## KNEC KCSE Mathematics Paper 1 Question Paper / 2016 KCSE MOKASA Joint Examination

### 2016 KCSE MOKASA Joint Examination

#### Mathematics Paper 1

### SECTION A (50 Marks)

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

Simplify without using table or a calculator to its lowest form.

3 marks

Abraham’s money box contains only sh. 5 coins and sh. 10 coins. There are 24 coins

and their total value is sh. 150. Find how many of each kind of coins there are in the

box.

3 marks

The gradient of a straight line M_{1} passing through the points (P(3, 4) and Q(x, y) is −3/2.

A line M_{2} is perpendicular to M_{1} and passes through the points Q and R(2, -1).

Determine the values of X and Y.

3 marks

Three inequalities define the region R shown below. Form the inequalities.

3 marks

Given that a = 10.5, b = 9.6 and c = 1.4 all correct to the decimal place. Find the

percentage error involved in the calculation of a𝑎+ b/c𝑏𝑏to 1 d. p .

3 marks

The GCD of three numbers is 4 while the LCM is 360. If two of the numbers are 24 and 40. List two other possible numbers.

3 marks

Given that *dy*/*dx*= 2*x*^{2} + 3 and that y = 3 when x = 0, find the value of y when x = 1/3.

3 marks

After buying 52 Sterling pounds, a businessman decided to exchange his money for US

dollars. Using the following currency exchange rate, calculate to 3s.f. the number of dollars he ended up with.

1 US dollars ($) = ksh. 89.75

1 Sterling pound (£) = ksh. 135.47

3 marks

Solve for X and Y in

4^{x}𝑥𝑥 × 4^{y}𝑦𝑦 = 1

3^{2x – y}𝑥𝑥 = 81

3 marks

Given the number 11055. Show that the number is divisible by 3, 5 and 11 using necessary divisibility tests.

3 marks

Calculate the area of the shaded region in the figure below, given that;

OX = OY = 10cm and XY = 16 cm. (Take 𝜋π𝜋 = 3.142)

4 marks

The figure below represents a model of a prism ABCDEF drawn to scale. Complete the

prism.

3 marks

Use reciprocal table to find the value of 1/0.325. Hence evaluate 3√0.000125/0.325 .

3 marks

A poultry farmer has brown and white chicken; the brown chicken are twice as many as

the white ones; of the white chicken 2/5 are layers while the rest are cockerels; of the

brown ones 5/12 are layers while the rest are cockerels. If the chicken is picked at

random in the dark, find the probability that it will be a cockerel.

3 marks

Triangle PQR has vertices P(3, 2), Q(-1, 1) and R(-3, -1). Under a rotation the vertices of

P^{1}Q^{1}R^{1} are P^{1}(1, 4), Q^{1}(2, 0) and R^{1}(4, -1). By construction find the centre and angle of

rotation.

4 marks

A metal cuboid of length 12cm, width 10cm and height 6cm has a density of 1.4g per cm^{3}. A

cylinder of different metal has a base radius of 6cm and height 12m, with a density of 1.8

g/cm^{3}. The two solids are melted down to recast in the shape of a cube without loss of metal.

(Take π = 3.142). Calculate the average density of the cube.

3 marks

### SECTION B (50 Marks)

**Answer any five questions in this section**

Three hundred and sixty litres of a homogenous paint is made by mixing three types of

paints A, B and C. The ratio by volume of paint A to paint B is 3:2 and paint B to paint

C is 1:2. Paint A costs shs. 180 per litre, paint B shs. 240 per litre and paint C shs. 127.50

per litre and paint C shs. 127.50 per litre.

(a) The volume of each type of paint in the mixture. (5 marks)

(b) The amount of money spent in making one litre of the mixture. (3 marks)

(c) The percentage profit made by selling the mixture at shs. 221 per litre.

(2 marks)

10 marks

The figure below is a solid frustum of a rectangular base ABCD and top rectangular

EFGH. Given that AB = 6cm, AC = 10cm, HF = 5cm, FG = 4cm and CF =25cm.

