Instructions to candidates
- This paper consists of two sections: Section I and Section II.
- Answer all the questions in Section I and only five questions from Section II.
- Show all the steps in your calculations, giving your answers at each stage in the spaces provided below each question.
- Marks may be given for correct working even if the answer is wrong.
- Non – programmable silent electronic calculators and KNEC Mathematical Tables may be used, except where stated otherwise.
SECTION I (50 Marks)
Answer all the questions in this section
- The length and width of a rectangular sheet of paper measured to the nearest millimetre are 22.3 cm and 15.7 cm respectively. Calculate to four significant figures, the percentage error in area of the paper. (3 marks)
- Make K the subject of the formula
(3 marks) -
Draw a line PQ of length 7 cm. On one side of the line PQ, construct the locus of a point R such that the area of triangle PRQ is 10.52cm. On this locus locate two positions of R,R1 and R2 such that ∠PR1Q=∠PR2Q = 90°. (3 marks)
-
A right – angled triangle has the length of its shorter sides as (2x+4) and (8x+8). If the length of the hypotenuse is 10x. Find its perimeter. (4 marks)
-
A piece of wire 360 metres long is to be used to fence rectangular plot. One end of the plot has a wall already erected. Calculate the maximum possible area of the plot. (3 marks)
-
Simplify
giving your answer in the form
where a, b and c are rational numbers. (3 marks) -
- Expand
(1 mark) - Use the expansion in (a) up to the fourth term to evaluate (1.96)6 correct to four decimal places. (2 marks)
- Expand
- A trader sells two brands of coffee P and Q. The coffee is packed in sachets of same size. The shelves can only accommodate 1000 sachets. He requires at least 200 sachets of P and more than 600 sachets of Q. If he orders x sachets of P and y sachets of Q. Write down all the inequalities in terms of x and y which satisfy the above information. (3 marks)
- Mr Wangombe a rice trader in Mwea mixed 8000kg of Pishori rice with 12000kg of ordinary rice. A kilogram of Pishori rice cost him Ksh.150 while a kilogram of ordinary rice cost him Ksh.80. He packed the rice in 2kg packets. Determine the price at which he sold each packet in order to realize a profit of 25% if 10% of the rice were damaged by rodents. (3 marks)
- Onyango wants to buy a radio on hire purchase. The cash price of the radio is Ksh.28000. Onyango makes a down payment of Ksh.6000 followed by 16 monthly instalments of Ksh.2100 each. Calculate the rate of compound interest per month to four significant figures. (3 marks)
- In the figure below, line TQ is a tangent to the circle at point Q. Line TRXP is a secant to the circle and line SXQ is a chord in the circle. The chord SXQ and the secant TRXP intersect at point X. The lines TR = 12 cm, XP = 9 cm, SX = 7.5 cm and XQ = 7.2 cm.
Calculate:- The length of RX (1 mark)
- Length TQ to two decimal places. (2 marks)
- Towns A and B lie on the same latitude south of equator. When it is 1.00 p.m. at A, the time at B is 7.00 p.m. Given that the longitude of town A is 20°E, find:
- The longitude of Q. (2 marks)
- The latitude where A and B lie given that the length of arc AB along the parallel of latitude is 3600 nautical miles. (2 marks)
- In the figure below, O is the centre of the circle and A if joined to B, passes through the centre of the circle O.
- Determine the centre and the radius of the circle. (1 mark)
- Express the equation of the circle in the form x2+y2+ax+bx+c=0 where a, b and c are constants. (2 marks)
- Without using logarithm tables or a calculator, evaluate: (3 marks)
log10200 - 1/3 log10512 + 2 log1020 - Solve the equation, 3 cos2x - 1 = 2 cos x for -180°≤x≤180° (3 marks)
-
Given that O is the origin, OA = 2i + 2j - 4k and OB = 6i + 10j + 2k. If R divides AB externally in the ratio 3:1. Find OR. (3 marks)
SECTION II: (50 Marks)
Answer any five the questions in this section -
Mr. Kosgei a chief inspector in the police service earns a basic salary of Ksh.56 000, house allowance of Ksh.30 000, commuter allowance of Ksh.12 000 and risk allowance of Ksh.10 000. He has a life insurance policy for which he pays Ksh.6 000 per month and for which he is allowed 15% as insurance relief. He is also entitled to a personal tax relief of Ksh.1 162 per month.
