3)

 

  1. 2x + 1£  10 – x  £    bx  -1

2x + 1 £ 10 – x            10 –x £  6x –1

3x £   9                                    11£   7x

x  £  3                               x   £ 11/7                                                             (3)

11/7 £  x   £   3

 

  1. a = 4000 r = 110/100   =      1.1   ( 4000, 4000 + 4000, 4400 + 0/100 (4400——)

(a and r)

Sn  =  a(r n – 1)       

                                    R  -1                                                     1.1 Log  = 0.04139

     X   5

0.20695

 

0.1                               (4)

= 4000 (1.15 –1)   (any)

1.1 –1                                                   4000 (1.6 – 1)

0.1

A  =  4000 ( 0.6105)

0.1

= Sh. 2442       =    Sh. 24,420       (A)                                       (4)

0.1

 

  1. (i) b2=  a2 + b2 – 2ab Cos B

= 642  + 562– 2(64) (56) cos 78

= 4096 + 3136  – 7168 (0.2079)

= 7232  – km 1490.3

 

b2  = 5741.7  = 5.77 km                  (5)

 

(ii)        b                a

            Sin B          Sin A

 

75.77    =      64

Sin 78         sin A         Sin A = 64 x 0.9781     

75.77                   

Sin A = 0.08262

A  = 55.70  (or B = 46.30)

 

Bearing = 90 – 28 – 55.7

= 0.06.30                       (A)

 

  1. (x + y) 6 =  1 (x) 6 (y)0 + 6 (x)5 (y)1+15(x)4 (y)2 + 20x3y3 + 15x2y4 + 6xy5 + y6

(1.02)6 = (1 +0.02)6 x = 1

y = 0.02

 

(1.02)6 = 1+6 (0.02) + 15 (0.02)2 + 15(0.02) + 20(0.02)3 + 15 (0.02)4                          

=  1 + 0.12  + 0.006 + 0.00016

= 1.12616

= 1.126  (to 3 d.p)                                                                                 (3)

 

  1.       3(1 +  3)                 =  3  +  3          3 + v3

(1-  3)(1+  3)                     1-3                          2

 

  1. Moving averages of order 4

M1        =  52 + 41 + 43 + 48                  184       = 146

4                                   4

M2            184 – 52 + 40   = 172  = 43                               for 7

4                 4                                   for > 4

M3             = 172– 40 + 38 = 170    = 42.5

4                     4

M4             170 – 38+36  = 168   = 42

4                  4

M5        = 168 – 36 + 40 = 173    = 43                (3)

4                4

M6             = 172 – 40 + 44 = 176    = 44

4              4

M7             = 176 – 44 + 45 = 177    = 44.25

4             4

 

  1. y = 3x + 4

A(0,4) B (1,7) Object points

                                                A         B          A         B

2          1          0          1          4          9

=

-1         2          4          7          8          13

Y =  Mx + C

M = 13 – 8  =  5  = 1

9-4                  5     1

 

y = x+c                                  y = x + 4

8 = 4 + c    c  = 4

 

  1. AB = 8     -4                        4                      BC =   16      – 8                        -8     for either

=

7     -8                      -1                                  5        – 7             -2

 

 

AB = ½   BC  and AB and BC share point B.

A,B,C  are collinear.                                                                (3)

 

  1. 2          -3

 

4          3          det. = 6 + 12 = 18

Inv.=     1         3          3

18

-4         2

1         3      3     2     -3   x       1           3   3       7

18                                            18

-4    2      4       2  y                     -4  2       5

x                       36

1

y          18        -18                    (3)

x = 2, y = -1      (A)

 

  1. 1/7 + 3/12.4 + 7/0.103

1/7 + 3/1.24 x 10-1 + 7/1.03 x 10-1

 

  0.1429 + 3(0.8064) + 7 x 10 (0.9709)

10

= 0.1429 + 0.2419 + 67.96                                 (3)

=70.52                             (A)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. (i) ADC = 2×73

= 1460

 

(ii) OCB = x = 180 – 146 = 34

(iii) 360 – 107 – 146 – 34

= 73 0

 

  1. Tan 300 = y/x y  =  x tan  30

Tan 600  = 1000 + y       ;     y = x tan 60 – 1000

X

X tan 300  = x tan 60 – 1000

0.5773 x = 1.732x – 1000

1.732x – 0.577 = 1000

1.155x = 1000

x = 1000

1.155           = 866.0 m         (A)                   (4)

 

 

  1. 5 km Slower speed = x km/hr

Time    = 5/x

Faster = (x+20) k/h

Time = 5/x=20            T1 – T2 = 5/x  – 5/x+20 = 30/3600

5 (x+20) –5x       1

x(x+20) 120

120 (5/x + 100 – 5x) = x2 + 20x             (5)

x2 + 20x – 12000

x = –20      400 + 48000

2

x = -20 ±  220

2

Spd = 100 km/h

And x = 120 km/h                                 (A)

  1. Log x = a log y = b

Log  x3  = Log x3  –  log y ½

y

= 3 Log x – ½ Log y

= 8a –  ½ ab

 

SECTION B

 

17.

Marks Mid point (x) d = x-44.5 F E = d/10 Ft T2 Ft2   v
0-9 4.5 -40 2 -4 -8 16 32
10-19 14.5 -30 4 -3 -12 9 36
20-29 24.5 -20 7 -2 -14 4 28
30-39 34.5 -10 19 -1 -19 1 19
40-49 44.5 -0 26 0 0 0 0
50-59 54.5 -10 15 1 15 1 15
60-69 64.5 20 12 2 24 4 48
70-79 74.5 30 5 3 15 9 45

=90                              =1                                =223

 

 

(a)   Mean = (1 / 90 x 10) + 44.5 = 44.5 + 0.111

= 44.610

 

(b)   Standard deviation = 10  233/90  – (1/90)2                        

                                                            10  2.478  – 0.0001                              (8)

10   2.478

10 x 1.574  = 15. 74    (A)

(c)    Median 45.5th value  = 39.5  + (13.5 x 10/ 26)

39.5 + 5.192                 (A)

44.69

 

(a)     The probability  = Shaded area

                                     Large circle area

Shaded area = ПR2 – П r2

= 22/7 (42 – 32) v  = 22/7 x 7  = 22

            Large area  = 22/7 x4x4 = 352/7 (A)

Probability = 22         = 22  x  7 =    7

352/7            352      16

 

(b)

  1 2 3 4 5 6
1 1,1 1,2 1,3 1,4 1,5 1,6
2 2,1 2,2 2,3 2,4 2,5 2,6
3 3,1 3,2 3,3 3,4 3,5 3,6
4 4,1 4,2 4,3 4,4 4,5 4,6
5 5,1 5,2 5,3 5,4 5,5 5,6
6 6,1 6,2 6,3 6,4 6,5 6,6

(M)

 

(i)    P(Product of 6) = P((1,6) or (2,3) or (3,2) or (6,1))

= 4/36   =  1/9

(4)

(ii)   P (sum of 8)   = P( (2,6) or (3,5) or (4,4) or (5,3) or (6,2) )

= 5/36               (A)

 

(iii)  P (same number)  = P (1,1) or (2,2) or (3,3) or (4,4) or (5,5) or (6,6)

6/36  = 1/6   (A)

 

 

 

 

 

 

 

 

 

 

(i)         Cos 60   = x/20 x = 20 x 0.5  = 10 cm

BD = 12 – 10 = 2 cm

 

(ii)          CD = y  Sin 60  = y/20                        y = 20x 0.8666

CD = 17.32 cm

 

 

 

 

(iii)        CHG  = 120        reflex  = 2400

CHG = 240/360 x 2 x p x r

= 50.27

DBF = 1200/360  x 2 x  П x  r  =  1/3 x 2 x 3.142 x 2

=  4.189                               (A)

Length C D E f G H C  =          50.27 + 2(17.32) + 4.189

= 89.189                     (A)

 

  1. (a) From the diagram, XO = 5/2 = 2.5

Tan 750 = VO/2.5          v m

VO  =  2.5 x 3.732

 

Perpendicular height  = VO  = 9.33 cm

2                      (A)

  1. Diagonal of base = 52 + 52  = 50
        Length of diag.   50       = 7.071    = 5.536

VA2 = AO2 + VO2     (m)

3.5362  + 9.32

12.50 + 87.05

= 99.55 = 9.98 cm2        (A)                  (8)

 

 

(c )                   = VAO  Tan =      9.33     = 2.639

3.536

VAO = 69.240                                                (A)

 

 

(d)                    Cos VBA = = 2.5 /9.98   = 0.2505

VBA = 75.490

Area VBA = ½  x 5  x 4.99 x sin 75.45             m  ( or other perimeter)

= 5 x 4.99 x   0.9681

= 24.15 cm2                  (A)

 

  1. Volume = cross – section Area x L

X-sec Area = (1 x 25)  +  (½  x 25 x 2.5)

=  25 + 31.25  =  56. M

Volume  = 56.25 x 12

= 675 m3                               

            Volume passed / sec  = cross section area x speed

= П r2 x l           = 3.14  x  9/100 x  9/100  x 3                 (8)

= 0.07635  m3 /sec         v (M)

Volume emptied in 2 minutes

= 0.07635 x 60 x 2

= 9.162 m2                (A)

1 m3  = 1000 l

= 9.162 litres

= 9160 litres                 (A)

 

 

 

 

 

24.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MATHEMATICS II

PART I

 

SECTION A (52 MARKS)

 

  1. Use tables to evaluate

3Ö 0.09122 + Ö 3.152                                                           (5mks)

0.1279 x 25.71

  1. Simplify  (a – b)2

a2 – b2                                                             (2mks)

 

 

  1. The gradient function of a curve that passes through point: (-1, -1) is 2x + 3.

Find the equation of the curve.                                                                               (3mks)

  1. Find the value of k for which the matrix k     3

has no inverse.   (2mks)                            3     k

  1. Without using tables, evaluate       log 128 – log 18

log 16 – log 6                                                          (3mks)

  1. Find the equation of the locus of points equidistant from point L(6,0) and N(-8,4). (3mks)
  2. The value of a machine is shs. 415,000. The machine depreciates at a rate of 15% p.a. Find how many years it will take for the value of the machine to be half of the original value. (4mks)
  3. Use reciprocal tables to evaluate to 3 d.p. 2             1   

0.321           n2.2                                          (4mks)

  1. Using the trapezium rule, estimate the area bounded by the curve y = x2, the x – axis and the co-ordinates x = 2 and x = 5 using six strips. (4mks)
  2. Solve the equation for 00 £ q £ 3600 and Cos2q + ½ Cosq = 0 (3mks)
  3. Point P divides line MK in the ratio 4:5. Find the co-ordinates of point P if K is point (-6,10) and M is

point (3,-8)                                                                                                                          (3mks)

  1. How many multiples of 3 are there between 28 and 300 inclusive. (3mks)
  2. The line y = mx – 1, where m is a constant , passes through point (3,1). Find the angle the line makes with the x – axis. (3mks)
  3. In the figure below, AF is a tangent to the circle at point A. Given that FK = 3cm, AX = 3cm, KX = 1.5cm and AF = 5cm, find CX and XN. (3mks)

 

 

 

 

 

 

 

 

 

 

 

  1. Make X the subject of the formula (3mks)

V = 3Ö k + x

sk – x

 

 

 

 

 

 

 

 

  1. Write down the inequalities that describe the unshaded region below. (4mks)

y

 

 

0.5         2                   x

 

-1.5

-2

 

      SECTION B (48 MARKS)

     

  1. Draw the graph of y = -x2 + 3x + 2 for –4 £ x £ 4. Use your graph to solve the equations

(i.) 3x + 2 – x2 = 0               (ii) –x2 – x = -2                                                       (8mks)

 

  1. The marks obtained by Form 4 students in Examination were as follows:

 

 

Marks 0-9 10-19 20-29 30-39 40-49 50-59
No. of students 2 8 6 7 8 10
Marks 60-69 70-79 80-89 90-99  
No. of Students 9 6 3    

      Using 74.5 as the Assumed mean, calculate:

(i) The mean mark

(ii) The standard deviation                                                                                      (8mks)

  1. In the figure below, a and b are the position vectors of points A and B respectively. K is a point on

AB such that the AK:KB = 1:1. The point R divides line OB in the ratio 3:2 and point S divides OK in

the ratio 3:1.

 

B

R

B                                 K

 

0               a                     A

(a) Express in terms of a and b

(i) OK       (iii) RS

(iii) OS      (iv) RA

(b) Hence show that R,S and A are collinear.                                                          (8mks)

 

  1. The figure below is the roof of a building. ABCD is a rectangle and the ridge XY is centrally placed.

 

 

 

 

 

 

 

 

 

 

 

Calculate:

(i) The angle between planes BXC and ABCD.

(ii) The angle between planes ABXY and ABCD.                                                          (8mks)

  1. On the same axis, draw the graph of y = 2cosx and y = sin ½x for 00 £ x £ 1800, taking intervals of 150

                                                                                                                                                                                                          (6mks)

From the graph, find:

(a) The value of x for which 2cosx = sin ½ x                                                                              (1mk)

(b) The range of values of x for which –1.5 £ 2cos x £ 1.5                                              (1mk)

  1. Two towns T and S are 300km apart. Two buses A and B started from T at the same time travelling towards S. Bus B travelled at an average speed of 10km/hr greater than that of A and reached S 1 ¼ hrs earlier.

(a) Find the average speed of A.                                                                                    (6mks)

(b) How far was A from T when B reached S.                                                                (2mks)

  1. P and Q are two ports 200km apart. The bearing of Q from P is 0400. A ship leaves port Q on a bearing of 1500 at a speed of 40km/hr to arrive at port R 7 ½ hrs later. Calculate:

(a) The distance between ports Q and R.                                                                        (2mks)

(b) The distance between ports P and R.                                                                  (3mks)

(c) The bearing of port R from port P.                                                                      (3mks)

  1. A farmer has 15 hectares of land on which he can grow maize and beans only. In a year he grows maize on more land than beans. It costs him shs. 4400 to grow maize per hectare and shs 10,800 to grow beans per hectare. He is prepared to spend at most shs 90,000 per year to grow the crops. He makes a profit of shs 2400 from one hectare of maize and shs 3200 from one hectare of beans. If x hectares are planted with maize and y hectares are planted with beans.

(a) Write down all the inequalities describing this information.                                      (13mks)

(b) Graph the inequalities and find the maximum profit he makes from the crops in a year.          (5mks)

 

 

MATHEMATICS II

PART II

 

  1. Use logarithm tables to Evaluate

3Ö 36.5 x 0.02573

1.938                                                                                                              (3mks)

  1. The cost of 5 shirts and 3 blouses is sh 1750. Martha bought 3 shirts and one blouse for shillings 850. Find the cost of each shirt and each blouse.             (3mks)
  2. If K = ( y-c  )1/2

4p

  1. a) Make y the subject of the formula.       (2mks)
  2. b) Evaluate y, when K = 5, p = 2 and c = 2                                                                   (2mks)
  3. Factorise the equation:

x + 1/x = 10/3                                                                                                             (3mks)

  1. DA is the tangent to the circle centre O and Radius 10cm. If OD = 16cm, Calculate the area of the shaded Region.       (3mks)

 

 

 

 

 

 

 

 

 

 

 

  1. Construct the locus of points P such that the points X and Y are fixed points 6cm apart and

ÐXPY =     600.                                                                                                            (2mks)

  1. In the figure below, ABCD is cyclic quadrilateral and BD is diagonal. EADF is a straight line,

CDF = 680, BDC = 450 and BAE = 980.

