## KNEC KCSE Mathematics Paper 2 Question Paper / 2016 KCSE MOKASA Joint Examination

### 2016 KCSE MOKASA Joint Examination

#### Mathematics Paper 2

### SECTION A (50 Marks)

Use logarithm tables to evaluate:

4 marks

The external and internal diameters of a cement pipe are 20cm and 14cm respectively.

Calculate the volume of cement required to prepare 1.4m long. Give your answer in cm^{3}.

3 marks

Make h the subject of the formula .

3 marks

Solve for x in the equation 2 Sin^{2} x – 1 = Cos^{2}x + sin for 0^{o} < x < 390^{o}

3 marks

The points A(-4,1) and B(-2,5) are the end points of a diameter of a circle. Determine the

coordinates of the centre of the circle, hence calculate the equation in the form:

x^{2} + y^{2} = 2ax – 2by + c + d —- = 0

3 marks

Solve the quadratic equation 3x^{2} – 4x = 2

3 marks

Evaluate by rationalizing the denominator and leaving your answer in surd form.

√8/1 + Cos45^{o}

3 marks

Expand (2 + 3x)^{6} up to the term x^{2}. Hence use your expansion to estimate (2.09)^{6}

3 marks

Two quantities M and N are such that M varies partly as N and partly as the square of N.

Determine the relationship between M and N given that when M is 1050, N = 10 and when

M = 2200, N = 20.

3 marks

A dealer has two types of grades of tea, A and B. Grade A costs sh 140 per kg. Grade B costs

sh. 160 per kg. If the dealer mixes A and B in the ratio 3:5 to make a brand of tea which he

sells at sh. 180 per kg, calculate the percentage profit that he makes.

3 marks

Onyango bought a refrigerator whose cash price is sh. 84,000 on hire purchase. He made a

cash deposit of sh. 20,000 and the 15 monthly instalments of shs. 6,000. Calculate the rate of

interest per month.

3 marks

Given that OA = 2i + 5k and OB = 71 – 5j. A point T is on AB such that 2AT = 3TB.

Calculate the magnitude of OT to 4 significant figures.

3 marks

Find the sum of A.P having 15 terms, the fourth term being -3.2 and the eight term 8.4

3 marks

Use matrix methods to solve the following simultaneous equation

4 marks

The volumes of two similar cylinders are 3240cm^{2} and 960cm^{3}. If the surface area of the

larger cylinder was 792cm^{2}. find the surface area of the smaller cylinder.

3 marks

Estimate the area bounded by the curve y = ½ x^{2} + 1, x = 0, x = 3 and the x-axis using the

mid-ordinate rule. Use three strips.

3 marks

### SECTION B (50 Marks)

The angle of elevation of the top of a flag post from a point P on a level ground is 20^{o}. The

angle of elevation of the top of the flag post from another point Q nearer to the flag post and

110m from P is 32^{o}. Q is between P and the flag post.

a) Draw a sketch diagram to show the above arrangement (2 marks)

b) calculate correct to 2 d.p

(i) The distance from the point Q to the top of the flag post (5 marks)

(ii) The length of the rope tied from the top of the flag post to the point P on the

ground, if 0.5m of the rope is used fro tying the knots. (3 marks)

10 marks

The diagram below shows a right pyramid with a square base ABCD and vertex V. O is the

centre of the base. AB = 14m, VA = 20m and N is the midpoint of BC.

Find;

a) The lengths of BO, VO and VN (3 marks)

b) The angle between VO and plane VBC (3 marks)

c) The angle between VB and base ABCD (2 marks)

d) The angle between VDC and VBC (2 marks)

10 marks

A number of students were asked to cut 30cm lengths of binding wire without measuring.

Later 100 pieces area collected and measured correct to the nearest 0.1 cm the data below

was collected.

Length l (cm) |
28.0 – 28.4 |
28.5 – 28.9 |
29.0 – 29.4 |
29.5 – 29.9 |
30.0 – 30.4 |
30.5 – 30.9 |
31.0 – 31.4 |
31.5 – 31.9 |

Frequency | 5 | 8 | 30 | x | 10 | 20 | 20 | 4 |

a) i) Calculate the value of x (1 mark)

ii) State the modal class (1 mark)

b) Using 29.7 as a working mean calculate;

i) the mean (4 marks)

ii) the standard deviation (4 marks)

10 marks

A transformation represented by matrix Maps A(0,0), B(2,0), C(2,3) and D(0,3) onto A^{1}B^{1}C^{1} and D1 respectively

a) Draw ABCD and its image A^{1}B^{1}C^{1}D^{1} (4 marks)

b) A transformation represented by maps A^{1}B^{1}C^{1}D^{1} on A^{11}B^{11}C^{11}D^{11}. Plot A^{11}B^{11}C^{11}D^{11}on the same graph. (3 marks)

c) Determine the matrix of a single transformation that maps A^{11}B^{11}C^{11}D^{11} onto ABCD (3 marks)

10 marks

In the figure K, L, M and N are points on the circumference of a circle centre O. The points

K, O, M and P lie on a straight line.

PT is a tangent to the circle at N. Given that <MKN = 40^{o}. find the values of the following

angles stating reasons.

a) <MLN (2 marks)

b) <OLN (2 marks)

c) <LNP (2 marks)

d) <MPN (2 marks)

e) <KLM (2 marks)

10 marks

The position of 3 cities P, Q and R are (15^{o}, 20^{o}W) (50^{o}N, 20^{o}W) and (50^{o}, 60^{o}E)

respectively.

a) Find the distance in nautical miles between:

(i) Cities p and Q (2 marks)

(ii) Cities P and R, via city Q (3 marks)

b) A plane left city P at 0250h and flew to city Q where it stopped for 3 hours then flew on

to city R, maintaining a ground speed of 900 knots throughout.

(i) The local time city R when the plane left city P (3 marks)

(ii) The local time (t the nearest minute) at city R when the plane landed at R.

(2 marks)

10 marks

The table below is for function y = x^{3} – 7x + 6 for the range -3 __<__ x __<__ 3.

x | -3 | -2 | -1.5 | -1 | 0 | 1 | 1.5 | 2 | 3 |

y | | | | | | | | | |

a) Complete the table above. (2 marks)

b) Draw the graph of the function y = x^{3} – 7x + 6 for the range -3 __<__ x __<__ 3 (3 marks)

c) Use the graph above to estimate the roots of the following;

(i) x^{3} = 7x -6 (1 mark)

(ii) –x^{3} + 8x – 2 = 0 (2 marks)

d) By drawing a tangent, estimate the gradient of the curve y=x^{3} – 7x + 6 at x = -2 (2 marks)

10 marks

a) The acceleration of a particle t seconds after passing a fixed point P is given by a = 3t – 3.

Given that the velocity of the particle when t = 2 is 5 m/s, find;

(i) Its velocity when t = 4 seconds (3 marks)

(ii) Its displacement at this time (3 marks)

(iii) find the exact area bounded by the graph x = 9y – y^{3} and the y-axis(4 marks)

10 marks