### SECTION A (50 Marks)

Answer ALL questions from this section
1.

Without using calculators evaluate 1/3 of (2¾ - 5½) x 36/7 ÷ 9/4

2 marks

2.

Use the method of completing the square to solve the quadratic equation
2x2 – 13x + 15 = 0

3 marks

3.

Solve for θ in the equation 6${\mathrm{cos}}^{2}$ θ - Sin θ - 4 = 0 in the range 0o < θ < 180o.

3 marks

4.

The sides of a rectangle are x cm and (x + 1) cm. A circle has radius of (x + 2) cm. If the sum of the area of the rectangle and the circle is 184 cm${}^{2}$. Using π as $\frac{22}{7}$ find the value of x.

4 marks

5.

Use binomial expansion to evaluate $\left[2+\frac{1}{\sqrt{2}}\right]+\left[2-\frac{1}{\sqrt{2}}\right]$

3 marks

6.

A line ${L}_{1}$ passes through point (1, 2) and has a gradient of 5. Another line ${L}_{2}$ is perpendicular to ${L}_{1}$ and meets it at a point where x = 4. Find the equation for ${L}_{2}$ in the form y = mx + c.

4 marks

7.

Find the value of x in the following equation.
${9}^{x}+{3}^{2x}-1=53$

3 marks

8.

The first and the last terms of an AP are 2 and 59 respectively. If the sum of the series is 610, find the number of terms in the series and the common difference.

4 marks

9.

The equation of a circle is $2{x}^{2}+2{y}^{2}+12x-20y-4=0$ . Determine the coordinates of the centre of the circle and state its radius.

3 marks

10.

Make b the subject of the formula $a=\frac{bd}{\sqrt{{b}^{2}-d}}$

3 marks

11.

Solve the inequality $3-2x\le x\le \frac{2x+5}{3}$ and show the solution on the number line.

4 marks

12.

Solve for x given that ${\mathrm{log}}_{2}5x-{\mathrm{log}}_{4}2x=3$

3 marks

13.

A salesman earns a basic salary of sh. 9,000 per month. In addition he is also paid a commission of 5% for sales above sh. 15,000. In a certain month he sold goods worth sh. 120,000 at a discount of 2½%. Calculate his total earnings that month.

3 marks

14.

A small cone of height 8 cm is cut off from a bigger cone to leave a frustum of height 16 cm. If the volume of the smaller cone is 160 $c{m}^{3}$, find the volume of the frustum.

3 marks

15.

Vector OP = 6i + j and OQ = -2i + 5j. A point N divides PQ internally in the ratio 3:1. Find PN in terms of i and j.

3 marks

16.

Without using mathematical tables or calculators express in surd form and simplify

$\frac{1+\mathrm{cos}{30}^{o}}{1-\mathrm{sin}{60}^{o}}$

3 marks

### SECTION B (50 Marks)

Answer any five questions in this section
17.

In the figure below, vector OP = p and OR = r. OS = 2r and OQ : OP = 3 : 2

(a) Express the following vectors in terms of p and r.

(i) QR

(ii) PS

(b) The lines QR and PS intersect at K. By expressing OK in two different ways, find the ratio PK : KS

10 marks

18.

On the graph paper provided, plot the triangle
(a) whose co-ordinates are A(1, 2) B(5, 4) and C(2, 6) {1 mark}

(b) On the same axes

(i) Draw the image ${A}_{1}{B}_{1}{C}_{1}$ of ABC under a rotation of ${90}^{o}$ clockwise about origin. {2 marks}

(ii) Draw the image ${A}_{11}{B}_{11}{C}_{11}$ of ${A}_{1}{B}_{1}{C}_{1}$ under a reflection in the line y = -x. State the
coordinates of ${A}_{11}{B}_{11}{C}_{11}$. {3 marks}

(c)${A}_{111}{B}_{111}{C}_{111}$ is the image of ${A}_{11}{B}_{11}{C}_{11}$ under the reflection in the line x = 0. Draw the image ${A}_{111}{B}_{111}{C}_{111}$ and state its coordinates. {2 marks}

(d) Describe a single transformation that maps${A}_{111}{B}_{111}{C}_{111}$ onto ABC.

