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** SECTION I**

Answer all the questions in this section

- In this question, show all the steps in your calculations, giving the answer at each stage. Use logarithms correct to four decimal place, to evaluate.

(3 marks) - Make
*h*the subject formula

(2 marks) - Line AB given below is one side of triangle ABC. Using a ruler and a compasses only:

- Complete the triangle ABC such that BC = 5 cm and angle ABC = 45°. (1 mark)
- On the same diagram construct atriangle touching sides AC, BA produced and BC produced. (2 marks)

- The position vectors of points A and B are respectively. A and P are divides AB in the ratio 2:3.

Find the position vector of point P. (3 marks) - The top of a table is a rectangular hexagon. each side of the hexagon measures 50.0 cm. Find the maximum percentage error in calculating the the perimeter of the top of the table. (3 marks)
- A student at a certain college has 60% chance of passing an examination at the first attempt. Each time a student fails and repeats the examination, his chances of passing are increased by 15%.

Calculate the probability that a student in college passes an examination at the second or at the third attempt. (3 marks) - An aeroplane flies at an average speed of 500 knots due East from point P(53.4
^{o}N, 40^{o}E) to another point Q. It takes 2¼ hours to reach point Q.

Calculate:- the distance in nautical miles it travelled; (1 mark)
- the longitude of point Q to 2 decimal places. (2 marks)

- Expand and simplify the expression. (2 marks)
- Use the expansion in (a) above to find the value of 14
^{5}. (2 marks)

- In the figure below, angles BAC and ADC are equal. Angle ACD is a right angle. The ratio of the sides AC: BC = 4:3.

Given that the area of the traingle ABC is 24 cm^{2}, find the area of triangle ACD. (3 marks) - Points A(2, 2) and B(4, 3) are mapped onto A'(2, 8) and B' (4, 15) respectively by the transformation T.

Find the matrix of T. (4 marks) - The equation of a circle given by

Determine the coordinates of the centre of the circle. (3 marks)

- Solve for
*y*in the equation (3 marks)

- Without using a calculator or mathematic tables, express in surd form and simplify. (3 marks)
- The figure below represents a triangular prism. The faces ABCD, ADEF and CBFE are rectangles.

AB = 8 cm, BC = 14 cm, BF = 7 cm and AF = 7 cm.

Calculate the angle between the faces BCEF and ABCD. (3 marks) - A particle moves in a straight line from a fixed point. Its velocity Vms
^{-1}after*t*seconds is given by V = 9*t*^{2}- 4*t*+ 1.

Calculate the distance travelled by the particle during the third second. (3 marks) - Find in the radians, the values of
*x*in the interval 0^{o}*≤ x*≤ 2π^{o }for which 2 cos^{2}x - sin x = 1

( Leave the answers in terms of π) (4 marks)**SECTION II**(50 MARKS)*Answer***any**five questions in this section. - A trader deals in two types of rice; type A and type B. Type A costs Ksh 400 per bag while type B costs Ksh 350 per bag.
- The trader mixes 30 bags of type A with 50 types of type B. If he sells the mixture at a profit of 20%, calculate the the selling price of one bag of the mixture. (4 marks)
- The trader now mixes type A with B in the ratio
*x:y*respectively. If the cost of the mixture is 383.50 per bag, find the ratio*x:y*. (4 marks) - The trader mixes one bag of the mixture in part (a)with one bag of the mixture in part (b) above.

Calculate the ratio of type A rice to type B rice in this mixture. (2 marks)

- Three variables
*p, q*and*r*are such that*p*varies directly as*q*and inversely as the square of*r.*- When
*p*= 9,*q*= 12 and*r*= 2.

Find*p*when*q*= 15 and*r*= 5. (4 marks) - Express
*q*in terms of*p*and*r*. (1 mark) - If
*p*is increased by 20% and*r*is decreased by 10%, find:- a simplified expression for the change in
*q*in terms of*p*and*r*. (3 marks) - the percentage change in
*q*. (2 marks)

- a simplified expression for the change in

- When
- Complete the table below, giving the correct values correct to two decimal places.

(2 marks) - On the grid provided, draw the graphs of
*y =*sin*2x*and*y = 3*cos*x - 2*for 0^{o}*≤ x*≤360^{o}on the same axes.

Use the scale of 1 cm to represent 30^{o}on the*x-*axis and 2 cm to represent 1 unit on the*y*-axis*.*(5 marks)

- Use the graph in (b) above to solve the equation
*y = 3*cos*x -*sin2*x = 2.*(2 marks) - State the ampitude of
*y = 3*cos*x - 2.*(1 mark)

- Complete the table below, giving the correct values correct to two decimal places.
- In the figure below, DA is the diameter of the circle ABCD centre O, radius 10 cm. TCS is a tangent to the circle at C, AB = BC and angle DAC = 38
^{o}.- F
- ind the size of angle:
- ACS; (2 marks)
- BCA. (2 marks)

- Calculate the length of:
- AC; (2 marks)
- AB. (4 marks)

- Two policemen were walking together at a road junction. each had a
*walkie talkie*. The maximum distance at which one could communicate with the other was 2.5 km.

One of the policemen walked due East at 3.2 km/h while the other walked North at 2.4 km/h. The policeman who headed East travelled for*x*km while the other one who headed North travelled for*y*km before they were unable to communicate.- Draw a sketch to represent the relative positions of the policemen. (1 mark)
- From the information above, form two simultaneous equations in
*x*and*y.*(2 marks) - Find the values of
*x*and*y.*(5 marks) - Calculate the time taken before the policmen were unable to communicate. (2 marks)

- From the information above, form two simultaneous equations in

- The table below shows the distribution of marks scored by 60 pupils in a test.

- On the grid provided, draw an ogive that represents the above information. (4 marks)
- Use the graph to estimate the interquartile range of this information. (3 marks)
- In order to pass the test, a pupil had to score more than 48 marks. Calculate the percentage of pupils who passed the test. (3 marks)

- Halima deposited Ksh 109 375 in a financial institution which paid simple interest at the rate of 8% p.a. At the end of 2 years, she withdrew all the money. She then invested the money in shares. The value of the shares depriciated at 4% p.a. during the first year of investment. In the next 3 years, the value of the shares appreciated at the rate of 6% every four months.
- Calculate the amount Halima invested in shares. (3 marks)
- Calculate the value of Halima shares:
- at the end of the first year; (2 marks)
- at the end of the fourth year to the next shilling. (3 marks)

- Calculate Halima's gain from the shares as percentage. (2 marks)

- The table below shows the values of x and some of the values of
*y*for the curve*y*=*x*^{3}+ 3*x*^{2}-4*x*-12 in the range -4 ≤*x*≤2.- Complete by filling in the missing values of
*y*.

(2 marks) - On the grid provided, draw the graph
*y*=*x*^{3}+ 3*x*^{2}-4*x*-12 for -4 ≤*x*≤2.

Use the scale Horizontal axis 2 cm for 1 unit and vertical axis 2 cm for 5 units. (3 marks)

- By drawing a suitable straigh line on the same grid as (b) above, solve the equation:
*x*^{3}+ 3*x*^{2}- 5*x*- 6= 0. (5 marks)

- Complete by filling in the missing values of