## 2013 KCSE Maths A Past Paper

**MATHEMATICS ALT. A (121)**

**Mathematics Alt. A Paper 1 ( 121/1)**

**SECTION I (50 marks)**

*Answer all the questions in this section in the spaces provided.*

1 Evaluate

(2 marks)

2 The production of milk, in litres, of 14 cows on a certain day was recorded as follows: 22, 26, 15, 19, 20,16, 27,15, 19, 22, 21, 20, 22 and 28. Determine:

(a) the mode; (1 mark)

(b) the median. (2 marks)

3 Use logarithms, correct to 4 decimal places, to evaluate:

4 Simplify the expression:

(4 marks)

5. A wholesaler sold a radio to a retailer making a proﬁt of 20%. The retailer later sold the radio for Ksh 1560 making a proﬁt of 30%. Calculate the amount of money the wholesaler had paid for the radio. (3 marks)

6 A point P on the line AB shown below is such that AP = %AB. By construction locate P. (3 marks)

7 Chelimo’s clock loses 15 seconds every hour. She sets the correct time on the clock at 0700h on a Monday. Determine the time shown on the clock when the correct time was 1900h on Wednesday the same week. (3 marks)

8 Given that sin (x + 20)° = — 0.7660, ﬁnd x, to the nearest degree, for 0°< x <360°. (3 marks)

9 A number m is formed by writing all the prime numbers between 0 and 10 in an ascending order. Another number n is formed by writing all the square numbers between 0 and 10 in a descending order.

(a) Find m — n; (2 marks)

(b) Express (m — n) as a product of its prime factors. (1 mark)

10 The ﬁgure below shows a net of a solid. (Measurements are in centimetres).

Below is a part of the sketch of the solid whose net is Shown above. Complete the sketch of the solid, showing the hidden edges with broken lines. (3 marks)

11 The interior angles of an octagon are 2x, 1/2x, (x + 40)°, 110°, 135°, 160°, (2x + 10)° and 185°. Find the value of x. (2 marks)

12 A straight line passes through points (-2, 1) and (6, 3). Find: (a) the equation of the line in the form y = mx + c; (2 marks)

(b) the gradient of a line perpendicular to the line in (a). (1 mark)

13 A triangle ABC is such that AB = 5 cm, BC = 6cm and AC = 7cm.

(a) Calculate the size of angle ACB, correct to 2 decimal places. (2 marks)

(b) A perpendicular drawn from A meets BC at N. Calculate the length AN correct to one decimal place. (2 marks)

14 A cylindrical pipe 2 1/2 metres long has an internl diameter of 21 millimetres and an external diameter of 35 millimetres. The density of the material that makes the pipe is 1.25 g/cm‘. Calculate the mass of the pipe in kilograms. (Take n = 22/7). (4 marks)

15 The ﬁgure below represents a pentagonal prism of length 12cm. The cross-section is a regular pentagon of side 5 cm.

Calculate the surface area of the prism correct to 4 signiﬁcant ﬁgures. (4 marks)

Given the inequalities x — 5 < 3x — 8 < 2x — 3.

(a) Solve the inequalities; (2 marks)

(b) represent the solution on a number line. (1 mark)

**SECTION ll (50 marks)**

*Answer only ﬁve questions in this section in the spaces provided.*

17 A farmer had 540 bags of maize each having a mass of 112 kg. After drying the maize, the mass decreased in the ratio 15: l 6.

(a) Calculate the total mass lost after the maize was dried. (3 marks)

(b) A trader bought and repacked the dried maize in 90 kg bags. He transported the maize in a lorry which could carry a maximum of 120 bags per trip.

(i) Determine the number of trips the lorry made. (3 marks)

(ii) The buying price of a 90 kg bag of maize was Ksh 1500. The trader paid Ksh 2500 per trip to transport the maize to the market. He sold the maize and made a proﬁt of 26%. Calculate the selling price of each bag of the maize.

(4 marks)

18 (a) Solve the equation, x + 3/24 = 1/x-2 (4 marks) (b) The length of a ﬂoor of a rectangular hall is 9m more than its width. The area of the ﬂoor is 136 ml. (i) Calculate the perimeter of the ﬂoor. (4 marks)

(ii) A rectangular carpet is placed on the ﬂoor of the hall leaving an area of 64 mi. If the length of the carpet is twice its width, determine the width of the carpet. (2 marks)

19 A trader bought 2 cows and 9 goats for a total of Ksh 98 200. If she had bought 3 cows and 4 goats she would have spent Ksh 2200 less.

(a) F om1 two equations to represent the above information. (2 marks)

(b) Use matrix method to determine the cost of a cow and that of a goat. (4 marks)

(c) The trader later sold the animals she had bought making a proﬁt of 30% per cow and 40% per goat.