 

 

 

 

 

 

 

 

Calculate the size of:                                                                                               (2mks)

  1. a) ÐABD                                       b) ÐCBD
  2. Otieno bought a shirt and paid sh 320 after getting a discount of 10%. The shopkeeper made a profit of 20% on the sale. Find the percentage profit the shopkeeper would have made if no discount was allowed?       (2mks)
  3. Calculate the distance:
  4. i) In nautical miles (nm)
  5. ii) In kilometres (km)

Between the two places along the circle of Latitude:

  1. a) A(300N, 200E) and B(300N, 800E) (Take Radius of Earth = 6371Km).                (2mks)
  2. b) X(500S, 600W) and Y(500S, 200E) (Take Radius of Earth = 6371Km).                  (2mks)
  3. A rectangular tank of base 2.4m by 2.8m and height 3m contains 3,600 litres of water initially. Water flows into the tank at the rate of 0.5m/s. Calculate the time in hours and minutes required to fill the tank. (4mks)
  4. Expand (1 + a)5 up to the term of a power 4. Use your expansion to Estimate (0.8)5 correct to 4 decimal places. (4mks)  
  5. A pipe is made of metal 2cm thick. The external Radius of the pipe is 21cm. What volume of metal is there in a 34m length of pipe (p = 3.14).       (4mks)
  6. If two dice are thrown, find the probability of getting: a sum of an odd number and a sum of scoring more than 7 but less than 10. (4mks)
  7. Find the following indefinite integral ò 8x5 – 3x dx                                                                  (4mks)

x3

  1. The figure below represents a circle of radius 14cm with a sector subtending an angle of 600 at the centre.

 

 

.

 

 

 

 

 

 

 

Find the area of the shaded segment.                                                                                         (3mks)

 

 

 

 

 

 

 

 

  1. Use the data below to find the standard deviation of the marks.

 

Marks (x ) Frequency (f)
5

6

7

8

9

3

8

9

6

4

(4mks)

 

SECTION II (48MKS)

 

  1. The figure below shows a cube of side 5cm.

 

 

 

 

 

 

 

 

 

 

 

 

 

Calculate:

  1. a) Length FC                                                                                                      (1mk)
  2. b) Length HB                                                                                                        (1mk)
  3. c) Angle between GB and the plane ABCD. (1mk)
  4. d) Angle between AG and the Base.       (1mk)
  5. e) Angle between planes AFC and ABCD. (2mks)
  6. f) If X is mid-point of the face ABCD, Find angle AGX. (2mks)
  7. Draw on the same axes the graphs of y = Sin x0 and y = 2Sin (x0 + 100) in the domain 00 £ x0 £ 1800
  8. i) Use the graph to find amplitudes of the functions.
  9. ii) What transformation maps the graph of y = Sin x0 onto the graph of : y = 2Sin (x0 +100).
  10. The table below shows the masses to the nearest gram of 150 eggs produced at a farm in Busiro

country.

Mass(g) 44 45 46 47 48 49 50 51 52 53 54 55
Freq.  1  2  2  1  6  11  9  7  10  12  16  16
Mass(g) 56 57 58 59 60 61 62 63 64 65 70  
Freq.  10  11  9  7   5  3  4  3  3  1  1  

 

Make a frequency Table with class-interval of 5g. Using 52g as a working mean, calculate the mean mass. Also calculate the median mass using ogive curve.

  1. A shopkeeper stores two brands of drinks called soft and bitter drinks, both produced in cans of same

size. He wishes to order from supplies and find that he has room for 1000 cans. He knows that bitter

drinks has higher demand and so proposes to order at least twice as many cans of bitter as soft. He

wishes however to have at least 90cans of soft and not more than 720 cans of bitter. Taking x to be

the number of cans of soft and y to be the number of cans of bitter which he orders. Write down the

four inequalities involving x and y which satisfy these conditions. Construct and indicate clearly by

shading the unwanted regions.

 

 

 

 

  1. Two aeroplanes, A and B leave airport x at the same time. A flies on a bearing 0600 at 750km/h and B flies on bearing of 2100 at 900km/h:
  2. a) Using a suitable scale draw a diagram to show the positions of Aeroplanes after 2hrs.
  3. b) Use your graph to determine:
  4. i) The actual distance between the two aeroplanes.
  5. ii) The bearing of B from A.

iii) The bearing of A from B.

  1. The Probabilities that it will either rain or not in 30days from now are 0.5 and 0.6 respectively. Find the probability that in 30 days time.
  2. a) it will either rain and not.
  3. b) Neither will not take place.
  4. c) One Event will take place.
  5. Calculate the Area of each of the two segments of y = x(x+1)(x-2) cut off by the x axis. (8mks)
  6. Find the co-ordinates of the turning point on the curve of y = x3 – 3x2 and distinguish between them.

 

MATHEMATICS II

PART I

MARKING SCHEME:

 

  1. 0.09122 = (9.12 x 10-2)2 = 0.008317

Ö 3.152 = 1.776

3Ö 1.776 + 0.008317

0.1279 x 25.91

= 3Ö 1.784317              No.             log      

0.1279 x 25.91           1.784         0.2514

0.1279    -1.1069

25.71           1.4101 +

0.5170

-1.7344

x 1/3

10-1 x 8.155(6)                    1-1.9115

Or 0.8155(6)

 

  1. (a – b)(a – b) a – b

(a – b)(a + b)       a + b

 

  1. dy = 2x + 3

dx

y = x2 + 3x + c

-1 = 1 – 3 + c

c = 1     ;     E.g  y = x2 + 3x + 1

 

  1. K2 – 9 = 0

K = ± 3

 

  1. log 128    =  log       64

18                    9

 

log   16        log     8 

6                    3

2 log (8/3)

log (8/3)

= 2

 

  1. Midpoint -8 + 6, 4 + 0         (-1, 2)

2         2

Gradient of LN = 4/-14 = -2/7

Gradient of ^ bisector = 7/2

y – 2  = 7/2

x + 1

y = 7/2X + 11/2

 

  1. 207,500 = 415,000(1 – 15 )n

100

0.5 = ( 85 )n

100

0.5 = 0.85n

log 0.5 = n log 0.85

log 0.5  = n

log 0.85

n = –1.6990   =    -0.3010 = 4.264yrs

-1.9294      -0.0706

 

  1. 2 x      1        =   1  . x 20 = 0.3115 x  20 = 6.230

3.21 x 10-1    3.21

   1     =         1      =  0.5807 = 0.005807

172.2    1.722 x 102           100

6.230 – 0.005807 = 6.224193

= 6. 224(3d.p)

 

X 2 2.5 3 3.5 4 4.5 5
y 4 6.25 9 12.25 16 20.25 25

h = ½

Area= ½ x ½[29+2(6.25+9+12.25+16+20.25+25)]

= ¼ [29 + 127.5]

= ¼  x 156.5  =  39.125  sq. units.

 

  1. Cos q (cos q + ½ ) = 0

cos q = 0        cos q = -0.5

q = 900, 2700    q = 1200, 2400              

\ q = 900, 1200, 2400, 2700

 

  1. MP = 4 MK MK =      -9

9                                   -18

MP = 4 ( -9  ) = ( -4 )

9  -18          8

\ P is ( -1,0 )

 

  1. a = 30 d = 3   l = 300

300 = 30 + 3 (n – 1 )

300 = 30 + 3n – 3

300 – 27 = 3n

273 = 3n

91 = n  

 

 

 

 

  1. y = mx – 1

1 = 3m – 1

m = 2/3 = 0.6667

tan q = 0.6667  ;     q = 33.690    

 

  1. FK x FC = FA2

FC = 25/3 = 8 1/3 cm

CX = 81/3 – 9/2 = 23/6 = 35/6 cm

CX x XK = XA x XN

33/6 x 3/2 = 3 x XN

\ XN = 111/12 cm

 

  1. V3 = k + x

k – x

V3k – V3x = k + x

V3k – k = x + V3x

V3k – k = x( 1 + v3)

V3k – k  = x

1 + V3

 

  1. (i.) x = 2 Þ x £ 2

(ii) y = -2 Þ y > -2

(iii)pts. (0.5,0)

(0,-1.5)

m = -1.5 – 0  = 3

0 – 0.5

Eq. Y = 3x – 1.5    y < 3x – 1.5

 

     

SECTION B

 

X -4 -3 -2 -1 0 1 2 3 4
Y -26 -16 -8 -2 2 4 4 2 -2

(i) Roots are x = -0.5   x = 3.6

 

(ii)  y = -x2 + 3x + 2

0 = -x2 – x + 2 

y = 4x     (-2, -8) (1, 4)

Roots are x = -2, x = 1

 

  1. class x f       d=x-74.5       fd             d2       fd2    

0 – 9        4.5    2         – 70         – 140       4900        9800

10 – 19    14.5     8         – 60         – 480       3600     28,800

20 – 29    24.5     6         – 50         – 300       2500     15,000

30 – 39    34.5     7         – 40         – 280       1600     11,200

40 – 49    44.5     8         – 30         – 240         900       7,200

50 – 59    54.5    10        – 20         – 200         400       4,000

60 – 69    64.5     9         – 10           – 90         100          900

70 – 79    74.5     6            0               0              0              0

80 – 89    84.5     3          10              30         100          300

90 – 99    94.5     1          20            20         400          400   

Sf =       Sfd =                                     Sfd2 =     77,600

60                        -1680

(i) Mean = 74.5 + -1680

60

= 74.5 – 28  =    46.5

(ii) Standard deviation = Ö 77600 – ( –1680 )2

60            60

= Ö 1283.3 – 784

= Ö 499.3 = 22.35

 

  1. a (i.) OK = OA + AK = ½ a + ½ b

(ii) OS = ¾ OK = 3/8 a + 3/8 b

(iii)RS = RO + OS = 3/8 a – 9/40 b

(iv) RA = RO + OA = – 3/5 b + a

 

  1. RA = a – 3/5 b   RS = 3/8 a + 9/40 b

= 3/8( a – 3/5 b)

\ RS = 3/8 RA

The vectors are parallel and they have a common

point R  \ point R, S and A are collinear

 

 

 

 

 

 

 

 

 

 

 

 

 

KB = 3m   NK = 1.5m   XB = 5m

(i)  XK = Ö 52 – 32  = Ö 16 = 4m

let ÐXKN = q

cos q = 1.5  = 0.375

4

q = 67.97(8)0

 

(ii) In DXNK

XN = Ö 42 – 1.52 = Ö 13.75 = 3.708

In D SMR; MR = KB = 3m

SM = XN = 3.708m

Let ÐSRM = a

tan a = 3.708  =1.236

3

a = 51.02(3)0

 

 

 

 

 

 

 

 

 

21.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

21.

 

  0 150 300 450 600 750 900 1050 1200 1350 1500 1650 1800
Y =2cosX 2.00 1.93 1.73 1.41 1.00 0.52 0.00 -0.52 -1 -1.41 -1.73 -1.93 -2.00
Y = sin ½ X 0.00 0.13 0.26 0.38 0.50 0.61 0.71 0.79 0.87 0.92 0.97  0.99 1.00

(a) X = 730 ± 10

(b) Between 40.50 and 139.50

 

 

  1. 300km

T                                               S

Let the speed of A be X km/hr

Speed of B = (X + 10) km/hr

Time taken by A = 300 hrs

X

Time taken by B = 300 hrs

X + 10

300300  =  5

x    x + 10    4

300(x + 10) – 300x  = 5

x(x + 10)    4

300x + 300 – 300x = 5

x2 + 10x

x2 + 10x – 2400 = 0.

x = 44.25

X = -54.25 N/A

(b) Distance covered by A in 1 ¼ hrs  = 44.25 x 5/4  = 55.3 km

Distance of A from T is 300 – 55.3 = 244.7 km

 

 

 

 

 

 

 

 

 

  1. (a) Distance = 15 x 40 = 300km

2

(b)

 

 

 

 

 

 

 

 

 

 

 

PR2 = 2002 + 3002 –2x 200 x 300 cos700

= 130,000 – 41040   =   88,960

PR = 298.3 km

 

(c) 298.3  = 300

sin 700    sin a

sin a = 300 sin 700

298.3

= 0.9344

a = 69.10

 

Bearing of R from P is

40 + 69.1 = 109.10

 

  1. (i.) X > y

(ii) 4,400X + 10,800Y £ 90,000

Simplifies to 11X + 27y £ 225

(iii) X + y £ 15

X > 0;  y > 0

Boundaries

x = y pts (6,6) (12,12)

11x + 27y = 225 pts (13,3) (1,8)

X + y = 15 pts (0,15) (8,7)

Objective function

2400 x 3200y

(pt (2,1)

2400X + 3200y = 8000

Search line ® 3X + 4y = 10

Point that give maximum profit is (12,3)

\ maximum profit

= 2400 x 12 + 3200 x 3 = 38,400 shs.

 

 

 

 

 

 

 

 

 

MATHEMATICS  II

PART II

MARKING SCHEME

 

  1. No log.

36.5        1.5623

0.02573   –2.4104 +

-1.9727

1.938         0.2874 –

-1.6853

 

-3  + 2.6853 

3         3

-1 + 0.8951

1.273(4) ¬ 0.1049

= 1.273(4)

 

  1. Let shirt be sh x,

let blouse be sh. y.

5x + 3y =1750 (i.)

3x + y = 850    (ii)

mult (ii) by 3

9x + 3y = 2550 (iii)

Subtract  (iii) – (i.)

– 4x = -800

Subt for x

  1. = 250

Shirt = sh 200  ;   Blouse = sh 250

     

  1. (a) K2 = y – c

4p

y – c = 4pK2

y = 4pK2 + c

(b)    y = 4 x 2 x 25 + 2   ;      y = 202

 

  1. x2 + 1 – 10x = 0

3

3x2 – 10x + 3 = 0

3x (x – 3) – 1(x – 3) = 0

(3x – 1) (x – 3 ) = 0

x = 1/3  or x = 3                                                                                                             

 

  1. Area D OAD pyth theorem AD =12.49cm

½  x 12.49 x 10  =   62.45cm2

Cos q = 10/16 = 0.625

q = 51.30                                     62.5

Sector 57.30  x 3.14 x 100    40.2 –

360                        = 22.3

 

 

 

 

 

 

 

 

  1. ÐXPY = 600

\ÐXC1Y = 1200

              B1             \ÐC1XY = ÐC1YX

= 1800 – 1200  = 300

2

 

 

 

 

Construct 300  angles

at XY to get centres

B1           C1 and C2  mojar arcs drawn

2            on both sides with C1X and C2X

as centres.

 

 

 

 

 

 

 

 

 

 

  1. DAB = 1800 – 980  = 820

ADB = 180 – (68 + 45 ) = 670

                                                                                                                                              ABD = 180 – (67 + 82)

= 310

 

(a) 1800 – (67 + 82)0 = 310

       ÐABD = 310                                                                                 Opp = 1800

(b) (180 – 82)0 = 980                                                                                   82 + 98 = 1800

        1800  – (980 – 450) =

ÐCBD = 370                                                                                  180 – (98 + 45)

= 370

  1. 10 x 320

100     Discount = sh 32

Sold at      sh 288

If no Discount = ( 320 x 20 ) % = 22.7%

288

 

  1. (a) Dist along circle of lat.

Long diff x 60 x cos q nm

100 x 60 x Cos 500

100 x 60 x 0.866

5196nm =      100 x 2pR Cos 500

                                               360

100  x 2 x 3.14 x 6371

360                       =  5780Km

 

 

 

 

 

 

(b) 80 x 60 Cos 50  = 3895 Km

 

  1. Vol =2.8 x 2.4 x 3 = 20.16m3

          1m3 = 1000 L

20.16m3 = 20160 L

20160

    3600       

16560 L to fill

0.5 L – 1 sec

16560 L – ?

 165600

5 x 3600

33120  hr

3600             @ 9.41 hrs     ;     @ 564.6 min.