Calculate:

(a) The height of the pyramid it was cut from giving your answer to the nearest

whole number. (2 marks)

(b) The surface area of the frustum. (5 marks)

(c) The volume of the frustum. (3 marks)

10 marks

(a) Complete the table below for the function 𝑦𝑦*y* = 1/2sin 2𝑥*x* where 0^{o} ≤ 𝑥*x* ≤ 360^{o}. (2 marks)

x | 0^{o} |
30^{o} |
60^{o} |
90^{o} |
120^{o} |
150^{o} |
180^{o} |
210^{o} |
240^{o} |
270^{o} |
300^{o} |
330^{o} |
360^{o} |

2x | 0^{o} |
60^{o} |
120^{o} |
180^{o} |
240^{o} |
300^{o} |
360^{o} |
420^{o} |
480^{o} |
540^{o} |
600^{o} |
660^{o} |
720^{o} |

sin 2x | 0^{o} |
0.866 | | 0 | | | | | 0.866 | 0 | | | |

y =½ sin 2x |
0 | 0.433 | | 0 | | | | | | | | | |

(b) On the grid provided, draw the graph of the function 𝑦𝑦*y =½* sin 2*x*𝑥 for 0^{o} ≤ *x*𝑥 ≤ 360^{o} using the scale 1 cm for 300 on the horizontal axis and 4cm for 1 unit of y-axis. (3 marks)

(c) Use your graph to determine the amplitude and period of the function *y =½* sin 2*x *. (2 marks)

(d) Use the graph to solve;

(i) ½sin 2*x*𝑥𝑥 = 0 (1 mark)

(ii) *½* sin 2*x*𝑥− 0.5 = 0 (2 marks)

10 marks

Using a ruler and a pair of compass only; Construct rectangle ABCD whose sides are

AB = 10cm and BC = 7cm.

Use the figure to:

(a) Find the point R and S on AD and DC respectively, such that R is equidistant

from AD and S is equidistant from DC. (2 marks)

(b) Shade the region within the rectangle in which a variable point X must lie given

that X satisfies the following conditions.

(i) X > 2𝑐𝑐𝑐𝑐 from RS (3 marks)

(ii) X is nearer to AB than to BC (3 marks)

(iii) BX is more than 5cm (2 marks)

10 marks

A youth group decided to raise ksh. 480,000 to buy a piece of land costing ksh. 80,000

per hectare. Before the actual payment was made, four of the members pulled out and

each of those remaining had to pay an additional ksh. 20,000.

(a) If the original number of the group members was X, write down;

(i) An expression of how much was contributed originally. (1 mark)

(ii) An expression of how much the remaining members were to contribute

after the four pulled out. (1 mark)

(b) Determine the number of members who actually contributed towards the

purchase of the land. (4 marks)

(c) Calculate the ratio of the supposed original contribution for the new

contribution. (2 marks)

(d) If the land was sub-divided equally, find the size of land each member got.

(2 marks)

10 marks

A train moving at 40km/h is moving in the same direction with a truck on a road

parallel to the railway line at a speed of 75km/h. The truck takes 1 ¼ min to overtake

the train completely.

(a) Given that the truck is 5m long. Determine the length of the train in metres.

(6 marks)

(b) The truck and the train continued moving parallel to each other at the original

speeds. Calculate the distance between them after 10 mins 15 sec from the time

the truck overtook the train. (2 marks)

(c) The truck stopped 50 minutes after overtaking the train. How long did the train

take to catch up with the truck? (2 marks)

10 marks

(a) Given that the position vectors of A, B and C are;

Find;

(i) **AC** (1 mark)

(ii) **AB** (1 mark)

(iii) **BC** (1 mark)

(b) In the figure below, OR = 5a, OP = 5b, and PQ = 2a – b

(i) Express as simply as possible, vectors **OQ** and **RQ** in terms of a and b.

(2 marks)

(ii) Given that line PQ produced meet at X and that **PX = k PQ** and

**OX = h OR **where k and h are constants; form an equation connecting k,

h, a and b. Hence deduce the values of k and h. (5 marks)

10 marks

Mr. Kamau earns a basic salary of ksh. 30,000 per month. He gets medical allowance of

ksh. 4000 per month. He occupies a company house for which he pays a nominal rent

of ksh. 1000 per month. He enjoys a tax relief of ksh. 600 per month. The following

PAYE is in operation.

Income K£ P.a.0 – 3600 3601 – 7200 7201 – 10800 10801 – 14400 14401 – 18000 18001 and over |
Rate of tax in Ksh. (£)2 3 5 7 9 10 |

(a) Calculate Kamau’s taxable income in Kenya pounds per annum. (3 marks)

(b) Calculate Kamau’s PAYE in Kenya shillings per month. (4 marks)

(c) In addition to the PAYE the following deductions were made on his salary every

month.

(i) WCPs at 2% of basic salary

(ii) NHIF of ksh. 300

(iii) Cooperatives shares and loan recovery totaling ksh. 2,500 per month.

Calculate Kamau’s net pay in ksh. per month. (3 marks)

10 marks