(i) Calculate the total amount of money she received. (2 marks)

(ii) Determine, correct to 4 signiﬁcant ﬁgures, the percentage proﬁt the trader made from the sale of the animals. (2 marks)

20 Two towns, A and B are 80 km apart. Juma started cycling from town A to town B at 10.00 am at an average speed of 40 km/h. Mutuku started his journey from town B to town A at 10.30 am and travelled by car at an average speed of 60 km/h.

(a) Calculate:

(i) the distance from town A when Juma and Mutuku met; (5 marks)

(ii) the time of the day when the two met. (2 marks)

(b) Kamau started cycling from town A to town B at 10.21am. He met Mutuku at the same time as Juma did. Determine Kamau’s average speed. (3 marks)

21 The displacement, s metres, of a moving particle from a point O, after t seconds is given by, s=t2—5t2 +3t + 10.

(a) Find s when t = 2.(2 marks)

(b) Determine:

(i) the velocity of the particle when t = 5 seconds;(3 marks)

(ii) the value of t when the particle is momentarily at rest.(3 marks)

(c) Find the time, when the velocity of the particle is maximum.(2 marks)

22 In the ﬁgure below, OABC is a trapezium. AB is parallel to OC and OC = SAB. D is a point on OC such that OD: DC = 3:2.

(a) Given that OA = p and AB = q, express in terms of p and q:

(i) OB;(1 mark)

(ii) AD;(2 marks)

(iii) CB.(2 marks)

(b) Lines OB and AD intersect at point X such that AX = kAD and OX = rOB, where k and r are scalars. Determine the values of k and r.

(5 marks)

23 (a) On the grid provided, draw the square whose vertices are A (6, -2), B (7, -2), C(7,— 1)andD(6,— 1). (1 mark)

On the same grid below, draw: (ii) A”B”C”D”, the image of A’B’C’D’ under a reﬂection in the line x = 0; (2 marks)

(iii) A'”B”’C”‘D’”, the image of A”B”C”D” under a rotation of +90° about (0,0). (2 marks)

(c) Describe a single transformation that maps A’B’C’D’ onto A”’B’”C’”D”’. (2 marks)

24 The ﬁgure below represents a cone of height 12 cm and base radius of 9 cm from which a similar smaller cone is removed, leaving a conical hole of height 4 cm.

(a) Calculate:

(i) the base radius of the conical hole; (2 marks)

(ii) the volume, in terms of 11:, of the smaller cone that was removed. (2 marks)

(b) (i) Determine the slant height of the original cone. (1 mark)

(ii) Calculate, in terms of 1:, the surface area of the remaining solid after the smaller cone is removed. (5 marks)

**4.1.2 Mathematics Alt. A Paper 2 (121/2)**

**SECTION l (50 marks)**

*Answer all the questions in this section in the spaces provided.*

1 The sum of n terms of the sequence; 3, 9, 15, 21, is 7500. Determine the value of n. (3 marks)

2 A quadratic curve passes through the points (—2, 0) and (1, 0). Find the equation of the curve in the form y = ax2 + bx + c, where a, b and c are constants. (2 marks)

3 Make d the subject of the fomula, P = 1/2mn — qd/n (3 marks)

4 Solve the equation

2 logx — log (x — 2) = 2 log 3. (3 marks)

5 (a) Using a pair of compasses and ruler only, construct an escribed circle to touch side XZ of triangle XYZ drawn below. (3 marks)

(b) Measure the radius of the circle. (1 mark)

6 The equation of a circle is given by x” + 4x + y” – 2y – 4 = 0. Determine the centre and radius of the circle. (3 marks)

7 (a) Expand (l — x)’. (1 mark)

(b) Use the expansion in (a) up to the term in x3 to approximate the value of (0.98)5. (2 marks)

8 The position vectors of points F, G and H are f, g and h respectively. Point H divides FG in the ratio 4:— 1. Express h in terms of f and g. (2 marks)

9 Two machines, M and N produce 60% and 40% respectively of the total number of items manufactured in a factory. lt is observed that 5% of the items produced by machine M are defective while 3% of the items produced by machine N are defective. If an item is selected at random from the factory, ﬁnd the probability that it is defective. (3 marks)

10 Two taps A and B can each ﬁll an empty tank in 3 hours and 2 hours respectively. A drainage tap R can empty the full tank in 6 hours. Taps A and R are opened for 5 hours then closed. (a) Determine the fraction of the tank that is still empty. (2 marks)

(b) Find how long it would take to ﬁll the remaining fraction of the tank if all the three taps are opened. (2 marks)

11 Simplify the expression

, leaving the answer in the fonn a Vb + c where a, b and c are integers. (3 marks)

12 A point P moves inside a sector of a circle, centre O, and chord AB such that 2cm < OP 5 3 cm and angle APB = 65°. Draw the locus of P. (4 marks)

13 The table below shows income tax rates in a certain year.

In that year, a monthly personal tax relief of Ksh 1056 was allowed. Calculate the monthly income tax paid by an employee who earned a monthly salary of Ksh 32500. (4 marks)

14 Solve the equation 6 cos’ x + 7 sin x — 8 = 0 for 0° < x <90°. (4 marks)

15 The positions of two towns are (2° S, 30° E) and (2° S, 37.4° E). Calculate, to the nearest km, the shortest distance between the two towns. (Take the radius of the earth to be 6 370 km) (2 marks)

16 The vertices of a triangle T are A(1 , 2), B(4, 2) and C(3, 4). The vertices of triangle T’ the image of T are A'(1/2, 1), B'(2, 1) and C’(3/2, 2).