 

  1. 15 + 5.14a + 10.13.a2 + 10.12a3 + 5.1.a4

a = -0.2

1 + 5(-0.2) + 10(-0.2)2 + 10(-0.2)+ 5 (-0.2)4

1 – 1.0 + 0.4 – 0.08 + 0.008  =   0.3277 (4d.p)                                                                                                                     

 

  1. Area of metal : Material – Cross section.

p(R2 – r2)

3.14 (21 –19)

Vol  6.28cm2 x 3400cm

= 215.52m3        

                                       

  1. Possibility space:

 

.            1  2  3  4  5  6 

1     2  3  4  5  6  7

2     3  4  5  6  7  8

3     4  5  6  7  8  9

4     5  6  7  8  9  10

5     6  7  8  9 10 11

6     7 8  9 10 11 12

 

P(odd) = 3/6 = ½

P(Sum > 7 but < 10)   =   9 /36

\ P(odd) and P(sum > 7 but < 10 )

= ½  x 9/36 = 9/72     =  1/8

 

  1. ò( 8x5/x3 – 3x/x3) d4

ò( 8x2 – 3x-2) d4                                                                

16x3/3 + 6x-3/-3  + C                                                 

16x3/3 – 2/x+ C

 

  1. Area of DAOB

½  x 14 x 14 x 0.866  =  84.866cm2

Area of sector  =  60  x3.14 x 14 x14 = 10.257

360

Shaded Area

84.666  –  10.257 = 74.409cm2                            

 

 

 

 

 

Marks F Fx fx2
5 3 15 75
6 8 48 288
7 9 63 441
8 6 48 384
9 4 36 324

 

åx =    åf=30   åfx=210   1512

S.d =  Ö åfx2  –  ( åfx )2

                             åf            åf

= Ö 1512   –  (210)

30            30

=  Ö 50.4 – 49

=   Ö 1.4  = 1,183                                                       

 

       SECTION II                                               .

 

  1. (a) FC = Ö 52 + 7.072 = Ö 50 = 7.071

(b) HB = Ö 52 + 7.072    = Ö 75 = 8.660

(c) q = Tan-1 5/5 = Tan-1   = 450                                                         

(d)  b = Tan-1 5/7.071 = Tan-1 0.7071  =  35.30                                                        

(e)  y = Tan-1 5/3.535   = Tan-1    = 54.70                                                        

(f) ÐAGX = 19.40

 

 

  1. y = Sin x
      x0 00 300 600 900 1200 1500 1800
sin x0 0 0.50 0.66 1.00 0.866 0.500 0

 

y = 2 Sin (x0 + 100)

      X0 00 300 600 900 1200 1500 700
2 Sin(x +100) 0.3472 1.286 1.8794 1.286 0.3472 -0.3472 -1.8794

Amplitudes for y = Sin x0 is 1

For

y = Sin(x+100) is 2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c.f X F
61 53 12
16   54
93 55 16
103 56 10
11   57
123 58 9
130 59 7
135 60 5
138 61 3
142 62 4
145 63 3
148 64 3
149 65 1
150 70 1

 

Mean =  x    + 52  + -4

150

52 –  0.02

=     51.08

Median  =     51.4g.

 

class interval 59

Class interval mid point Freg. c.f
44-48 46 12 12
49-53 51 49 61
54-58 56 64 125
59-63 69 22 147
64-68 66 3 130
69-73 71 1 150

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. X + Y £ 1000

X £ 2Y

Y < 720

X > 90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

21.(a)    1cm = 200Km/h

A = 200 x 7.5  =  1500 Km

B =  200 x 9  = 1800Km.

 

(b) (i.) 15.8cm x 200                     (ii) Bearing 2240

= 3160 Km.                              (iii) Bearing 0490

 

  1. (a) P(R) x P(R)1                         (b) P(R)¢ x P(R)                        (c) P(R) x P(R’)

= 0.5 x 0.6                                     0.5 x 0.4                                          P(R)’ x P(R)

= 0.3                                     =  0.2                                            0.5 x 0.6 = 0.3

0.5 x 0.4 = 0. 2= 0.5

  1. y = x(x + 1)(x – 2)

= x3 – x2 – 2x

A1 = ò(x3 – x2 –2x) d4                                

-1[¼ x4 –  1/3 x2]-1

= 0 – ( ¼ + 1/3 – 1)    =  5/12

A2 = 2ò(x3 – x2 –2x) d4

0ò ¼ x4 – 1/3 x3 – x2)-20                     

= ( ¼ .16 – 1/3 .8 – 8 )

= 4-0 – 8/3 – 4  =   – 8/3

              A1 = 5/12= A2 = 2 2/3         

                            

  1. y = x3 – 3x2

dy  = 3x2 – 6x

At stationary

Points      dy = 0

dx

i.e   3x2 – 6x = 0

3x(x – 2) = 0

x = 0 or 2

Distinguish

dy = 3x2 – 6x

dx

d2y  =  6x – 6

dx2

    (i)    x = 0  dy2 = 6x – 6 = -6                 (ii)       x = 2

dx2                                                 d2y  =  6

-6 < 0 – maximum.                               dx2

\ (0,0) Max Pt.                                                6 > 0 hence

Minimum Pt.

x = 2,  y = 8 – 12 = -4

(2, -4)     minimum point.

 

MATHEMATICS II

PART I

 

SECTION 1 (52 Marks)

  1. Without using tables evaluate:

 

Ö7.5625 x 3Ö3.375

15                                                                                                        (5 mks)

 

  1. Make k the subject of the formula.

y = 1  Ök + y                                                                            

T2      k                                                                                                       (3 mks)

 

  1. If A = (x, 2) and xB     =     x     and if AB = (8), find the possible values of x.

-2                                                                                 (3 mks)

  1. Simplify completely. (3 mks)

rx4 – r

2xr – 2r

 

  1. Solve the equation. (3 mks)

Log 3 (8-x)  –  log 3 (1+x) = 1

 

  1. Under an enlargement scale factor -1, A(4,3) maps onto A1 (4,-5). Find the co-ordinates of the centre of enlargement. (3 mks)

 

  1. Find the equation of the line perpendicular to the line 4x-y = -5 and passing through the point (-3,-2).       (2 mks)
  2. Find the standard deviation of the data below:

3,5,2,1,2,4,6,5                                                                                                   (4 mks)

 

  1. What is the sum of all multiples of 7 between 200 and 300? (4 mks)

 

  1. Solve the equation.

½ tan x  =  sin x for -1800  £  x  £  3600.                                                            (3 mks).

 

  1. Expand (1-2x)4. Hence evaluate (0.82)4 correct to 5d.p. (4 mks)

 

  1. The line y = mx – 3 passes through point (5,2). Find the angle that the line makes with the x-axis. (2 mrks)
  2. A two digit number is such that 3 times the units digit exceed the tens digit by 14. If the digits are reversed, the value of the number increases by 36. Find the number (4 mks)

 

 

 

 

 

 

  1. In the figure below, O is the centre of the circle, OA = 7 cm and minor arc AB is 11 cm long. Taking P = 22/7, find the area shaded. (3 mks)

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. A box contains 36 balls, all identical except for colour. 15 of the balls are black, 15 are brown and the rest are white. Three balls are drawn from the box at random, one at a time, without replacement. Find the probability that the balls picked are white, black and brown in that order. (2 mks)

 

  1. Find the inequalities that describe the unshaded region R below. (4 mks)

y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SECTION  2 (48 Marks)

 

  1. Draw the graph of y = x2 + x – 6 for -4 £ x £

Use your graph to solve the equations.

(i)  x2 + x – 6 = 0                       (ii) x2 + 2x – 8 = 0                                             (8 mks)

 

  1. The diagram below represents a bucket that has been placed upside down. The radius of the top surface is 15cm and that of the bottom is 40cm. The vertical height of the bucket is 50cm.

 

 

 

 

 

 

 

 

 

 

 

 

 

Determine:-

  • The volume of the bucket.
  • The curved surface area of the bucket. (leave your answers in terms of p)

 

  1. Draw, on the same axes, the graphs of y = cos q and y = 5 sin q for – 1800 £ q £ 1800
  • From your graph, determine the amplitude of each wave.
  • For what value(s) of q is cosq – 5 sin q = 0 (8 mks)

 

  1. A point P lies on a coast which runs from West to East. A ship sails from P on a bearing of 0320. When it reaches Q, 7km from P, a distress signal is observed coming from another ship at R. Given that R is N.E of P and on a bearing of 0660 from Q, calculate:
  • Ð
  • The distance QR, between the two ships.
  • The shortest distance from R to the shore. (8 mks)

 

  1. A bag contains x red balls and y yellow balls. Four times the number of red balls is equal to nine times the number of yellow balls and twice the total number of balls exceeds the number of yellow balls by 44.
  • How many balls of each colour are three in the bag?
  • If two balls are drawn out of the bag at random one at a time with replacement what is the probability that the two balls are red? (8 mks)

 

  1. A Kenyan businessman goes on a trip to West Germany through Italy and back to Kenya. In Kenya he is allowed to take Ksh. 67,000 for sales promotion abroad. He converts the Kenya currency into US dollars. While in Italy, he converts 2/5 of his dollars into Italian lire, which he spends in Italy. While in West Germany, he converts 5/8 of the remaining dollars into Deutsche marks which he uses up before coming to Kenya. Using the conversion rates 1 US dollar = 1.8 Deutsche marks = 16.75

Ksh = 1340 Italian lire. Answer the following questions:

  • How many US dollars did he take out of Kenya?
  • How many Italian lire did he spend in Italy?
  • How much money, in Deutsche marks did he spend in West Germany?
  • How much money in Ksh. did he have on his return to Kenya? (8 mks)

 

  1. PQRS is a parallelogram in which PQ = r and PS = h. Point A is the midpoint of QR and B is a point on PS such that PS : PB = 4:3. PA and QB intersect at M.

 

 

 

 

 

 

 

 

 

 

Given that PM = kPA and BM = tBQ where k and t are scalars, express PM in two different ways and hence find the values of k and t.

Express PM in terms of r and h only.                                                                                   (8 mks)

 

 

 

 

 

 

 

 

  1. Two variables T and X are connected by the equation T = abx where a and b are constants. The values of T and X are given in the table below:

 

T 6.56 17.7 47.8 129 349 941 2540 6860
X 2 3 4 5 6 7 8 9

 

 

Draw a suitable straight line graph and use it to estimate the values of a and b.              (8 mks)

 

 

MATHEMATICS III

PART II

 

Section I:   (52 Marks)

 

  1. Use mathematical tables to evaluate:

 

8.67                                                                                                                        (3 mks)

Ö 0.786 x (21.72)3

 

  1. Simplify completely. (3 mks)

4      –    1

x2 – 4        x-2

 

  1. An Indian on landing at Wilson Airport changes Re 6000 into Kenya shillings when the exchange rate is Re = Ksh. 1.25. He spent Ksh. 5000 when in Kenya and converted the remaining amount to Rupees at the same rate as before. Find out how much the Indian is left with in Rupees. (3mks)

 

  1. The last of three consecutive odd numbers is (2x+3). If their sum is 105, find the value of x. (4 mks)

 

  1. a S  b is defined by:           a S b  =  (a + b)

ab

If B S   (2  S   3)  =  4  S   1, Find B.                                                                                   (3 mks)

  1. Find the value of M. (3 mks)

 

 

M

 

850

 

1600

 

 

  1. (a) Expand (1+2x)6 upto the term containing x3 .                                                                (2 mks)

 

(b)  By putting x = 0.01, find the approximate value of (1.02)6 correct to 4 S.F.                    (2 mks)

 

 

  1. Show that x is the inverse of : Y =    3          -3      1           X =       2      1                       (3 mks)

-5        2                     5      3

 

 

 

 

 

 

  1. The probabilities of three candidates K, M and N passing an examination is 2/3, ¾ and 4/5 Find the probability that :

(a)  All pass:                                                                                                           (1 mk)

(b)  At least one fails:                                                                                              (2 mks)

 

  1. In the figure, PR is tangent to the circle centre O. If ÐBQR=300, ÐQBC=270,and ÐOBA=370, find ÐBAC and Ð

 

C                        A

 

 

 

 

B                                                                                            P                                                                                 R

  1. A frustrum of height 10cm is cut off from a cone of height 30cm. If the volume of the cone before cutting is 270cm3 , find the volume of the frustrum. (3 mks)

 

  1. Evaluate 0 (2 mks)

( 3x2 –  1 ) dx

4 x 2

1

  1. If one litre of water has a mass of 1000g, calculate the mass of water that can be held in a rectangular tank measuring 2m by 3m by 1.5m. (give your answer in tonnes). (2 mks)
  2. Write down the three inequalities which define the shaded region. (3 mks)

 

 

 

(3,2)

 

 

 

 

 

 

(2,1)                                   (4,1)

 

 

 

 

  1. The depth of sea in metres was recorded on monthly basis as follows:

 

Month March April May June July
Depth (m) 5.1 4.9 4.7 4.5 4.0

Calculate the three monthly moving averages.                                                               (3 mks)

  1. A number of women decided to raise sh. 6300 towards a rural project for bee keeping. Each woman had to contribute the same amount. Before the contribution, seven of them withdrew from the project. This meant the remaining had to pay more. If n stands for original number of women, show that the increase in contribution per woman was: 44100                   (3 mks)

n(n-7)

 

 

 

 

 

SECTION II:   (48 Marks)

 

  1. Find the distance between points A(500 S, 250 E) and B(500 S, 1400 E) in:

(i)   Km                  (ii)   nm                                                                                                (8 mks)

(take radius of earth to be 6400km, P =  3.14)

 

  1. The distance S in metres, covered by a moving particle after time t in seconds, is given by :

S  =  2t3 + 4t3– 8t + 3.

Find:

(a)  The velocity at :            (i)  t  =  2                      (ii)  t  =  3

  • The instant at which the particle is at rest. (8 mks)

 

  1. A car starts from rest and its velocity is measured every second for six seconds. (see table below).
Time (t) 0 1 2 3 4 5 6
Velocity v(ms -1) 0 12 24 35 41 45 47

 

Use trapezium rule to calculate the distance travelled between t = 1 and t = 6.                (8 mks)

 

  1. Using a pair of compass and ruler only, construct triangle ABC such that AB=9cm, BC=14cm and ÐBAC = 1200 . Draw a circle such that AB, BC and AC are tangents. What is the radius of this circle?                                                                                                                                (8 mks)
  2. The marks scored by 100 students in mathematics test is given in the table below:
Marks 10-19 20-29 30-39 40-49 50-59 60-69 70-79
No. of students 8 15 15 20 15 14 13

 

(a)  Estimate the median mark.                                                                               (2 mks)

(b) Using 44.5 as the assumed mean, calculate:-

(i)         The mean mark:                                                                                   (2 mks)

(ii)        The variance:                                                                                        (2 mks)

(iii)       The standard deviation:                                                                         (2 mks)

 

  1. (a) On the same axes, draw the graphs of : y  =  sin x  ;  y  =  cos x

y  =  cosx  +  sin X for 00 Ð X Ð 3600 .

(b)  Use your graph to deduce

(i) The amplitude

(ii) The period of the wave y = cos x + sin x.

(c) Use your graph to solve:

Cos x  = – sin x for 00 Ð X Ð 3600 .

 

  1. Given a circle of radius 3 units as shown in the diagram below with its centre at O(-1, 6). If BE and DE are tangents to the circle where E (8,2). Given further that Ð DAB = 800.

B

 

 

A                                                                              E

C

 

 

D

(a)  Write down the equation of the circle in the form ax2 + bx + cy2 + dy + e = 0 where a, b, c,             d, e are constants.                                                                                       (2 mks)

(b)  Calculate the length DE.                                                                                   (2 mks)

(c)  Calculate the value of angle BED.                                                                     (2 mks)

(d)  Calculate the value of angle DCB.                                                                     (2 mks)

 

  1. A building contractor has to move 150 tonnes of cement to a site 30km away. He has at his disposal 5 lorries. Two of the lorries have a carrying capacity of 12 tonnes each while each of the remaining can carry 7 tonnes. The cost of operating a 7 tonne lorry is sh. 15 per km and that of operating a 12 tonne lorry is sh. 25 per km. The number of trips by the bigger lorries should be more than twice that made by smaller lorries.                                                                                     (8 mks)

 

(a)  Represent all the information above as inequalities.

