**PAPER 1**

**SECTION A**

1. Simplify

2. If , find the values of **a **and **b.**

3. **ABCD** is a quadrilateral in which angles **ABC** and **CDA** are 90^{0} each. If =6cm, = 10cm and = 5cm, find

a) length

b) Angle **ACD.**

4. Factorize x^{3} - 9xy^{2}completely.

5. Given that **A** = and **B**= , find

a) the matrix **P** such that **AB** =** P**,

b) ** P**^{-1}

6. Use the fact that log_{10}2 = 0.301 and x = 4, to find the value of log _{10}x^{2}.

7. Given that f(x) = , find the value of **x** for which f(x) = 17.

8. find the equation of the line passing through the points )-1, 3) and (4, 2)

9. A food store has enough food to feed 200 students for 15 days. For how long will the food last if 50 more students join the group

10. The pie chart below represents the number of students who attend various courses in a commercial college.

CIRCLE

If the number of students studying accountancy is 120,

a) Determine the student population of the college.

b) Find the number of students who study marketing.

** **

**SECTION B**

11. a) The diagram below shows a rectangle **ABCD** of length 44 cm and width 15cm.

RECTANGLE

If it was curved in such a way the **AD** and **BC** come together to hollow cylindrical figure, find the volume of the cylindrical figure formed.

b) A rectangular piece of cardboard measuring 27 cm long and 15 cm wide rests against a vertical wall as shown in the diagram below.

RECTANGLE

If angle DAY =25^{0}, find the height of **C **above the ground.

12. Using a pair of compasses and ruler only,

a) construct triangle **ABC** such that = 10.6 cm and angles **ACB** = 75^{0} and **ABC** = 60^{0},

b) construct a circumcircle of triangle **ABC **with **O** as its centre,

c) Measure lengths and the radius of the circle.

13. Three secondary schools football teams **X, Y** and **Z** qualified for a football tournament, which was played on two rounds with other teams. In the first round:

Team **X** won one game, drew one and lost three games.

Team **Y** won three and lost two games.

Team **Z** won two drew two and lost one game.

In the second round:

Team **X** won two drew two and lost one game.

Team **Y** won four and drew one game while team **Z **won three games, drew one and lost one.

a) write down

(i) A 3x3 matrix to show the performance of the three teams in each of the two rounds.

(ii) A matrix which shows the overall performance of the teams in the two rounds.

b) If three points are awarded for a win, two points for a draw and no point for a loss, use matrix multiplication to determine the winner of the tournament.

c) Given that shs1, 475,000 is to be shared by the three teams according to the ratio of the points scored in the tournament, find how much money each team will get.

14. The distance from town **A** to town **B** is 360 km. An express bus leaves town **A** at 6.30 a.m. And travels at a steady speed of 80kmh^{-1} towards town **B.** at the same time, a taxi omnibus leaves town **B** travelling non-stop towards town **A**, at a steady speed of 100km^{-1}. On the same axes draw a distance time graph for the journey of two vehicles. Use a scale of 2cm to represent 1 hour and 2 cm to represent 50kmh^{-1}. From the graph

a) Find the difference in the time of arrival of the bus and the taxi.

b) Determine when, and at what distance from town **A** the two vehicles will meet.

15. A packet has 60 different vitamin tablets. Each tablet contains at least one of the vitamins **A, B** and **C**. twelve of the tablets contain only vitamin Am seven contain vitamin **B** only and eleven contain only vitamin **C**; six contain all the three vitamins.

Given that n(A'∩ B ∩ C) = n(B' ∩ A ∩ C) = n(C' ∩A ∩ B), find the

a) number of tablets that contain vitamins **A**,

b) probability that a tablet picked at random from the packet contains vitamins **C,**

c) Probability both vitamins **A** and **B**.

16. The table below shows the weight (in kg) of 40 students of a class and their corresponding cumulative frequencies.

Weight (kg) cumulative frequency

30 - 34 2

35 - 39 7

40 - 44 12

45 - 49 21

50 - 54 28

55 - 59 34

60 - 64 38

65 - 69 40

a) Draw a cumulative frequency curve. Use your graph to estimate the

(i) median weight of the students,

(ii) 25^{th} and 75^{th} percentile weights.

b) Calculate the mean weight of the students.

17. A private car park is designed in such a way that it can accommodate x pickups and y mini-buses at any given time. Each pick up is allowed 15m^{2 }of space and each mini-bus 25m^{2 }of space. There is only 400m^{2} of space available for parking. Not more than 35 vehicles are allowed in the park at a time. Both types of vehicles are allowed in the park, but at most 10 mini-buses are allowed at a time.

a) (i) Write down all the inequalities to represent the above information.

(ii) On the same axes plot graphs to represent the inequalities in (i) above. Shading out the unwanted regions.

b) If the parking charges for a pick up is shs500 and that for a minibus is shs800 per day, find how many vehicles of each type should be parked in order to obtain maximum income. Hence find the maximum parking income per day.