Monthly income in
Kenya shillingsPercentage tax rate
in each shilling0 - 10164 10 10165 - 19740 15 19741 - 29316 20 29317 - 38892 25 38893 and above 30 - Using the tax table above:
- Determine Mr Kosgei’s monthly taxable income. (2 marks)
- Calculate net tax paid by Mr Kosgei per month. (4 marks)
- Mr Kosgei also had the following monthly deductions from his salary: Sacco loan repayment Ksh.1 000, WCPS 2% of basic salary, NHIF Ksh.1 500. Determine his net monthly salary. (2 marks)
- If Mr. Kosgei got an annual increment of 20% in his basic salary, determine the percentage increase in net tax paid per annum. (2 marks)
- Using the tax table above:
- Complete the table given below by filling in the blank spaces. (2 marks)
x 0 15 30 45 60 75 90 105 120 135 150 165 180 y = 4 cos 2x 4.00 2.00 0 -2.00 -4.00 -2.00 0 2.00 4.00 y = 2 sin(2x+30) 1.00 1.73 2.00 1.73 0 -1.00 -2.00 -1.73 0 1.00 - On the grid provided, draw on the same axes, the graph of y=4cos2x and y=2sin(2x+30) for 0°≤x≤180°. Take the scale, 1 cm for
15° on the x – axis and 1 cm for 1 unit on the y – axis. (5 marks) - From the graph:
- State the amplitude of y=4 cos 2x (1 mark)
- Find the period of y=2sin(2x+30) (1 mark)
- Use your graph to solve the equation 4 cos2x-2sin(2x+30)=0 (1 mark)
- Complete the table given below by filling in the blank spaces. (2 marks)
- A wedding committee consisting of three people is to be chosen from five men and seven women. Draw a tree diagram to represent the above information. (2 marks)
Using the tree diagram above, find the probability that:- All committee members are of the same gender. (2 marks)
- At least two of the committee members are men. (3 marks)
- A tetrahedron is biased such that the probability of a face showing up is given by P(t) = mt where m is a constant and t = 1, 2, 3 and 4 (number of the faces). Find the probability that
when the tetrahedron is tossed twice the sum of the faces that will show up is 7. (3 marks)
- A wedding committee consisting of three people is to be chosen from five men and seven women. Draw a tree diagram to represent the above information. (2 marks)
-
The figure below shows a tetrahedron PQRV. PQ = QR = RP = 12 cm and VP= VQ=VR=15 cm.
Calculate to one decimal place:- Height of the tetrahedron. (3 marks)
- Angle between VQ and the base PQR. (2 marks)
- Angle between VPQ and the base PQR. (2 marks)
- Angle between VPQ and VQR. (3 marks)
-
- A carpenter wishes to make a ladder with cross – pieces. The cross – pieces are to diminish uniformly in lengths from 63 cm at the bottom to 28 cm at the top. Calculate:
- The length in centimetres of the seventh cross – piece from the bottom. (3 marks)
- The length in centimetres of the fourth cross – piece from the top. (2 marks)
- The third, fifth and eighth terms of another Arithmetic Progression (A.P.) form the first three consecutive terms of a Geometric Progression (G.P.). If the common difference of the AP is 3, find:
- The first term of the Geometric Progression. (3 marks)
- The sum of the first eleven terms of the Geometric Progression. (2 marks)
- A carpenter wishes to make a ladder with cross – pieces. The cross – pieces are to diminish uniformly in lengths from 63 cm at the bottom to 28 cm at the top. Calculate:
- The data below shows the marks obtained by 50 students in a certain class.
Marks 25-34 35-44 45-54 55-64 65-74 75-84 85-95 No of students 3 6 16 12 8 4 1 - Using an assumed mean of 59.5, calculate:
- The mean (3 marks)
- The standard deviation of the distribution (3 marks
- Estimate:
- The lower quartile of the distribution. (2 marks)
- The pass mark if 34 students passed the exam. (2 marks)
- Using an assumed mean of 59.5, calculate:
-
- The speed V m/s of a moving particle is partly constant and partly varies as time t seconds. It is given that V = 28 m/s when t=2 and V = 53 m/s when t = 7 seconds.
Find the speed of the particle when t =11seconds. (4 marks) - A quantity R varies directly as T and inversely as the cube root of S. Given that S = 64 when T = 6 and R = 30;
- Find the formula connecting R, S and T. (3 marks)
- Find the percentage change in R when T is decreased by 10% and S increased by 25%. (3 marks)
- The speed V m/s of a moving particle is partly constant and partly varies as time t seconds. It is given that V = 28 m/s when t=2 and V = 53 m/s when t = 7 seconds.
- The vertices of a square PQRS are P(1,1),Q(1,3),R(3, 3) and S (3,1). The vertices of its image under a transformation T are P' (1 ,-2),Q' (1 ,-6),R' (3 ,-6) and S' (3 ,-2).
- On the grid provided, draw PQRS and its image P'Q'R'S' under T (2 marks)
- Determine the matrix of transformation represented by T. (3 marks)
- Find the vertices of P''Q''R''S'' the image of P'Q'R'S' under transformation represented by
On the same grid as in (a) above, draw P''Q''R''S''. (3 marks) - Describe fully the transformation U. (2 marks)
- On the grid provided, draw PQRS and its image P'Q'R'S' under T (2 marks)
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