  • How should the contractor deploy his fleet in order to minimise the cost of moving the cement?                                                                                                                                   (8 mks)

 

 

MATHEMATICS III

PART I

MARKING SCHEME

 

 

 

 

 

 

SOLUTION MRK AWARDING  
1. Ö7.5625 = 2.75

 

3Ö3.375 = 3Ö3375 X 3Ö10-3

 

3 Ö33 x 53 x 10-1 = 3 x 5 x 10-1 = 1.5

 

= 2.75 x 1.5  =  2.75  =  0.275

1.5 x 10          10

 

1

 

1

 

1

1

1

 

 

Method for Ö7.5625

Square root

 

Method for 3Ö

3Ö

Answer

 
    5    
2. T2y  =  Ö k+y

K

T4y2k =  k+y

T4y2k – k  =  y

K(T4y2-1) =  y

K  =  y

T4y2 – 1

 

 

1

 

 

1

 

1

 

 

Removal of square root

 

Rearrangement of terms

Answer

 
    3    
3. (x 2)         x      =  (8)

-2

 

x2 – 4  =  8

 

x  =  +Ö12 = + 2Ö3 = + 3.464

 

1

 

 

1

 

1

 

 

Matrix equation

 

 

Quadratic equation

Answers in any form

 
    3    
4. r(x2 – 1)

2r(x – 1)

 

r(x2 – 1)(x2 + 1)

2r (x – 1)

 

r(x – 1)(x + 1)( x2 + 1)

2r (x – 1)

 

=   (x + 1)( x2  + 1)

2

 

 

 

 

1

 

 

 

1

 

 

1

 

 

 

 

Complete factorisation of numerator

 

Factorisation of denominator

 

Answer

 
    3    
5.       1  =  log3 3

8 – x    =   3

1+x

 

-4x  =  -5

 

x = 5

4

1

 

 

 

1

 

1

 

 

 

Logarithic expression.

 

 

Equation

 

Answer

 

 

 
    3    
6. Let the centre be (a,b)

 

4-9        =  -1      4-a

-5-b                  3-b

 

4-a  =  -4+9           -5-b  =  -3+b

a  =  4                     b  =  -1

centre is (4,-1)

 

 

 

1

 

 

1

 

1

 

 

 

 

 

Equation

 

 

Linear equations

 

Centre

 

 
    3    
7. Y  =  4x + 5

Gradient = 4

Gradient of ^ line – ¼

y + 2  =  – 1

x + 3        4

4y + x  =  -11

 

 

1

1

 

 

Gradient of ^ line.

Equation.

 

 
    2    

8.

X  = 28  =  3.5

8

 

 

standard deviation = Ö 22 = Ö2.75  =  1.658

8

 

 

1

 

 

1

1

 

1

 

Mean

 

 

d values

d2 values

 

Answer

 
    4

 

   
9. a = 203    d = 7   L = 294

 

294  =  203 + 7(n-1)

n  =  14

 

S 14  =  14 (203 +  294)

2

 

=  7 x 497

=  3479

 

1

 

1

 

 

1

 

 

 

1

 

For both a and b

Equation

 

 

For n

 

 

 

Sum

 

 
    4    
10. Sin x  =  2 sin x

Cos x

 

Sin x  =  2 cosx

Sin x

 

2 cos x  =  1

cos x  =  0.5

 

x  =  600, 3000, -600

 

 

 

1

 

 

 

1

 

1

 

 

 

 

Simplification

 

 

 

Equation

 

All 3 values

 
    3    
11. (1 +-2x)4  =  1-8x + 24x2 – 32x3 + 16x4

 

(0.82)4  =  (1 + -2 x 0.09)4

x     =  0.09

(0.82)4  = 1 – 0.72 + 0.1944 – 0.023328 + 0.00119376

= 0.35226576

@  0.35227 (5 d..p)

1

 

 

1

1

 

1

 

Expansion

 

 

Value of x

All terms

 

 

Rounded

 
    4    
12.   2  =  5m – 3

m =  1

tan q  =  1                    q  =  450

 

1

1

 

 

Value of m.

Angle

 
    2    
13.  Let the number be xy

3y  =  x + 14

10y + x  =  10x + y + 36  =  9y – 9x  Þ  36

3y – x  =  14

9y – 9x  =  36

y  =  5

x  =  1

the number is 15.

 

1

1

 

1

 

 

1

 

 

1st equation

2nd equation

 

method of solving

 

Answer

 

 
   

 

S

4    
14. Let ÐAOB  =  q

  q  x  2  x   22  x  7  =  11

360              7

q  =  900

 

Area shaded  =   90 x 22 x 7 x 7 – 1 x 7 x 7

360    7                2

77 49

2     2

= 28  =  14cm2

2

 

 

 

1

 

1

 

 

1

 

 

 

 

Value of q

 

Substitution

 

 

Answer

 
    3    
15. P(WBb)  =  6 x 15 x 15

36    35   34

 

=   15

476

1

 

 

1

 

Method

 

 

Answer

 
    2    
16. Equation                                  inequality

L1    y =  x                                   y  £  x

L2    y = -2                                   y  ³ -2

L3    2y + 5x = 21                        2y + 5x < 21

1

1

1

1

 

1 mark for each inequality.

Method for obtaining L3

 

 
 

 

 

 

 

 

 

 

 

 

 

 

(i)  roots are x = -3

x = 2

(ii)  y = x2 + x-6

0 = x2 + 2x-8

y = -x + 2

roots are x = -4

x =  2

4 2

 

 

 

 

 

 

 

1

 

1

 

 

1

 

1

 

1

 

For all correct points.

1 for atleast five correct points.

 

 

 

Correct plotting.

 

Scale

 

 

Smoothness of

curve

 

Both roots

 

 

Linear equation

 

 

Both roots

 

   

 

  8  
         
18.    h     =  15

h+50     40

 

h   = 30cm

H  =  80cm

 

(a)  Volume  =  1/3 p x 40 x 40 x 80 – 1/3  p x 15 x 15 x 30

 

128000 p  –  6750 p

3               3
=   121,250p cm3

3

 

(b)   L2  =  802 + 402                      L    =  152 + 302

= 6400 + 1600                      = 225 + 900

=  8000                                   = 1125

L    =  89.44 cm                    L    =  33.54 cm

Curved surface area of bucket = p x 40 x 89.44

p x15x33.54

= 3577.6p – 503.1p

=  3074.5cm2

1

 

 

1

 

 

1

 

 

 

 

1

 

1

 

1

 

1

 

1

 

Expression

 

 

Value of H

 

 

Substitution

 

 

 

 

Volume

 

L

 

L

 

 

Substitution

 

Area

 

 
    8    
 

 

19.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     
         
 

20.

 

 

 

(i)  ÐRPQ  =  130

        ÐPQR  =  320+900+240 =  1460

ÐPRQ  =  1800 – (1460 + 130)

=  210

 

(ii)    P      =        7

sin130         sin 210

P    =   7 sin 130

Sin 210

=  4.394km

 

 

 

 

 

 

 

 

 

 

 

P                                                               T

 

(iii)    Let PR  =  q

 

q       =       7

sin 1460      sin 210

 

q     =  7 sin 1460

sin 21

q       =  10.92 km

 

sin 450  =    RT

10.92

 

RT  =  10.92 sin 450

 

= 7.72 km (2 d..p)

 

 

 

 

1

 

 

 

 

 

 

1

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

1

 

 

 

 

 

1

 

 

1

 

1

 

 

 

Fair sketch

 

 

 

 

 

 

ÐPRQ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Equation

 

Method

 

 

 

Equation

 

 

 

 

 

 

Distance PR

 

 

Equation

 

RT

 

 

 

 

 
  8    
21. (a)  4x  =  9y

2(x+y)  =  y+44  Þ  2x + y  =  44

 

4x – 9y = 0

4x + 2y = 88

11y = 88

y   =  8

 

x  =  18

(b)  P(RR)  =   18  x  18   =  81

26      26      169

 

1

 

1

 

2

 

1

1

1

1

 

 

 

Equation

 

Equation

 

Method of solving

Value y

Value x

Method

Answer

 
    8    
22. (a)  67,000 Ksh  =  67,000 US dollars

16.75

= 4,000 dollars

 

(b)  2 x 4,000  =   1600 US dollars

5

1600 US dollars  =  1600 x 1340

=  2,144,000 Italian lire

(c)  Remainder  =  2400 US dollars

5  x  2400   =  1500 US dollars

8

1500 US dollars = 1500 x 1.8

= 2700 Deutche marks

(d)  Remainder  =  900 US Dollars

900 US Dollars = 900 x 16.75 Ksh.

=  15,075 Ksh.

 

1

 

1

 

1

 

1

1

 

 

1

1

1

 

 

Method

 

Answer

 

Method

 

Answer

 

For 1500

 

 

Answer

 

Method

Ksh.

 
    8    
23. PM  =  kPA

=  k(r + 1h)

2

=  kr + 1kh

2

PM  =  PB +  BM

3h + t BQ

4

=   3h + t(-3h + r)

4          4

 

3h – 3t h + tr

4     4

3 –   3t    h + tr

4     4

 

t = k           33t  =  1k

4   4       2

33t = 1 t

4    4     2

5t  =   3

4       4

t  =  3 + 4

4    5

= 3

5

\   k = 3

5

\   PM  =  3r  +  3h

5       10

 

 

 

1

 

1

 

 

1

 

 

1

 

1

 

 

1

 

1

1

 

 

 

 

 

PM

 

PM

 

 

PM simplified

 

 

 

 

 

 

 

Both equations

 

method

 

 

 

 

Value of k

 

Value k

PM

 

 
    8    
         
 

 

 

24.

Y

LogT

 

 

 

Log T  =  log a + x log b

Log T  Þ  0.82, 1.25, 1.68, 2.11, 2.54, 2.97, 3.40, 3.84

 

y – intercept = log a = 0

a = 1

gradient  =  3.84 – 0.82  =   3.02

9 – 2                  7

= 0.4315

 

log b = 0.4315   =  0.4315

b = antilog 0.4315

b  =  2.7

 

1
1

 

 

 

 

 

 

 

1

2

 

1

 

1

 

 

 

 

1

8

Plotting
Labeling of axis

 

 

 

 

Linear

All correct logs

 

Value of a

Method of gradient

 

Value of  b

 

MATHEMATICS III

PART II

MARKING SCHEME

 

  1. SOLUTION MARKS    AWARDING
1.    No                                      log

 

8.69                                   0.9390

0.786                                 1.8954

21.72                                 1.3369

1.2323

1.7067 – 2

 

21.7067

2           2

– 1  +  0.8533

0.7134 x 10 -1     =  0.07134

 

 

 

 

 

M1

 

 

M1

 

 

 

A1

 

 

ü reading to 4 s.f

 

 

 

 

 

 

Rearranging

    3  
2.  

 4                   –         1

(x-2)(x+2)                  (x-2)

 

 – x+2

(x-2(x+2)

– (x-2)

(x-2(x+2)

 

-1

x+2

 

 

 

M1

 

 

M1

 

 

A1

 

 

 
    3  
3.  

Re6000  =  Ksh. 75000

Spent 5000 Rem 2500

Rem    2500

1.25

Re 2000

M1

 

 

M1

 

A1

 

 
    3  
4. 2x – 1  ,  2z + 1  ,  2x + 3

6x +  3  =  105

6x  =  102

x  =  17

M1

M1

A1

A1

 

Allow M1 for us of different variable.
    4  
5.  

4 * 1  =  5

4

2 * 3  =  5

6

A * 5  =  5

6      4

A + 5  =  5  x  5A

6      4       6

A +  5  =  25 A

6       24

A   =  20

 

 

M1

 

 

 

 

M1

 

 

A1

3

 
6.  

 

 

 

 

180 – M + 20 + 95  =  180

295  –  M  =  180

– M  =  – 115

M  =  115

 

 

 

B1

 

 

B1

 

 

A1

 

 
    3  
 

7.

 

1 + 2x + 60x2 + 160x3 +

1 + 0.2 + 0.006 + 0.00016

=  1.20616

=  1.206

 

M1

M1

M1

A1

4

 

Only upto term in x3.

Correct substitution

 

Only 4 s.f.

 

8.  

3   -1      2    1    =    I

-5   2       5    3

 

6   -5             3    -3

-10 +10         -5 + 6

 

1      0

0       1

 

 

M1

 

M1

 

 

A1

 

 

Matrix multiplication gives :

 

I       1   0

0   1

  3  
9. (a)   2  x  3  x  4      =  2

3      4      5           5

(b)

2  x  3  x 1     +     2  x  1  x  4     +     1  x  3  x  4
3      4     5            3      2      5             3      4      5

 

1  +  4  +  1

10     15     5

 

=     17

10

M1

 

 

M1

 

 

 

 

A1

 

 

 
    3  
10. ÐQCB  =  300

180 – (27 + 30)  =  1230

\     BAC  =  570.

 

 

 

 

OBA  =  370

OAB  =  370

 

 

AOB  =  1060

\ ACB  =  530

 

 

 

M1

 

 

 

 

 

M1

 

 

A1

 

 

 

 

 

 

 

 

 

Isosceles triangle.

 

Angle at centre is twice angle at circumference.

    3  
11. V  =  1  x  3.14  x  r 2  x 10  =  270

L.S.F.      20   =  2

30       3

V.S.F  =    2   3        =     8

3                   27

Vol. of cone  =  8  x  270

27               =      80cm3

\ Vol. Of frusturm  = (270 – 80)  =  190cm3

 

 

 

M1

 

M1

 

 

A1

 

 
    2  
12.

 

 

 

 

 

 

 

 

3x 3  –  x  -1          2

3       -1         1

 

x 3  +  1     2

x     1

 

8  +  1     –   ( 1  –  1)

2

8 1  –  2     =         6  1

2                           2

 

 

 

 

 

 

M1

 

 

 

A1

2

 
13. (2 x 3 x 1.5)  volume

9 m3

1L  º  1000 cm3

1000 L  =  1 m3

9000 L  =  9 m3

1000 L  =  1 tonne

9000 L  =  9 tonnes.

 

 

M1

 

 

 

A1

 

 
    2  
14.      y   ³ 1            (i)

y   <  x – 1     (ii)

y   <  5 – x     (iii)

 

B1

B1

 

 
    3  
15. M1  =  5.1  +  4.9  +  4.7  =  4.9

3

M2  =  4.9 + 4.7 + 4.5  =  4.7

3

M3  =  4.7 + 4.5 + 4.0  =  4.4

3

M1

M1

M1

 

 

 
    3  
16. Original contribution per woman  =  6300

N

Contribution when 7 withdraw  =  6300

(n-7)

Increase   –  Diff.

6300   –   6300

n-7          n

6300n  –  6300(n-7)

n(n-7)

6300n – 6300 + 44100

n(n-7)

44100

n(n-7)

 

 

 

M1

 

 

M1

 

1

3

 
SECTION II (48 Marks)

 

17. (i)

1150

 

A                                B

 

Centre of circles of latitude 500 S.  R Cos 500

AB  =  115  x  2p R Cos 50o

115  x  40192  x  0.6428

360

=  8252.98  km

 

(ii)   Arc AB 60 x 115  Cos 50 nm

60 x 115 x 0.6428 nm

4435 nm

 

 

 

 

 

M1

M1

 

 

M1

 

A1

 

M1

M1

M1

A1

 

 

 

 

 

 

 

No.                     log

60                      1.7782

1+5                    2.0607

0.6428               1.8080

4435nm             3.6469

    8  
18. (a)  V  =  ds  =  6t2 + 8t – 8

dt

(i)  t  =  2

V  =  6×4 + 8×2 – 8

= 32 ms-1

(ii)  t  =  3

V =  6×9 + 8×3 – 8

= 70ms-1

 

(b)  Particle is at rest when V = 0

6t2 + 8t – 8 = 0

2(3t – 2) (t+2) = 0

t  =  2                   t  =  -2

3

particle is at rest at t = 2 seconds

3

   

 

 

 

 

 

 

 

 

 

 

 

Do not accept t = -2. Must be stated.

