## KNEC KCSE Mathematics Paper 2 – 2014 Gatundu Mock

### 2014 Gatundu Mock

#### Mathematics Paper 2

### SECTION I (50 Marks)

**ATTEMPT ALL QUESTIONS IN THIS SECTION**

Simply without using tables or calculators:

3 marks

Using a ruler, a set square and a pair of compasses only,

(a) Draw a line AB = 5cm (1 mark)

(b) Divide the above line into 9 equal parts. (2 marks)

3 marks

Simplify

4 marks

(a) Expand and simplify (2 marks)

(b) Hence use the above expansion to evaluate (1.03)^{4} to 4 s.f (2 marks)

4 marks

A circle whose equation is (x – 1)^{2} + (y – k)^{2} + (y-k)^{2} = 10 passes through the point (2, 5). Find the

coordinates of the two possible centres of the circle.

3 marks

(a) Find the inverse of the matrix (2 marks)

(b) Hence determine the point of intersection of the:

lines:

y + x = 7

3x + y = 15 (2 marks)

4 marks

Waiguru invested shs.6000 for 1 ½ years at 12% p.a compounded quarterly. Determine:

(a) The amount accruing at the end of 1 ½ years. (3 marks)

(b) Interest earned. (1 mark)

4 marks

Given that y^{2} is inversely proportional to x and y = 1.5 when x = 2/3, find the value of y when x = 6.

3 marks

A single unbiased dice is thrown once. Find the probability that the score will be:

(a) even (1 mark)

(b) a multiple of 3 (1 mark)

2 marks

Two types of tea which cost ksh.200 per kg and ksh 250 per kg are mixed so that their weights are in

the ratio 5:3 respectively. Calculate the cost of 20kg of the mixture.

3 marks

A plane flies from Cairo (40^{o}N, 30^{o}E) to Harare ( S^{o}, 30^{o}E) a distance of 3,300nm. Determine the

latitude θS^{o}.

3 marks

Find the percentage error in evaluating (x + y) – z ; if x = 3.2cm, y = 5.6cm and z = 2.4cm.

3 marks

Without using calculator or mathematical tables; Find the sums of the first 20 terms of the series:

log2 + log 4 + log 8 + log 16 + —————-

3 marks

Q represent a positive quarter turn about (0, 0) and R is the reflection in the line y-x=0. Find QR (2, 1)

3 marks

Find the acute angel y if cos 4y = sin 2y.

2 marks

In the diagram below, O is the centre of the circle SPQ. Angle SOQ = 130^{o}

Calculate angle:

(a) r^{o} (1 mark)

(b) p^{o} (2 marks)

3 marks

### SECTION II (50 Marks)

**ATTEMPT ANY 5 QUESTINS IN THIS SECTION**

The vectors a and b are not parallel. Given that

(a) 3**a** + h**b** = k**a** + 2**b**. Find the values of the scalars h and k. (3 marks)

(b) The position vectors of points A and B relative to a point of origin O are **a** and **b **respectively. C

is the point on AB such that AC:CB = 5:3. Express in terms of a and b the vectors.

(c) (i) D is a point on OB such that OD = b. Express in terms of **a** , **b** and . (2 marks)

(ii) Find the value of ; if OA is parallel to DC. (1 mark)

10 marks

18. Complete the table below for graphs of:

y = 3 sin (2x – 30^{o}) and y = sin x for -120^{o}< x < 120^{o}

(a) Correct to 2 d.p.

x^{o} |
-120^{o} |
-90^{o} |
-60^{o} |
-30^{o} |
0^{o} |
30^{o} |
60^{o} |
90^{o} |
120^{o} |

y=sin x^{o} |
-0.87 | -1.00 | -0.50 | 0 | 0.87 | 1.00 | |||

y = 3sin (2x-30^{o}) |
1.50 | -3.00 | -1.50 | 1.50 | 1.50 |

(b) On the grid provided; draw the graphs on the same axes of: y = sin x and y = 3 sin (2x – 30^{o}) (4 marks)

Take the scale: 1cm for 15^{o} on x-axis and 2cm for 1 unit on the y-axis.

(c) State the amplitude and period of each wave. (2 marks)

(d) Find the value of x: for which Sin (2x – 30^{o}) – 1/3 sin x = 0. (2 marks)

10 marks

The heights to the nearest centimeter of 100 boys were recorded as shown below

Heights | 140-145 | 146-151 | 152-157 | 158- 163 | 164-169 | 170-75 | 176-181 | 182-187 | 188-193 | 194-199 |

Frequency | 3 | 4 | 7 | 10 | 10 | 15 | 25 | 16 | 8 | 2 |

Using an assumed mean of 166.5

(a) Calculate the mean (5 marks)

(b) The variance (3 marks)

(c) The standard deviation (2 marks)

10 marks

P and V are connected by the law P = KV^{n} where k and n are constants. The table below shows values

of P and V.

V | 2 | 2.3 | 3 | 3.5 | 4 |

P | 288 | 503.7 | 1458 | 2701 | 4608 |

_ | _ | _ | _ | _ | _ |

_ | _ | _ | _ | _ | _ |

(a) Express P = KV^{n} in linear form. (1 mark)

(b) Complete the table and draw a linear graph to represent the information given above. (4 mark)

(c) Find the values of k and n (2 marks)

(d) Find the law connecting p and v. (1 mark)

(e) Find the value of p when v = 6. (2 marks)

10 marks

An institute offers two types of courses, veterinary and Agriculture. The institute has a capacity of 500 student. There must be more Agriculture students than Veterinary students. At least 200 students must take veterinary courses. If x represent the number of veterinary students and y the numbers of

Agriculture students.

(a) Write down three inequalities that describe the given conditions. (3 marks)

(b) On the grid provided, draw the three inequalities. (3 marks)

(c) If the institute makes a profit of shs.25,000 to train one veterinary student and shs. 10,000 to

train one Agriculture student, determine:

(i) The number of students that must be enrolled in each course to maximize the profit.

(2 marks)

(ii)The maximum profit. (2 marks)

10 marks

Use a ruler and compasses only in this question.

The diagram below shows a square ABCD.

(a) Construct two circles with centre O, with one passing through the vertices ABCD and the other one,

touching the four sides of the square ABCD (4 marks)

(b) (i) Locate point P if BPC is the tangent of the smaller circle. (1 mark)

(ii)Show the locus of a point T inside the square such that <APB = <ATB (3 marks)

(c) (i) Point Q is the point where the locus b(ii) above meets side AD. Label the locus of point Q.

(1 mark)

(ii) Shade the locus of a region R inside the square such that <QRP 90^{o} and above the locus of point T. (1mark)

10 marks

In the figure below Angle CBD = 37^{o}, angle BCD = 20^{o} and ABC is a tangent to the circle at B.

(a) Find:

(i) Angle BED (2 marks)

(ii) Angle ABE (1 mark)

(iii) Angel BDC (1 mark)

(b) With reasons state whether triangle BDC is similar to triangle CBE. (2 marks)

(c) Given that the radius of the circle is 2.5 cm and the length BC= 6cm, find the length EC using the

cosine rule. (4 marks)

10 marks

The displacement in metres of a particle from a point O after t seconds is given by s=t(t-2)t-1); Find :

(a) The speed of the particle when t = 2 seconds. (3 marks)

(b) Its minimum speed. (2 marks)

(c) The time when the particle is momentarily at rest. (3 marks)

(d) It acceleration when t = 3 seconds. (2 marks)

10 marks