Determine the transformation matrix M = (3 that maps T onto T’. (3 marks)

**SECTION ll (50 marks)**

*Answer only ﬁve questions from this section in the spaces provided.*

The Hire Purchase (H.P.) price of a public address system was Ksh 276000. A deposit of Ksh 60000 was paid followed by 18 equal monthly installments. The cash price of the public address system was 10% less than the H.P. price.

(a) Calculate:

(i) the monthly instalment; (2 marks)

(ii) the cash price. (2 marks)

(b) A customer decided to buy the system in cash and was allowed a 5% discount on the cash price. He took a bank loan to buy the system in cash. The bank charged compound interest on the loan at the rate of 20% p.a. The loan was repaid in 2 years. Calculate the amount repaid to the bank by the end of the second year. (3 marks)

(c) Express as a percentage of the Hire Purchase price, the difference between the amount repaid to the bank and the Hire Purchase price. (3 marks)

In the ﬁgure below, PR is a diameter of the circle centre O. Points P, Q, R and S are on the circumference of the circle. Angle PRQ = 72°, QS = QP and line USV is a tangent to the circle at S

Giving reasons, calculate the size of:

(a)< QPR; (2 marks)

(b)< PQS;(2 marks)

(c)< OQS;(2 marks)

(d)< RTS;(2 marks)

(e)< RSV.(2 marks)

19 (a) Complete the table below for y = x‘ + 4x” — 5x — 5. (2 marks)

(b) On the grid provided, draw the graph of’y = x‘ + 4×3 — 5x — 5 for – 5 < x < 2. (3 marks)

(c) (i) Use the graph to solve the equation x’ + 4x — 5x — 5 = 0. (2 marks)

(ii) By drawing a suitable straight line on the graph, solve the equation x”‘ + 4r‘ — Sx — 5 = – 4x — 1. (3 marks)

20 The ﬁgure ABCDEF below represents a roof of a house. AB = DC = 12m, BC = AD = 6m, AE = BF = CF = DE = 5m and EF=8m.

Calculate, correct to 2 decimal places, the perpendicular distance of EF from the plane ABCD. (3 marks)

Calculate the angle between:

(i) the planes ADE and ABCD; (2 marks)

(ii) the line AE and the plane ABCD, correct to 1 decimal place; (2 marks)

(iii) the planes ABF E and DCFE, correct to l decimal place. (3 marks)

(a) Complete the table below, giving the values correct to l decimal place. (2 marks)

(b) On the grid provided, using the same scale and axes, draw the graphs of y = 2 sin (x + 20)° and y = \/5 cos x for 0° Sx S 240°. (5 marks)

(c) Use the graphs drawn in (b) above to determine:

(i) the values of x for which 2 sin (x + 20) = x/5 cos x; (2 marks)

(ii) the difference in the amplitudes of y = 2 sin (x + 20) and y = \/3 cos x. (1 mark)

22 Three quantities R, S and T are such that R varies directly as S and inversely as the square of T.

(a) Given that R = 480 when S = 150 and T = 5, write an equation connecting R, S and T. (4 marks)

(b) (i) Find the value of R when S = 360 and T = 1,5. (2 marks)

(ii) Find the percentage change in R if S increases by 5% and T decreases by 20%. (4 marks)

23 The equation of a curve is given by y = 5x — 1/2×2.

(a) On the grid provided, draw the curve of y = 5x — 1/2×2 for 0 < x < 6. (3 marks)

(b) By integration, ﬁnd the area bounded by the curve, the linex = 6 and the x-axis. (3 marks)

(c) (i) On the same grid as in (a), draw the line y = 2x. (l mark)

(ii) Determine the area bounded by the curve and the line y = 2x. (3 marks)

24 The table below shows marks scored by 42 students in a test

(a) Starting with the mark of 25 and using equal class intervals of 10, make a frequency distribution table. (2 marks)

(b) On the grid provided, draw the o-give for the data. (4 marks)

(c) Using the graph in (b) above, estimate:

(i) the median mark; (2 marks)

(ii) the upper quartile mark. (2 marks)

**2013 KCSE Maths A Past Paper-Marking Scheme/Answers**

**MATHEMATICS ALT. A (121)**

**Mathematics AIternative a Paper 1 (121/1)**

**Mathematics Alternative A Paper (2/121/2)**