    8  
19. Area under velocity – time.

graph  gives distance.

 

A  = { h ½  (y1 + y6 ) + y2 + y3 + y4 + y5 )}

 

= 1 { ½ ( 12+47) + 24 + 35 + 41 + 45)}

=  29.5 + 14.5

=  174.5m

 

 

B1

B1

M1

M1

B1

B1

A1

 

Trapezium rule only accepted.

Formula.

 

Substitution into formular.

    8  
20.                  Drawing actual

Scale 1cm  =  2cm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Radius      1cm

=  2cm

 

M1

 

M1

 

M1

 

M1

 

M1

M1

 

M1

M1

 

 

Bisect ÐA

 

Bisect Ð B

 

Intersection at centre of inscribed circle.

Draw circle.

 

Measure radius.

Arcs must be clearly shown.

  8  
 

 

 

21.

 

 

 

 

mean = 44.5 +  130

100

=  44.5  +  1.3

=  45.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)  Variance  S (x – A) 2  =  2800

Sf               100

= 28

S.D.  =  Ö 28  =  5.292

 

 

 

 

M1

 

 

 

 

 

A1

 

M1

 

A1

M1

A1

 

 
    8  
 

 

 

 

 

22.

y = sin x

x    0        60        120        180     240      30      360

sin x 0    0.866     0.866      0     -0.866   -0.866    0

y = cos x

x     q        60        120        180     240    300  360

cos x 1     0.5       -0.5       -1.0     -0.5     0.5   1.0

y = cosx + sinx

x            q        60       120        180     240      30     360

cosx + sinx 1  1.366   0.366       -1   -1.366  -0.366 1.0

(c)      Cos x = – sin x

x  =  450 , 2250

   
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

23.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(i)  amplitude   =  1.366

(ii)  Period  =  3000

 

 

 

(a)  (x+1) 2  +  (y-6)2  =  32

x2 + 2x + 1 + y2 – 12y + 36  =  9

x2 + 2x + y2 – 12y + 28  =  0

 

(b)  cos 10  =  OD             DE  =  3

DE                   0.9848

DE  =  3.046

 

(c)  Twice ÐOED

100 x 2  =  200

 

(d)  DAB  =  800

\ DCB  =  1000

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M1

 

A1

 

M1

A1

 

 

M1

A1

 

M1

A1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Formular

(x-a)2 + (y-b)2 = r2

 

 

 

 

 

 

 

 

 

Cyclic quad.

 

 

    8  
24. Let number of trips by 12 tonne lorry be x.

Let number of trips by 7 tonne lorry be y.

 

(a)   x > 0  ;  y > 0

24x + 21y  £  150

 

12 x 25 x X + 15 x 7 x y £ 1200

300x + 105y  £  1200

x > 2y

 

(b)  Ref. Graph paper.

Minimising:

3 – 12 tonne lorry and 2 – 7 tonne lorries should be deployed.

 

 

 

B1

 

 

 

B1

B1

 

 

 

 

 

 

 

MATHEMATICS IV

PART I

 

SECTION 1 (52MKS)

 

  1. Evaluate using logarithms 3Ö7.673 – 15.612

12.3                                                              (4mks)

 

  1. Solve x   –  3x  –  7    =  x – 2                                                                                   (3mks)

3            5             5

 

  1. In the given figure CD is parallel to BAC, calculate the values of x and y. (3mks)

 

 

C                                       D

 

 

 

 

 

 

B                                                A

 

  1. The surface area and volume of a sphere are given by the formulars S = 4pr2 and V= 4/3 pr3.

Express V in terms of S only.                                                                                (3mks)

 

  1. A line perpendicular to y = 3-4x passes through (5,2) and intercepts y axis at (0,k)

Find the value of K.                                                                                              (3mks)

 

  1. An alloy is made up of metals P,Q,R, mixed in the ratio 4:1: 5: A blacksmith wants to make 800g of the

alloy. He can only get metal P from a metallic ore which contains 20% of it. How many Kgs of the ore

does he need.                                                                                                           (3mks)

 

 

  1. The co-ordinate of point A  is (2,8) vector AB =   5    and vector BC  =  4   Find the

-2                                 3

co-ordinate of point C.                                                                                             3mks)

 

  1. Two buildings are on a flat horizontal ground. The angle of elevation from the top of the shorter building to the top of the taller is 200 and the angle of depression from the top of the top of the shorter building to the bottom of the taller is 300. If the taller building is 80m, how far apart are they

(4mks)

  1. The given figure is a quadrant of a piece of paper from a circle of radius 50cm. It is folded along AB

and AC to form a cone . Calculate the height of the cone formed.

(4mks)

 

 

 

 

5Ocm

 

 

50cm

 

 

  1. Express 3.023 as a fraction                                                                                      (2mks)
  2. Point A (1,9), Point B(3,5) and C (7,-3). Prove vectorically that A,B and C are collinear.       (4mks)
  3. A salesman gets a commission of 4% on sales of upto shs 200,000 and an additional 2% on

sales above this. If in January he got shs 12,200 as commission, what were his total sales    (4mks)

  1. Water flows through a cylindrical pipe of diameter 3.5cm at the rate of 2m/s. How long to the nearest minute does it take to fill a spherical tank of radius 1.4m to the nearest minute? (4mks)
  2. Rationalize the denominator in Ö3

Ö 7 – 2

Leaving your answer in the form Öa + Öb

C

Where a ,b, and c are integers                                                                              (3mks)

  1. For positive values of x, write the integral solutions of 3£ x2  £  35                 (4mks)
  2. 8 girls working 5 hours a day take 12 days to drain a pool. How long will 6 girls working 8 hours a day take to drain the pool?( Rate of work is equal) (2mks)

 

SECTION II  (48 mks)

 

  1. In the given circle centre O , A,E,F, is target to the circle at E. Angle FED = 300  <DEC = 200 and  <BC0  = 150

 

 

 

 

A                                                                       F

 

 

 

 

Calculate   (i) <CBE                                                                                              (3mks)

(ii)  <BEA                                                                                            (2mks)

(iii) <EAB                                                                                            (3mks)

 

  1. The sum of the 2nd and third terms of a G.P is 9/4 If the first term is  3,

(a) Write down the first 4 terms of the sequence .                                              (5mks)

(b) Find the sum of the first 5 terms using positive values of the common ratio (r)

(3mks)

  1. E and F are quantities related by a law of the form E = KFn Where k and n are

constants. In an experiment , the following values of E and F were obtained .

 

E 2 4 6 8
F 16.1 127.8 431.9 1024

 

Use graphical method to determine the value of k and n (Graph paper provided)      (8mks)

 

  1. In the domain –2 £ x £ 4 draw the graph of y = 3x2 + 1 –2x .Use  your graph to solve the equation.  6x2 4x + 4 = 0 (graph paper provided)                                                                 (8mks)
  2. A solid sphere of radius 18cm is to be made from a melted copper wire of radius 0.4mm . Calculate the length of wire in metres required to make the sphere.                                       (5mks)

(b) If the density of the wire is 5g/cm3. Calculate the mass of the sphere in kg.        (3mks)

 

  1. A right cone with slant  height of 15cm and base radius 9cm has a smaller cone of height 6cm chopped off to form a frustum. Find the volume of the frustum formed                    (8mks)

 

 

 

 

 

 

 

 

9cm

 

  1. PQRS are vertices of a rectangle centre. Given that P(5,0) and Q and R lie on the line x+5 = 2y, determine

(a) The co-ordinates of Q,R,S,                                                                                                   (6mks)

(b) Find the equation of the diagonal SQ                                                                                     (2mks)

  1. A tap A takes 3 hours to fill a tank. Tap B takes 5 hours to fill the same tank. A drain tap C takes 4 hours to drain the tank. The three taps were turned on when the tank was empty for 1½ hours. Tap A is then closed. Find how long it takes to drain the tank.

(8mks)

 

 

 

 

 

 

 

 

MATHEMATICS IV

PART II

 

SECTION   I  (52MKS)

 

  1. Without using mathematical tables, evaluate                                                                    (3mks)

 

Ö 0.0784 x 0.27                                              (leave your answer in standard form)

0.1875

 

  1. A father is three times as old as his son. In ten years time , the son will be half as old as the father . How

old are they now?                                                                                                                                      (3mks)

 

  1. A,B,C,D, is a parallelogram diagram. ADE is an equilateral triangle. AB and CD are 3cm apart.

AB = 5cm. Calculate the perimeter of the trapezium ABCE                                               (3mks)

 

E                            D                                    C

A                                   B

  1. Given that a = -2, b = 3 and c = -1, Find the value of   a3 – b – 2c2                                    (2mks)

2b2 – 3a2c

 

  1. The exchange rate in January 2000 was US $ 1 = Ksh 75.60. and UK £1 = Ksh 115.80.    A tourist  came to Kenya with US $ 5000 and out of it spent ksh.189,000. He changed   the balance in UK £ . How many pounds did he receive?                                                                                                   (4mks)

 

  1. ABC is a cross – section of a metal bar of uniform cross section 3m long. AB = 8cm and  AC = 5cm.

Angle BAC = 600 . Calculate the total surface area of the bar in M2.                                     (4mks)

 

  1. The bearing of a school chapel C, from administration block A, is 2500 and 200m  apart.

School flag F is 150m away from C and on a bearing of 0200. Calculate the distance and

bearing of A from F.                                                                                                               (5mks)

  1. A box has 9 black balls and some white balls identical except in colour. The probability of picking a white ball is 2/3

(i) Find the number of red balls                                                                                       (2mks)

(ii) If  2  balls are chosen at random without replacement, find the probability that they are of different colour.                                                                                                                          (2mks)

  1. Under an enlargement of linear scale factor 7, the area of a circle becomes 441.p

Determine the radius of the original circle.                                                              (3mks)

  1. A circle has radius 14cm to the nearest cm . Determine the limits of its area.                     ( 3mks)
  2. Expand (1 + 2x)5 up to the term with x3. Hence evaluate 2.045 to the nearest 3 s.f. (4mks)
  3. The nth term of a  G.P is given by  5 x 2 n-2

(i) Write  down the first 3 terms of the G.P                                                                (1mk)

(ii) Calculate the sum of the first 5 terms                                                                            (2mks)

  1. 3 bells ring at intervals of 12min, 18min and 30min respectively. If they rang together at 11.55am, when will they ring together again.                                                         (3mks)
  2. On a map scale 1:20,000 a rectangular piece of land measures 5cm by 8cm. Calculate its actual area in hectares.                                                                                                                                      (3mks)
  3. It costs Maina shs. 13 to buy 3 pencils and 2 rubbers; while Mutiso spent shs.9 to buy one pencil and 2 rubbers. Calculate the cost of a pencil and one rubber                      (3mks)

 

  1. Three angles of a pentagon are 1100, 1000 and 1300. The other two are 2x and 3x respectively. Find their values .                                                              (2mks)

 

SECTION II (48MKS)

 

  1. Members of a youth club decided to contribute shs 180,000 to start a company. Two members withdrew their membership and each of the remaining member had to pay shs. 24,000 more to meet the same expense. How many members remained? (8mks)
  2. A box contains 5 blue and 8 white balls all similar . 3 balls are picked at once. What is the probability that

(a)  The three are white                                                                                         (2mks)

(b)  At least two are blue                                                                                                    (3mks)

(c) Two are white and one is blue                                                                                         (3mks)

 

  1. A rectangular tennis court is 10.5m long and 6m wide. Square tiles of 30cm are fitted on the floor.

(a) Calculate the number of tiles needed.                                                                             (2mks)

(b) Tiles needed for 15 such rooms are packed in cartons containing 20 tiles. How many cartons are

there in total?                                                                                                                 (2mks)

(c)  Each carton costs shs. 800. He spends shs. 100 to transport  each 5 cartons. How  much would one

sell each carton to make 20% profit ?                                                                             (4mks)

  1. The following was Kenya`s income tax table in 1988.

Income in K£ P.a             Rate (Ksh) £

1          –   2100                  2

2101    –   4200                  3

4201     –  6303                  5

6301     –  8400                  7

 

(a) Maina earns £ 1800 P.a. How much tax does he pay?                                         (2mks)

(b) Okoth is housed by his employer and therefore 15% is added to salary to make  taxable income. He

pays nominal rent of Sh.100 p.m His total tax relief is Shs.450. If he earns K£3600 P.a, how much

tax does he pay?                                                                                              (6mks)

  1. In the given figure, OA = a , OB =b,  OP: PA =3:2,  OQ:QB = 3:2

Q

B
R

O                                                                            A

(a) Write in terms of a and b vector PQ                                                                                       (2mks)

(b) Given that AR = hAB where h is a scalar, write OR in terms h, a. and b                    (2mks)

(c) PR  =  K PQ Where K is a scalar, write OR in terms  of k, a and b                           (1mk)

(d) Calculate the value of k and h                                                                                               (3mks)

 

  1. A transformation P = and maps A(1,3) B(4,1) and C(3,3) onto A1B1C1. Find the

 

 

co-ordinates of A1B1C1 and plot ABC and A1B1C1 on the given grid.

Transformation Q maps A1B1Conto A11 (-6,2) B11(-2,3) and C11(-6,6). Find the matrix Q and plot

A11B11C11on the same grid. Describe Q fully.                                                           (8mks)

 

  1. By use of a ruler and pair of compasses only, construct triangle ABC in which AB = 6cm,

BC = 3.5cm and AC = 4.5cm. Escribe circle  centre 0 on BC to touch AB and

AC produced at P and Q respectively. Calculate the area of the circle.                       (8mks)

  1. The following were marks scored by 40 students in an examination

330       334      354     348     337     349     343    335    344    355

392       341      358     375     353     369     353    355    352    362

340       384      316     386     361     323     362    350    390    334

338       355      326     379     349     328     347    321    354    367

 

(i) Make a frequency table with intervals of 10 with the lowest class starting at 31          (2mks)

(ii) State the modal and median class                                                                         (2mks)

(iii) Calculate the mean mark using an assumed mean of 355.5                                        (4mks)

 

 

MATHEMATICS IV

PART 1

MARKING SCHEME

 

1.  

Ö –  7.939

12.3

 

=      No             log

7.939                       0.8998

12.3              1.0899

T.8099   1/3 = 3 + 2.8099                                T.9363                   3

 

=  -0.8635

B1

 

 

 

 

B

 

M1

 

A1

4

 

 Subtraction

 

 

 

 

Logs

 

Divide by 3

 

Ans

2. 5x – 3 (3x –7 )    =  3(x – 2 )

5x – 9x + 21    =   3x – 6

-7x             = -27

x              =  36/7

 

M1

M1

 

A1

3

Multiplication

Removal ( )

 

Ans

3. 3x +5y + x =  180

9x   =  180

x    =   20

y   =    60

M1

A1

B1

3

Eqn

X

B

 

 

4.  

.                               r   =       3v      1/3

4P

 

.                              r   =        S       ½

4P

 

\ 3V      1/3              =            ½

4P                                 4P

 

3V                         =       S       3/2

4P                                 4P

 

V             =       4P      S     3/2

3            4P

 

 

 

B1

 

 

 

 

 

 

 

M1

 

 

 

A1

3

 

 

 

Value r

 

 

 

 

 

 

 

Equation

 

 

 

Expression

5.

 

 

 

 

6.

Grad  line          = ¼

y – 2        = ¼

x – 5

y            =  ¼ x + ¾

k             =      ¾

P in Alloy         = 4/10  x 800

= 320g

100 x 320

20

=  3.2 kg

 

M 1

 

A1

A 1

3

 

B1

 

M1

 

A 1

 

Equation

 

Equation

K

 

 

P in alloy

 

Expression

 

Ans

 

 

 

 

7.