10 marks

19.

A bus left Kitale at 10.45 a.m and travelled towards Nairobi at an average speed of 60 km/h. A Nissan left Kitale on the same day at 1.15 p.m and travelled along the same road at an average speed of 100 km/h. The distance between Kitale and Nairobi is 500 km.
(a) Determine the time of the day when the Nissan overtook the bus. {6 marks}

(b) Both vehicles continued towards Nairobi at their original speed. Find how long the Nissan had to wait in Nairobi before the bus arrived. {4 marks}

10 marks

20.

The table below shows how income tax was charged in a certain year.

 (Kenya pounds) (Ksh. per Kenya pound) 1 – 3630 2 3631 - 7260 3 7261 - 10890 4 10891 - 14520 5 14521 - 18150 6 18151 - 21780 7 21781 and above 7.5

During the year Mwadime earned a basic salary of Ksh. 25,200 and a house allowance of Ksh. 12,600 per month. He was entitled to a personal tax relief of Ksh. 1,162 per month.

(a) Calculate:
(i) Mwadime’s taxable income in Kenya pounds per annum. {2 marks}

(ii) The net tax he pays per month. {6 marks}

(b) Apart from income tax he also contributes monthly NHIF Ksh. 1600, WCPS Ksh. 1000. Calculate his net monthly pay. {2 marks}

10 marks

21.

X, Y and Z are three quantities such that X varies directly as the square of Y and inversely as the square root of Z.
(a) Given that X = 18 when Y = 3 and Z = 4, find X when Y = 6 and Z = 16. {5 marks}

(b) If Y increases by 10% and Z decreases by 19%, find the percentage increase in X. {5 marks}

10 marks

22.

(a) A port B is on a bearing 080o from a port A and a distance of 95 km. A Submarine is stationed at a port D, which is on a bearing of 200o from A, and a distance of 124 km from B. A ship leaves B and moves directly Southwards to an Island P, which is on a bearing of 140o from A. The Submarine at D on realizing that the ship was heading to the Island P, decides to head straight for the Island to intercept the ship. Using a scale of 1 cm to represent 10 km, make a scale drawing showing the relative positions of A, B, D and P. {4 marks}

Hence find:
(b) The distance from A to D. {2 marks}

(c) The bearing o the Submarine from the ship when the ship was setting off from B. {1 mark}

(d) The bearing of the Island P from D. {1 mark}

(e) The distance the Submarine had to cover to reach the Island P. {2 marks}

10 marks

23.

The data below represent the heights taken to the nearest centimeters of 40 lemon trees in a garden. (NB: A = Assumed mean)

 Height (cm) f X d = x - A fd ${d}^{2}$ $f{d}^{2}$ 131 – 140 3 141 – 150 4 151 – 160 7 161 – 170 11 171 – 180 9 181 – 190 5 191 – 200 1

(a) Complete the table. {6 marks}

(b) Using 165.5 as the assumed mean, calculate the mean height. {2 marks}

(c) Calculate the standard deviation of the distribution. {2 marks}

10 marks

24.

The line segment BC = 7.5 cm long is one side of triangle ABC.
(a) Use a ruler and compasses only to complete the construction of triangle ABC in which
∠ABC = 45o, AC = 5.6 cm and angle BAC is obtuse. {3 marks}

(b) Draw the locus of a point P such that P is equidistant from a point O and passes through the vertices of triangle ABC. {3 marks}

(c) Locate point D on the locus of P equidistant from lines BC and BO. Q lies in the region enclosed by lines BD, BO extended and the locus of P. Shade the locus of Q. {4 marks}

10 marks

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