 

 

 

 

B (a,b) ,            C (x ,y)

.a – 2          =    5

.b – 8               -2

.a  = 8     b = 6      B(8, 6 )

x – 8          =   3

y – 6               4

x = 11,  y = 10 c(11,10)

 

 

 

 

 

B1

 

M1

 

 

A1

3

 

 

 

 

B conduct

 

Formular

 

 

C

8.  

 

 

 

 

 

80 – x

 

 

 

 

 

.h = x tan 70

h = (80 – x ) tan 60

\   x tan 70 = 80 tan 60-x tan 60

2.7475x + 1.732x = 138.6

4.4796 x       =   138.6

.h     =    138.6 x tan 60

4.4796

 

= 53.59

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B1

M1

 

 

 

M1

 

A1

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Expression for  h both

Equation

 

 

 

Expression for h

 

Ans

9.                 2pr    =  90  x 2p x 50

360

r    =  12.5

h     =  Ö2500 –  156.25

=   Ö2343.75

=   48.41 cm

 

M1

P

A1

M1

 

A1

4

Equation

 

.r

expression for h

 

ans

 

 

10.

 

100 n      =   302.323

     n      =      3.023   

99n       =   299.3

n      =    2993

990

=    323/990

 

M1

 

 

A1

4

 

 

Equation

 

 

Ans

 

11. AB        =     3-1

5-9

=     2

-4

BC         =     4

-8

AB         = ½   BC

\ AB // BC

But B is common

\ A,B,C are collinear.

 

 

 

 

B1

 

 

 

 

 

B1

 

 

B1

3

 

 

A B &  BC

 

 

 

 

 

Both

 

 

Both

 

12.       4% of 200,000  = 8000/=

balance                   = 4200/=

6% of  x                 = 4200/=

x                 = 4200 x 100

6

=  70,000

sales                 =  sh. 270,000

B1

 

 

M1

A1

B1

4

 

 

Both

 

 

Expression

Extra sales

Ans

 

 

 

 

 

 

13 .

 

 

 

 

 

Time          =   22/7 x 3.5/2x 3.5/2 x 200   hrs

22/7x 140x140x 140x 3600

 

8960

3600

= 2 hrs 29min

 

 

 

 

 

M1

M1

 

M1

 

A1

4

 

 

 

 

 

 

Vol tank

Vol tank

 

Div x 3600

 

Tank

 

 

14.

 

 

 

 

 

 

 

    Ö3                      =     Ö3           Ö7 + Ö2

Ö7Ö2                         Ö2Ö2         Ö7+ Ö2

 

= Ö3 Ö7 + Ö2

5

 

= Ö21 + Ö6

5

M1

 

M1

 

A1

3

Multi

 

Expression

 

 

 

Ans

15.           3 £ x 2                   x2 £ 35

±1.732 £x                 x £ ± 5.916

1.732 £ x           £ 5.916

integral x : 2, 3, 4, 5

 

B1

B1

B1

B1

4

Lower limit

Upper limit

Range

Integral values

 

16.  No of days   =  8/6 x 5/8  x 12

=   10 days

M1

A1

2

Expression

days

17. (i)  ÐCED      =  ÐECD   = 30

Ð CDE     =  180 – 60

=  120

Ð CBE    =  180-120

=60

(ii) Ð AEC  = 90+30

= 120

Ð EAB  = 180-(120+45)

= 150

(iii) ÐBEO  = 90-45

= 45

B1

B1

B1

B1

 

B1

 

B1

B1

 

B1

8

 

 

 

 

 

 

 

ÐA EB = 450

 

ÐBEO

18.   .ar + ar2    =  9/4

3r + 3r2   =  9/4

12r2  + 12r – 9 = 0

4r2  + 3r – 3   = 0

4r2 + 6r – 2r –3 = 0

(2r – 1) (2r + 3)  = 0

r  = ½  or r   = -11/2

 

Ss      = 3(1- (1/2 )5)

1 – ½

 

= 3 (1-12/3 2)

½

= 6 ( 31/32)

= 6 31/32

 

B1

B1

 

B1

 

M1

A1

 

M1

 

 

 

M1

 

 

A1

8

 

 
19.

LOG  E.    0.3010   0.6021     0.7782     0.9031

LOG  F      1.2068   2.1065     2.6354     3.0103

 

Log E =n log F  + Log K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.n  = gradient    = 2        2.4 – 1.4   =  12  =  3

Log k.             =  0.3       0.7 – 0.3       4

.k              = 1.995

¾ 2

‹         E     =  2F 3

B1

B1

 

 

S1

 

 

P1

 

 

L1

 

 

M1

A1

 

B1

8

 

Log E

Log F

 

 

Scale

 

 

Plotting

 

 

Line

 

 

Gradient

 

 

K

 

 

 

20  

.x       -2     -1     0    1     2    3      4

.y      17      6      1    6     9  22     41

 

.y  =  3x 2  – 2x + 1    –

0       =  3x 2 – 3x – 2

y   =  x     +  3

 

 

 

 

 

 

 

B2

 

B1

 

B1

 

S1

P1

C1

 

L1

 

B1

 

8

 

 

 

All values

 

At least  5

 

Line

 

Scale

Plotting

Smooth curve

 

Line drawn

 

Value of r

 

 

21. .h          = ¾ p x 18 x 18x 18

p x 0.04 x 0.04

= 24 x 18x 18x 18

0.04   x 0.04 x 100

 

=  48,600m

 

density  = 4/3 x 22/7 x 18 x 18x 18x 15 kg

1000

= 122.2kg

M1

M1

M1

M1

 

A1

 

M1

M1

A1

8

N of wire

¸ to length in cm

¸ for length

conversing to metres

 

length

 

expression for density

conversion to kg

ans

 

 

22.  

H = Ö152 – 92

= Ö144

= 12

 

X/6  = 9/12

X    = 4.5

Volume   = 1/3 x 22/7x (81 x 12 –20.25×6 )

 

= 22/21  (972 – 121 -5)

 

=   891  cm3

 

 

M1

 

 

A1

 

M1

A1

M1

M1

M1

 

A1

8

Method

 

 

 

 

Method

Radius

Small vd

Large vol

Subtraction of vol.

 

Ans

23. R(-a , b) , Q (c,d), S(x , y) ,P (5,0)

PR is  diagonal

(a)    Mid point  PR  (0,0)

a + 5    = 0

2

.a         =   -5

b- 0     =   0

2
b = 0

R (-5,0)

Grad  PQ   = -2

Grad RS   = -2

.d – 0   =  -2

c –5

.d – 0      = ½

c+5

.d+ 2c     = 10

2d – c     = 5×2         –

4d – 2c   = 10

5d         = 20

d         = 4

c         = 3

Q (3, 4)

x + 3  ,    y+4   =  (0,0)

2           2

x  =  -3 , y = -4   \ s(-3 -4)

 

(b) y – 4   =   8

x – 3        6

3y  = 8x – 12

 

 

 

 

 

 

 

B1

 

 

 

M1

 

 

 

 

M1

 

 

 

 

A1

 

M1

A1

 

M1

 

A1

8

 

 

 

 

 

 

 

Ans .

 

 

 

Expression both correct

 

 

 

Equation

 

 

 

 

Ans

 

 

 

 

Expression

 

Equation

 

       

MATHEMATICS IV

PART II

MARKING SCHEME

 

 

1.                784 X 27        =

187500

Ö 784 x 9           =    4 x 7x 3

62500                      250

=       42

125

=       0.336

 

 

 

M1

 

M1

 

 

A1

 

 

 

Factors for

Fraction or equivalent

 

C.A.O

    3  
2.      Father 3x ,  r son  = x

2(x +10)        = 3x + 10

2x +20       =  3x + 10

x        = 10

father            = 30

M1

 

 

A1

B1

 

Expression

 

 

 

 

 

 

    3  
3. 3   = sin   60

AE

AE  = 3

Sin 60

= 3.464

perimeter  = 5×2 + 3.464 x 3

= 10+10.393

= 20.39

M1

 

 

 

A1

 

 

B1

Side of a triangle

 

 

 

 

 

 

Perimeter

    3  
4.    .a3 – b-2c2  =  (-2)3 – 3 –2(-1)2

2b2 – 3a2c      2(3)2 –3(-2)2(-1)

= -8 –3-2

18 + 12

= -13

30

M1

 

 

M1

 

A1

Substitution

 

 

Signs

 

C.A.O

    3  
5.        Ksh  189,000          =   $ 189,000

75.6

= $ 2500

balance                    = $ 2500

=  Kshs. 189,000

Kshs. 189,000          =             189,000

115.8

Uk    ₤1632

M1

 

A1

 

M1

A1

 

A1

4

 

Conversion

 

 

 

Conversion

 

6. Area of 2 triangles  =   2 (½ x 8x 5 sin 60)

=   40 sin 60

=   40x 0.8660

= 34.64 cm2

Area of rectangle    = 300 x 8 + 300 x 5 +300 x BC

BC              = Ö64 +25 – 2 x 40cos 60

= Ö89 – 80 x 0.5

= Ö89 – 40

= Ö49

= 7

Total   S.A.              = 300 (8+5+7) + 34.64 cm2

= 6000 + 34.64

= 6034.64 cm2

M1

 

 

 

 

M1

 

 

 

 

M1

 

A1

Areas of D

 

 

 

 

B.C. expression

 

 

 

 

Area

 

    4  
7.    AF2    = 32+42+-2+12x cos 50

= 25 – 24 x 0.6428

= 25-15.43

= 9.57

AF      =  3.094 x 50

AF      =  154.7m

Sin Q  =  200 sin 50o

154.7

= 0.9904

Q   = 82.040

Bearing = 117.96

M1

 

 

 

 

A1

M1

 

 

A1

B1

 

 

 

 

 

 

 

 

 

 

 

Bearing

    5  
8. (i)  No. of white  = w

w       = 2

w+9         3

3w       = 2w + 18

w      =  18

(ii)  p(different colour )  = p(WB N  BW)

= 2   x   9   + 918

3      25     27    25

= 12/25

M1

 

 

 

 

A1

M1

 

A1

 
    4  
9. A.sf                =  1

49

smaller area       = 1 x 441 p

49

=  9p

pr2          = 9p

r2         =  9

r           = 3

 

 

 

M1

 

M1

 

 

A1

 

 
    3  
10.  Largest area         = 22 x (14.5)2

7

=  660.8 cm 2

smallest area          =  22/7 x (13.5)2

= 572.8

572.8    £ A  £ 660.81

M1

 

 

M1

 

A1

 
    3  
11. (1 +2 x)5  =  1 + 5 (2x) + 10 (2x)2 + 10 (2x)3

=  1 + 10x   + 40x2  + 80x3

2.0455    =   1+2 (0.52)5

= 1+10 (0.52)+ 40(0.52)2+80(0.52)3

= 1+5.2 + 10.82 + 11.25

= 28.27

M1

A1

 

M1

 

A1

 
    4  
12.          Tn           =  5x 2n –2

(i)               T1 , T2, T3 = 2.5, 5, 10

(ii)                      S5      =  2.5(25-1)

2-1

= 2.5 (31)

= 77.5

 

B1

M1

 

 

A1

 

All terms

 

    3  
13. 12         = 22 x 3

18         = 2 x 32

30         = 2x3x5

Lcm         = 22 x 32x 5 = 180 min

=  3hrs

time they ring together =11.55 +3 = 2.55 p.m

M1

 

 

 

A1

B1

 
    3  
14.  Map area      = 40cm 2

Actual area   =  200x200x40m2

= 200x200x40ha

100×100

= 320ha

M1

M1

 

 

A1

Area in m2

Area in ha

 

 

CAO

    3  
15.     3p + 2r    = 13

p + 2r    =   9  –

2p           =   4

p     = sh 2

r     = 3.50

M1

 

 

A1

B1

 
    3  
16. 110 + 100+130+2x +3x = 540

5x  = 200

x  = 400

2x , 3x     = 80 and 1200 res

M1

 

A A1

2

 
17. Contribution / person    = 180,000

X

New contribution    = 180,000

x – 2

180,000   – 180,000  = 24,000

x –2               x

180,000x – 180,000x +360,000 = 24,000(x-2)x

24,000x2  –  48,000x – 360,000 =0

x2  – 2x – 15 = 0

x2 – 5x + 3x – 15 = 0

x (x – 5)+ 3 (x – 5) = 0

(x + 3 )(x – 5)  = 0

x     = -3

or     = 5

remaining members            = 5-2

= 3

B1

 

B1

 

M1

M1

 

 

A1

M1

 

 

A1

 

B1

 

‘C’

 

 

 

eqn

mult

 

 

eqn

factor

 

 

both ans

 

remaining members

    8  
18. (a) P (3 white)         =  8   x  7  x   28

13      12     11    143

(b) P(at least 2 blue)=p(WBBorBBWorBWB)orBBB

= 8  x   5  x   4   +  5  x   4  x  8

13     12     11      13     12    11

+ 5  x   8  x   4 +   8 x   7 x   6

13     12     11    13     12    11

= 204

429

= 68

143

(c) p(2 white and one blue )= p(WWB or WBW or BWW)

= 8  x  7  x  5  +  8  x  57  +  587

13     12    11   13     12   11   13    12   11

= 3 x 8 x 7 x 5

13 x 12 x 11

 

=  70

143

M1

A1

 

 

M1

 

M1

 

 

 

A1

 

 

 

M1

M1

 

 

 

A1

 

 
    8  
19. (a) recourt area    =  10.5 x 6  m2

title  area       =    0.3 x 0.3 m2

No of tiles     =    10.5 x 6

0.3 x 0.3

=  700

(b) No of cartons = 700 x 15

20

= 52.5

 

(c) Cost of 525 cartons  =   525 x 100 + 800 x 525

+ transport                        5

=  10,500+420,000

=   430,500

sale price                  =  120 x 4.30,500

100

=  sh    516,600

s.p of a carton            =  516,600

525

= sh. 984

 

 

M1

A1

 

M1

 

A1

 

 

 

B1

 

M1

 

 

M1

 

A1

 

 

 
    8  
20. (a) Maina`s tax dues       = 1800 x 10

100

=        180

(b) Taxable income        = 3600 x 115 – n rent

100

= 36 x 115 – 100 x 12

20

= 4140 – 60

=         4080

Tax dues                         = 10    x 2100  + 15  x 1980

100                 100

= 210 + 297

=        507

Tax  relief                      =        270-

Tax  paid                        =        237

M1

 

A1

 

 

M1

 

 

A1

M1

M1

 

A1

 

B1

 

 

 

 

 

 

 

 

 

 

1st slab

2nd slab

    8  
21.  (a)            PQ                 =  –3/5 a   +  3/1b

=  31/23/5 a

(b)             OR                 =   h a + h b

=   a – ha + hb

=  (1-h) a + h b

(c)              OR                =  3/5 a   + k (31/2 b – 3/5a)

=  (3/53/5k)a +3k b

(d)                      1 – h     =  3/53/5k    (i)

3k    =  h                   (ii)

Sub (i)              1 – 3k    =  3/53/5k

5- 15k    =  3-3k

12k    =  2

k    =   1/6

h     =  ½

 

 

B1

 

M1

A1

M1

A1

 

 

M1

 

 

A1

B1

 
     

8

 
 

22.

 

P(ABC) =     0  – 1      1  4  3      =  -3  -1  -3

1    0      3  1  3            1   4   3

A1 (-3,1)B1 (-1,4)C1(-3,3)

Q(A1B1C1) =  a  b    -3 –1 -3    =        -6 –2 –6

c d       1   4  3                2   8   6

 

=> -3a + b =  -6                -3c + d = 2

-a + 4b   =  -2 x 3         -c + 4d = 8 x 3

– 3a  + 12b = -6              – 3c + 12d = 24

11b  = 0                     -11d  = -22

b = 0                           d = 2

a = 2                           d = 2

c  = 0

Q =    2     0

0       2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M1

A1

 

 

M1

 

 

M1

 

 

 

 

 

A1

 

 

 

B1

 

 

 

 

B1

 

 

 

B1

 

 

A1 B1 C1

 

 

 

 

 

 

 

 

 

 

 

L Q

 

 

 

A1 B1 C1 drawn

 

 

 

 

All BII CII

Ploted

 

 

 

 

Destruction

 

 

 

    8  
23.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

24.

R     = 2.2CM ± 0.1

Area = 22 x  2.2 x 2-2

7

= 15.21cm2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ef =40                        efd = -80

(ii) model class    = 351- 360

modern class  = 341 – 350

(iii) mean             = 355.5  – 80

40

=  355.5 – 2

=  353.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B1

 

B1

 

B1

 

B1

 

B1

 

B1

 

M1

 

 

 

1

1

 

 

8

 

B1

B1

 

M1

 

 

A1

 

B1

 

B1

B1

B1

 

 
    A1  
    8  

 

 

 

MATHEMATICS V

PART I

 

SECTION 1 (52 MARKS)

 

 

 

  1. Use logarithms to evaluate 6 Cos 40   0.25
    63.4                                                                                                                                                                                                       (4mks)
  2. Solve for x in the equation (x + 3) 2 – 5 (x + 3) = 0 (2mks)
  3. In the triangle ABC, AB = C cm. AC = bcm. ÐBAD = 30o and ÐACD = 25o. Express BC in terms of b and c.                                                                                                     (3mks)
  4. Find the equation of the normal to the curve y = 5 + 3x – x3 when x = 2 in the form
    ay + bx = c                                                                                                             (4mks)
  5. Quantity P is partly constant and partly varies inversely as the square of q. q= 10 and p = 5 ½  when q =20. Write down the law relating p and q hence find p when qs is 5.            (4mks)
  6. Solve the simultaneous equation below in the domain 0  £ x £  360 and O£  y £ 360
    2 Sin x + Cos y = 3
    3 Sin x – 2 Cos y = 1                                                                (4mks)
  7. Express as single factor 2     –     x + 2         +       1
    x + 2    x2 + 3x + 2         x + 1                                       (3mks)
  8. By use of binomial theorem, expand (2 – ½ x )5 up to the third term, hence evaluate (1.96)5
    correct to 4 sf.                                                                                                        (4mks)
  9. Points A(1,4) and B (3,0) form the diameter of a circle. Determine the equation of the circle and write it in the form ay2 + bx2 + cy + dy = p where a, b, c, d and p are constants.                                                                                                                              (4mks)
  10. The third term of a GP is 2 and the sixth term is 16. Find the sum of the first 5 terms of the GP. (4mks)
  11. Make T the subject of the formulae 1       –  3m   +  2
    T2         R         N                        (3mks)
  12. Vectors, a =   2     b =   2   and   c –   6
    2              0                   4
  13. By expressing a in terms of b and c show that the three vectors are linearly dependent.                                                                                                                              (3mks)
    A cylindrical tank of base radius 2.1 m and height is a quarter full. Water starts flowing into this tank at 8.30 a.m at the rate of 0.5 litres per second. When will the tank fill up? (3mks)
  14. A piece of wood of volume 90cm3 weighs 54g. Calculate the mass in kilograms of 1.2 m3 of the wood.      (2mks)
  15. The value of a plot is now Sh 200,000. It has been appreciating at 10% p.a. Find its value 4 years ago.
    (3mks)
  16. 12 men working 8 hours a day take 10 days to pack 25 cartons. For how many hours should 8 men be

working in a day to pack 20 cartons in 18 days?                                                     (2mks)

SECTION II (48MARKS)

  1. The tax slab given below was applicable in Kenya in 1990.
    Income in p.a.                           rate in sh
    1  – 1980                                  2
    1981 – 3960                              3
    3961 – 5940                              5
    5941 – 7920                              7
    Maina earns Sh. 8100 per month and a house allowance of Sh. 2400. He is entitled to a tax relief of Sh.

800 p.m. He pays service charge of Sh 150 and contributes Sh 730 to welfare. Calculate Mwangis net

salary per month.                                                                                                    (8mks)

  1. OAB is a triangle with OA = a , OB = b. R is a point of AB. 2AR = RB. P is on OB such that
    3OP = 2PB. OR and AP intersect at Y, OY = m OR and AY = nAP. Where m and n are scalars.    Express in terms of a and b.
    (i) OR                                                                                                                    (1mk)
    (ii)AP                                                                                                                    (1mk)

    (b) Find the ratio in which  Y divides AP                                                                (6mks)

  2. The table below gives related values of x and y for the equation y = axn where a and n are constants
X 0.5 1 2 3   10
Y 2 8 32   200 800

By plotting a suitable straight line graph on the graph provided, determine the values of a and n.

20.       Chalk box x has 2 red and 3 blue chalk pieces. Box Y has same number of red and blue

pieces. A teacher picks 2 pieces from each box. What is the probability that
(a)        They are of  the same colour.                                                                            (4mks)
(b)        At least one is blue                                                                                           (2mks)
(c)        At most 2 are red                                                                                              (2mks)

21.  Point P(50oN, 10oW) are on the earth’s surface. A plane flies from P due east on a parallel of

latitude for 6 hours at 300 knots to port Q.
(a) Determine the position of Q to the nearest degree.                                                    (3mks)
(b)  If the time at Q when the plane lands is 11.20am what time is it in P.                      (2mks)
(c) The plane leaves Q at the same speed and flies due north for 9 hours along a longitude to

airport R. Determine the position of R.                                                                       (3mks)
22.       Using a ruler a pair of compasses only, construct :
(a)        Triangle ABC in which AB = 6cm, AC = 4cm and Ð ABC = 37.5o.                                (3mks)
(b)        Construct a circle which passes through C and has line AB as tangent to the circle at A.             (3mks)
(c)        One side of AB opposite to C, construct the locus of point P such that  ÐAPB = 90o.              (2mks)
23.       A particle moves in a straight line and its distance is given by S = 10t2 – t3 + 8t where S is

distance in metres at time t in seconds.
Calculate:
(i) Maximum velocity of the motion.                                                                             (4mks)
(ii) The acceleration when t = 3 sec.                                                                              (2mks)
(iii) The time when acceleration is zero.                                                                                   (2mks)

 

 

 

  1. A rectangle ABCD has vertices A(1,1) B(3,1), C(3,2) and D(1,2). Under transformation

matrix M =   2  2   ABCD is mapped onto A1B1C1D1

1   3
under transformation M =   -1  0    A1B1C1D1 is mapped onto  A11B11C11D11. Draw on the given grid
0 –2

(a)       ABCD, A1B1C1D1 and A11B11C11D11                                                                  (4mks)
(b)        If area of ABCD is 8 square units, find area of A11B11C11D11.                              (3mks)
(c)        What single transformation matrix maps A11B11C11D11 onto A1B1C1D1               (1mk)

MATHEMATICS V

PART II

 

SECTION 1 (52 Marks)

 

  1. Evaluate without using mathematical tables (2.744 x 15 5/8)1/3                              (3mks)
  2. If 4 £ x £ 10 and 6 £ y £5, calculate the difference between highest and least
    (i) xy                                                                                                                    (2mks)
    (ii)  y/x                                                                                                                     (2mks)
  3. A 0.21 m pendulum bob swings in such a way that it is 4cm higher at the top of the swing than at the bottom. Find the length of the arc it forms.       (4mks)
  4. Matrix 1        2x   has on inverse, determine x                                                     (3mks)
    x +3      x2
  5. The school globe has radius of 28cm. An insect crawls along a latitude towards the east from A(50o, 155oE) to a point B 8cm away. Determine the position of B to the nearest degree.                                                                                                                                                 (4mks)
  6. The diagonals of triangle ABCD intersect at M. AM = BM and CM = DM. Prove that triangles ABM and CDM are Similar.       (3mks)
  7. Given that tan x = 5/12, find the value of 1  –   sinx
                                                                         Sin x + 2Cos x,   for 0 £ x £ 90           (3mks)

 

  1. Estimate by MID ORDINATE rule the area bounded by the curve y = x2 + 2, the x axis and the lines x = O and x = 5 taking intervals of 1 unit in the x. (3mks)
  2. MTX is tangent to the circle at T. AT is parallel to BC. Ð MTC = 55o and Ð XTA = 62o. Calculate Ð (3mks)
  3. Clothing index for the years 1994 to 1998 is given below.
Year 1994 1995 1996 1997 1998
Index 125 150 175 185 200

Calculate clothing index using 1995 as base year.                                                          (4mks)

  1. A2 digit number is such that the tens digit exceeds the unit by two . If the digits are reversed, the number formed is smaller than the original by 18. Find the original number. (4mks)
  2. Without using logarithm tables, evaluate log5 (2x-1) –2 + log5 4 = log5 20             (3mks)
  3. Mumia’s sugar costs Sh 52 per kg while imported sugar costs Sh. 40 per kg. In what ratio should I mix the sugar, so that a kilogram sold at Sh. 49.50 gives a profit of 10%. (4mks)
  4. The interior angles of a regular polygon are each 172o. Find the number of sides y lie polygon.                                                                                                                            (2mks)
  5. Evaluate 2x   =       2    +        3
    341       9.222                                                                           (2mks)
  6. A water current of 20 knots is flowing towards 060o. A ship captain from port A intends to go to port

B   at a final speed of 40 knots. If to achieve his own aim, he has to steer his ship at a course of 350o.

Find the bearing of A from B.                                                                                (3mks)

SECTION II  (48 MARKS)

  1. 3 taps, A, B and C can each fill a tank in 50 hrs, 25 hours and 20 hours respectively. The three taps are turned on at 7.30 a.m when the tank is empty for 6 hrs then C is turned off. Tap A is turned off after four hours and 10 minutes, later. When will tap B fill the tank? (8mks)
  2. In the domain –5 £ x £ 4, draw the graph of y = x2 + x – 8. On the same axis, draw the graph of y + 2x = -2. Write down the values of x where the two graphs intersect. Write down an equation in x whose roots are the points of intersection of the above graphs. Use your graph to solve. 2x2 + 3x – 6 = 0.                                                                                            (8mks)
  3. The average weight of school girls was tabulated as below:
Weight in Kg 30 – 34 35 – 39 40 – 44 45 – 49 50 – 54 55-59 60-64
No. of Girls 4 10 8 11 8 6 3

(a) State the modal class.                                                                                           (1mk)
(b) Using an assumed mean of 47,
(i) Estimate the mean weight                                                                                (3mks)
(ii) Calculate the standard deviation.                                                                      (4mks)

 

 

 

 

 

 

 

 

 

 

  1. The table below shows values of y = a Cos (x – 15) and y = b sin (x + 30)
X 0 15 30 45 60 75 90 105 120 135 150
a Cos(x-5) 0.97       0.71 0.5       -0.5 -0.71
b sin(x+3) 1.00       2.00       1.00   0.00

(a) Determine the values of a and b                                                                               (2mks)
(b) Complete the table                                                                                                  (2mks)
(c) On the same axes draw the graphs of y = across(x – 15) and y = b sin(x + 30)            (3mks)
(d) Use your graph to solve ½ cos (x – 15) = sin(x + 30)                                                 (1mk)

21.    The diagram below is a clothing workshop. Ð ECJ = 30o AD, BC, HE, GF are vertical

walls. ABHG is horizontal floor. AB = 50m, BH = 20m,  AD=3m

 

 

 

(a) Calculate DE                                                                                                           (3mks)
(b) The angle line BF makes with plane ABHG                                                              (2mks)
(c) If one person requires minimum 6m3 of air, how many people can fit in the workshop         (3mks)

  1. To transport 100 people and 3500 kg to a wedding a company has type A vehicles which take          10 people and 200kg each and type B which take 6 people and 300kg each. They must not use more

than 16 vehicles all together.
(a)     Write down 3 inequalities in A and B which are the number of vehicles used and plot them

in a graph.                                                                                                           (3mks)
(b)     What is the smallest number of vehicles he could use.                                          (2mks)

(c)     Hire charge for type A is Sh.1000 while hire for type B is Sh.1200 per vehicle. Find the cheapest

hire charge for the whole function                                                                        (3mks)

A circle centre A has radius 8cm and circle centre B has radius 3cm. The two centres are

12cm apart. A thin  tight string is tied all round the circles to form interior common tangent. The tangents CD and EF intersect at X.

(a) Calculate AX                                                                                                           (2mks)
(b) Calculate the length of the string which goes all round the circles and forms the tangent.
(6mks)

 

  1. Airport A is 600km away form airport B and on a bearing of 330o. Wind is blowing at a speed of

40km/h from 200o. A pilot navigates his plane at an air speed of 200km/h from B to A.
(a)     Calculate the actual speed of the plane.                                                                (3mks)
(b)     What course does the pilot take to reach B?                                                          (3mks)
(c)     How long does the whole journey take?                                                                (2mks)

 

MATHEMATICS V

PART I

MARKING SCHEME

 

1 SOLUTION MKS AWARDING
  No         Log

13.6        1.1335   +

Cos 40    1.8842

1.0177   –

63.4       1.8021

1. 2156

(4 + 3.2156) 1/4

1.8039

Antilog    0.6366

 

B1

 

M1

 

 

M1

 

A1

 

Log

 

+

 

 

divide by 4

 

C.A.O

    4  
2. (x + 3) (x + 3 – 5) = 0

(x +3)b (x – 2) = 0

x = -3 or x = 2

M1

 

A1

 

Factors

 

Both answers

3 BD = C Sin 30  = 0.05

CD = b Cos 25

= 0.9063b

‹ BC = 0.9063b + 0.5 C

B1

 

B1

B1

 

BD in ratio from

 

CD in ratio form

Addition

    3  
4  Dy  = 3 – 3x2
dx
x = 2, grad = 1
9
Point (2,3)
y – 3  = 1
x – 2     9

9y – 27  = x – 2
9y – x   =  25

B1

 

B1

 

M1

 

 

A1

 

Grad equ

 

Grad of normal

 

Eqn

 

 

Eqn

 

    4  
5   700 = 100 + n
2200 = 400 + n

1500 = 300m

m = 5

n = 200

P = 5 + 200
q2
When q = 5 P = 13

M1

 

 

A1

 

 

B1

B2

Equan

 

 

Both ans

 

 

Eqn (law)

Ans (P)

    4  
 

6

 

4 Sin x + 2 cos y = 6

3 Sin x – 2 Cos y = 1
7 sin x                  = 7

Sin x            = 1

X                = 90

Cos y          = 1

Y        = 0o

 

M1

M1

 

 

A1

 

B1

 

Elim

Sub

 

 

 

 

 

7 2(x +1) – 1(x + 2) + x + 2

(x+2) (x +1)
= 2x +2 – x – 2 + x = 2

(x +2) (x + 1)

=     2x + 2

(x + 2)  (x + 1)

=     2
x + 2

M1

M1

 

 

 

A1

Use of ccm

Substitution

 

 

 

Ans

8 (-2 – ½ x)5  = 25  – 5 (2)4 ( ½ x) + 10(2)3( ½ x)2

=  32 – 40x + 20x2

= 32 – 4 (0.08) + 20 (0.08)2

= 32 – 0.32 + 0.128
= 3

M1

A1

 

M1

A1

 

 

 

 

 

    4  
9. Circle centre C = (3 +1,   0 + 4)

2                 2

C( 2, 2)

R =Ö (2 – 0)2 + (2 – 3)2

=Ö 5

(y – 2)2 + (x – 2)2 = Ö5

y2 + x2 – 4y – 4x =  8 + Ö5

B1

 

B1

 

M1

 

A1

Centre

 

Radius

 

 

 

 

    4  
10  ar2 =2,  ar5 = 16

a  = 2  \ 2 r5 = 16

r2       r2

2r3 = 16

r3 = 8

r = 2, a = ½

 

S5= ½ (1 – ( ½ )5)

½

= 1 – 1/32

= 31/32

M1

 

 

 

 

A1

 

M1

 

 

A1

 

 

 

 

 

Both

 

Sub

 

 

CAO

    4  
11 NR – 3MT2  = 2RT2

T2(2R + 3M) = NR

T2   =   NR

2R + 3m

T =  ! Ö  NR
2R + 3m

M1

 

M1

 

A1

X mult

 

72

 

ans

    3  
12  2  = m   2   + n    6

2            0           4

2 = 2m + 6n

2 = 0 + 4n

n = ½

m = – ½

\a = – ½ b + ½ c

\a b c are linearly dep

M1

 

 

 

 

A1

 

B1

 
    3  
13 Volume = 22 x 2.1 x 2.1 x 2 x ¾ m3

7

Time = 11 x 0.3 x 2.1 x 3 x 1,000,000

500 x 3600

= 11.55

= 11.33 hrs

time to fill = 8.03 pm

M1

 

 

M1

 

 

 

A1

 
    3  
14 Mass = 54   x  1.2 x 1,000,000

90              1000

= 720kg

M1

 

A1

 
    2  
15 V3 = P

P(0.9)3     = 200,000

P = 200,000

0.93

= 200,000

0.729

= Sh 274,348

M1

 

M1

 

 

 

A3

 
    3  
16 No of hours = 8 x 12 x10 x 20

8 x 18 x 25

= 19200

3600

= 5hrs, 20 min

M1

 

 

 

A1

 
    2  
17  Taxable income = 8100 + 2400

= sh. 10,500

=   ₤6300

Tax dues      = Sh 1980 x 2 + 1980 x 3 + 1980 x 5 + 3670 x 7

12

= 22320

12

= Sh 1860

net tax = 1860 – 800 p.m.

= Sh 1060

Total deduction = 1060 + 150 + 730

= 1940

Net salary = 10,500 – 1940

= Sh 8560 p.m.

B1

 

 

M1

M1

 

A1

 

B1

 

B1

 

M1

A1

Tax inc

 

 

2

2

 

 

 

net tax

 

total dedu.

    8  
18 OR = 2/3 a + 1/3b or (1/3 (2a + b)

AP = 2/5 b – a

OY = m OR = A + n (2/5b – a)

2/5m b + ma = (1 – n)a + 2/5 n b

2/5m = 2/5n
m = n

\m = 1 – m

2m = 1

m  = ½ = n

½ AP = Ay

AY:AP = 1:1

B1

B1

 

B1

M1

M1

A1

A1

 

 

B1

 

 

 

EXP, OY

Eqn

M = n

Sub

CAO

 

 

Ratio

    8  
 

19

 

 

 

 

Log y = n log x + log a

Log a = 0.9031

A = 8

Grad = 1.75 – 0.5

0.4 + 0.2

= 1.25
0.6

= 2.08

n = 2

\y = 8x2

x = 3  y = 8 x 32   = 72

y = 200           x = 5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B1

B1

 

 

 

B1

 

B1

B1

S1

P1

L1

 

 

 

 

Log x

Log y

 

 

 

A

 

N

Missing x and y

Scale

Points

Line

    8  
 

 

 

20

 

 

 

P (same colour) = P (XRRrr orXBB or YXX or YBB)

= ½ (2/5 x ¼ + 3/5 x 2/4)  x 2

2  +  6
20     20

=    8
20

2/5

(b) P(at least 1B) = 1 – P(non blue)

= 1 – P (XRR or YRR)

= 1 – ½ (2/5 x ¼) x 2

= 1 – 1/10

= 9/10

(c) P(at most 2 Red) = 1 – P (BB)

= 1 – ½ (3/5 x 2/4)2

= 1 – 6/20

= 14/20 or 7/10

 

 

 

M1

M1

 

M1

 

A1

 

 

M1

 

A1

M1

 

 

A1

 

 

 

Any 2

Any 2

 

Fraction

 

 

    8  
21 (a) PQ  = 1800nm

q     =     1800

60 x 0.6428

= 46.67

= 47o

Q (50oN, 37oE)

 

(b) Time diff = 47 x 4
60

= 3.08

Time at P = 9.12am

(c) QR = 2700 nm

x o   = 2700

60

= 45o

R (85oN, 133oW)

M1

 

 

 

A1

 

 

M1

 

A1

 

M1

 

 

A1

B1

 
    8  
 

 

22

   

 

 

 

B1

B1

 

B1

B1

B1

B1

B1

B1

 

 

 

 

 

Bisector of 150

Bisector 75

 

AB  AC

^ at A

Bisector AC

Circle

Ð AB

Locus P with A  B excluded

    8  
24                           A1B1 C1D1

2  2  1 3 3 1   =  4  8 10 6

1  3  1 1 2 2       4  6  9  7

 

A11 B11 C11  D11

-1   0     4  8 10  6       =   -4  –8   -10   -6

0 –2     4  6  9   7            -8   -12  -18  -14

 

NM =   -1  0        2  2

0 –2       1  3

 

=  -2  -2

-2   -6

 

 

(b)      det  = Asf  =  12 – 4    = 8

Area A11 B11 C11 D11  = 8 x 8

= 64  U2

(c) Single matrix = Inv N
= ½    -2 –  0

0       –1

 

=     -1     0

0       – ½

 

 

B1

 

 

B1

 

 

 

 

 

 

 

 

 

 

B1

M1

A1

 

 

 

 

B1

 

 

Product

 

 

Product

 

 

 

 

 

 

 

 

 

 

Det

 

 

 

 

 

 

Inverse

    6  
23  

Ds  = 20t  – 3t2 + 8 =0

Dt     3t2 – 20t – 8 = 0

T =  20 !  Ö400 + 4 x 3 x 8

6

t = 7.045 sec

max vel          = 148.9 – 140.9 – 8

= 0.9 m/s


d2 s
  = 6t – 20

dt2

when t = 3   a = -2m/s2

6t – 20 = 0

6t  = 20

t = 3 2/3 sec

 

 

M1

 

A1

M1

A1

M1

 

A1

M1

 

A1

 
    8  
       

 

 

 

 

 

 

 

 

MATHEMATICS V

PART II

MARKING SCHEME

 

No Solution Mks Awarding
1  2744 x 125   1/3

1000            8

 

2744  1/3  x   53     1/3

1000            23

 

23 x 73  1/3  x   5

103                         2

 

2 x 75   = 3.5

10      2

 

 

 

M1

 

 

 

 

M1

A1

 

 

 

Factor

 

 

 

 

Cube root

 

    3  
2 (i) Highest – 10 x 7.5 = 75

Lowest  – 6 x 4 =  24

51

(ii) Highest = 7.5 = 1.875

4

Lowest = 6   = 0.600

10   1.275

M1

A1

 

M1

 

A1

Highest

 

 

Fraction

 

 

    4  
3 Cos q  =  17  = 0.8095

21

 

q = Cos 0.8095

= 36.03o

 

Arc length = 72. 06 x 2 x 22 x 21

360                       7

= 26.422cm

M1

 

 

A1

 

 

M1

 

A1

 

 

 

q

    4  
4  x2 – 2x(x +3) = 0

x2 – 2x2 – 6x = 0

-x2 – 6x = 0

either x = 0

or  x = 6

M1

 

M1

 

A1

Equ

 

Factor

 

Both A

    3  
 

 

5

 

8  = x  x 2 x 22 x 28 Cos 60o

360            7

 

8 =  x    x 44 x 28 x 0.5

360         7

x =   8 x 360 x 7
        44 x 28 x 0.5

= 32.73o

= 33o

 

 

M1

 

 

 

 

M1

 

A1

B1

 

 

 

 

 

 

 

x exp

 

 

 

6

 

 

 

ÐDMC = Ð AMB vert. Opp = q

ÐMAB  = Ð MDC = 180 – q BASE Ls of an isosc. <

2
Ð MBA = Ð MAC   180 – q base angles of isos <

2

<’s AMC and < CDM are equiangle

 

\ Similar proved

 

 

 

 

B1

 

 

 

 

 

B1

 

B1

 
    3  
7 Tan x = 5/12

h = Ö b2 + 122

= Ö25 + 144

= Ö169

= 13

 

1 – Sinx               =       1 – 5

sin x + 2 Cos x      5/13 + 2 x 12/13

 

12/13      = 12 x 13  =  12

29/13          13   29      29

 

 

 

 

 

 

 

 

 

 

 

 

 

M1

M1

 

A1

 

 

 

 

 

 

 

 

 

 

 

 

 

Hypo

Sub

 

    3  
8 Y = x 2 + 2

 

 

 

 

 

Area = h (y1, = y2 +……..yn)

= 1(2.225 + 4.25 + 8.25 +14.25 + 22.25)

= 51.25 sq units

 

 

 

B1

 

 

M1

 

A1

 

 

 

Ordinals

    3  
 

9

ÐCBA = 117o

Ð ACD = 55

Ð BAC = 180 – (117 + 55) = 8o

 

 

 

 

 

 

 

 

 

 

 

 

 

B1

B1

B1

 

3

 
10  

 

 

 

B1

B1

B1

B1

1994

1996

1997

1998

    4  
11. Xy = 35

y = 35/x

9x – 9y = -18

Sub x2 + 2x – 35 = 0

x2 + 7x – 5x – 35 = 0

x (x + 7) – 5(x + 7) = 0

(x – 5) (x + 7) = 0

x  = -7

x = +5

y = 7

Smaller No.

= 57

= 75

B1

 

M1

 

 

 

 

A1

 

 

 

B1

 

 
    3  
12 Log5 (2x – 1 )4  = log552

20

4(2x – 1)  = 52

20

2x – 1 = 25

5

2x – 1 = 125

2x = 126

x = 63

M1

 

M1

 

 

 

 

 

A1

 
    3  
13 C.P = 100 x 49.50

110

= 45/-

52x + 40y = 45

x + y

45x + 45y  = 52x + 40

-7x  = -54

x/y  = 5/7

x : y = 5 : 7

 

 

B1

M1

 

 

M1

 

A1

 
    4  
14  

2n – 4 it angle = 172

n

(2n – 4) x 90 = 172n

n

90 (2n – 4) x 90 = 172

n

180 n – 360 = 172n

 

180n – 172n = 360

8n = 360

n = 45

 

M1

 

A1

 

M1

 
    2  
15 2 x = 2.    1    +    3.    1

6.341                  9.22

2x = 2 x 0. 1578 + 3 x 0.1085

= 0.3154 + 0.3254

= 0.6408

x = 0.3204

 

 

B1

 

 

A1

 

 

Tables

    2  
16 Bearing 140o

Sin q = 20 Sin 110

40

= 0.4698

= 228.02

Bearing of A from B = 198.42

 

M1

 

 

A1

B1

 
    3  
17 Points that each tap fills in one hour

 

A =  1   B  = 1       C – 1
          50         25            20

In one hour all taps can fill = 1  +  1   +  1   =  11

50    25      20     100

In 6hrs all can fill =  11  x 6 = 33 parts

100                 50

taps A and B can fill =  = 1  +  1  = 3 part in 1 hr

50    25    50

In 4 1 hrs, A and B =  25 x 3  +  1

6                           6     50     4

Parts remaining for B to fill = 1 – 33  +  = 1  – 91   = 9 parts

50         4           100    100

Time  taken =  9  x  25  hrs = 2 ¼ hrs

100          1

7.30 am

6.     hrs

13.30

  4.10

5.40pm

  2.15

  7.55 pm

 

 

 

M1

 

 

 

B1

 

 

 

 

 

B1

 

B1

 

 

 

 

 

M1

 

A1

 

 
 

 

18

 

 

 

 

 

 

 

 

 

x2 + x – 8 = -2 – 2x

y = x2 + 3x – 6

Points of intersection (-4, 1.4)

y = x2 + x – 8 = 2x2 + 3x – 6

x2 + 2x + 2

y = x2 + x – 8 x 2

2y = 2x2 + 2x – 16

0 = 2x2 + 3x – 6

2y = -x  – 10

y = – 2.6

Ny = 1.2

 

8

 

 

 

 

 

 

 

 

 

B1

B1

 

 

 

 

 

B1

 

B1

 

 

 

 

 

 

 

 

 

 

Eqn

Point of inter

 

 

 

 

 

Line eqn

 

Both

 

 

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)    Modal class = 45 – 49

(i)               Mean = 47 + -55

50

= 47 – 1.1

= 45.9

 

(ii) Standard deviation = Ö 3575 –  –55 2
50         50

=  Ö71.5   – 1.21

=Ö 70.29

= 8.3839

 

4

 

 

 

 

 

 

 

 

 

 

 

 

 

B1

B1

 

 

 

 

M1

 

 

A1

B1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fd

fd2

    8  
20  

 

 

 

 

 

 

 

 

(a)    a =   1
b = 2

½ cos (x – 15) = Sin (x + 30)

has no solution in the domain

 

 

 

 

 

 

 

 

B1

B1

B1

 

B1

 

 

 

 

 

 

 

 

All

All

A & b

 

 

    8  
21 (a)       O Cos 30 = 20

X

X =  20

0.866

= 23.09

 

DE = Ö 502   + 23.092

= Ö 2500 + 533.36

= Ö 3033.36

= 55.076m

 

(b)       GB =  Ö 202  + 502

= 53.85

Tan q = 14.55
53.85

=  0.27019

q    = 15.12o

 

 

 

 

B1

 

M1

 

 

 

A1

 

 

M1

 

 

A1

 
  8  
  (c)       Volume of air = 50 x 20 x 3 + ½ x 20 x 11.55 x 50

= 3000 + 5775

= 8775

No. of people  =   8775
                               6

= 1462.5

j 1462

 

M1

 

M1

 

 

A1

 
    8  
22 (a)    A + B [ 16

5A + 3B ³ 50

2A + 3B [ 35

 

 

(b)   14 vehicles

 

(c)    A – 6 vehicles

B –  8

Cost = 6 x 1000 + 8 x 1200

= 6000 + 9600

= 15,600/=

 

 

B1

 

 

B1

 

B1

 

M1

 

A1

 

 

 

In equation 3

 

 

Vehicles

    8  

23

 

 

 

 

 

 

 

 

 

 

 

 

 

x        =      8

12 – x           3

 

= 8.727

FBX =    3    =  0.9166   = 23.57
3.273

 

3FBX = 47.13

 

Reflex  Ð FBD = 312.87

 

Reflex arc FD = 312.87   x 22  x 6
360           7

 

= 16.39cm

Reflex Arc CE = 312.87 x 22 x 16
360         7

 

=  43.7cm

 

FE (tangent) =  Ö144 – 121

= Ö 23

= 4.796cm

2 FE            =  9.592

 

Total length = 9.592 + 4.796 + 43.7 + 16.39

= 74.48 cm2

 

 

 

 

 

 

 

 

 

M1

 

 

A1

 

 

 

 

 

 

 

M1

 

 

A1

 

M1

 

 

A1

 

 

 

 

 

 

M1

A1

 
    8  

24

 

 

 

 

 

 

 

 

 

 

 

 

(b)         200      =    40

Sin 50       Sin q

 

Sin q =  40Sin 50
                200
= 0.7660
5
=0.1532

q         = 8.81o

Ð ACB = 180 – (50 + 8.81)o

= 121.19o

    x             =   200
Sin 121.19     Sin 50

 

= 200 x Sin 121.19
Sin 50

= 200 x 0.855645
0.7660

= 223.36Km/h

 

(b)  Course = 330o – 8.81o

= 321.19o

 

(c) Time  =    600
321.19o

 

= 2.686 hrs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M1

 

 

 

 

 

 

A1

 

 

 

M1

 

 

M1

 

 

A1

 

B1

 

 

 

M1

 

A1

 

 

8

 

 

 

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