**PAPER 2**

**SECTION A**

1. Express 784 as a product of prime factors. Hence find the square root of 784.

2. If the exchange rate of a Kenya shilling to Uganda shilling is 1 K.shs=24 Ushs. And an American dollar to Uganda shilling is $1=ush1, 950 how many American dollars would one get in exchange for Ksh9, 750?

3. In the diagram below, is a tangent to the circle with centre **O** and angle **BAO **= 30.

CIRCLE

Find the size of the angles **x** and **y**.

4. Given that the representative fraction of a map is , find the length of a horizontal road on the map whose length on the ground is 66.25 km long.

5. The transformation described by the matrix maps the point **P** (-1, 3) onto its image **P"** (10, 8). Find the values of **b** and **c**.

6. In the figure below, = 8.0cm, =12.0cm and =5.0 cm.

TRIANGLE

Given that is parallel to , find length .

7. Solve the equation 3x^{2} + 10x = 8

8. Given that 133_{n }= 43_{ten}, find the value of **n**.

9. A fair doe with faces marked 1, 2, 3, ....6 and a fair coin with one side showing a court of arms (c) and the other side a fish (F) are tossed together once.

a) Construct a possibility space showing all the possible outcomes.

b) Find the probability that a six and a fish will show up.

10. The angle of depression of the sun's rays to a man's head is 14^{0}. If the man, whose height is 1.7 m, is standing upright on horizontal ground, find the length of his shadow, correct to 2 significant figures.

** **

**SECTION B**

11. a) At the beginning of the year 2000, a customer deposited shs1, 900,000 in a bank which offers a compound interest rate of 2.75% per four months. Find how much interest he earned at the end of the year.

b) a cooking oil factory offers a trade discount of 2% to its customer. It also offers a 1% cash discount to any customer who pays cash for the oil bough. If the factory price for a 20-litre jerrican of cooking oil is shs30, 000. Find the amount of money a customer saves by paying cash for 100 jerricans of the oil.

12. The coordinates of the vertices of a triangle **OAB **are **O** (0, 0), **A** (1, 0) and **B** (1, 1).

a) find the coordinates of the image formed when

(i) Triangle **OAB** undergoes a translation of to form **O'A'B'.**

(ii) O'A'B' is transformed by the matrix to form **O''A''B''**.

b) (i) plot triangle **OAB** and its images on the same graph.

(ii) use your graph to find the area of triangle **O''A''B''.**

13. Two cyclists **C _{1}**

_{ }and

**C**start at the same time from trading centre

_{2 }**P**travelling to trading centre

**Q**which is 24 km apart. Cyclist

**C**starts at a steady speed of 10kmh

_{1}^{-1}greater than that of cyclist

**C**who also travels at a steady speed. When

_{2}**C**has covered half the distance, he delays for three quarters of an hour, after which he travels at a speed 25% less his original speed and arrives in trading centre

_{1}**Q**fifteen minutes earlier than cyclist

**C**.

_{2}_{a) }determine the speeds of cyclists **C _{1}** and

**C**

_{2}b) If cyclist **C _{2}** started from trading centre

**Q**at the same time as cyclist

**C**started from trading centre

_{1 }**P**, both of them travelling non stop on the way, find how far from

**Q**the two cyclists would meet. After how long would they meet?

14. a) Plot the graph y = 3x^{2} + 2x - 16 of for values of

b) Use your graph to solve the equation 3x^{2} + 2x - 8 = 0.

15. The bearing of tower **A** from point **O** is 060^{0 }and that of tower **B** from **O**, 200^{0}. =24km and =33km. Tower **C** is exactly halfway between towers **A** and **B**.

a) Using a scale of 1 cm to represent 5km, draw an accurate diagram showing the positions of the towers.

b) use your diagram to find

(i) distances and .

(ii)the hearing of tower **B** from tower **A**,

(iii)the bearing of tower **C** from **O**.

c) Find (i) the average speed of a cyclist who takes 2^{1/}_{4} hours to travel directly from **A** to **B**

(ii) How long it takes another cyclist to travel from **A** to **B** via **O** at a steady speed of 4.5kmh^{-1} faster than that of the cyclist in (c)(i) above.

16. The diagram below shows a quadrilateral **OSRQ. OS** = **q**, **OP** = **p** and **SX** = **k (SP)**

QUADRILATERAL

(i) Express vectors **SP** and **OX** in terms of **p, q** and **k**.

(ii) If **OQ** = 3p and **QR** = 2**OS**, and **OX** = ** lOR**, find the values of

**k**and

**. hence find the ratio**

*l***SX: XP**.

17. In the diagram below **VABCD** is a rectangular base **ABCD** and **V**, the vertex. **O **is the centre of the base **ABCD**. = 8m. =6m. =13m. **M** is a point on **VO** such that . **M** is also the centre of base **EFGH** of a small pyramid **VEFGH** similar to **VABCD** which is to be cut off from the original pyramid **VABCD.**

PYRAMID

Find the:

(i) Dimensions of the base **EFGH**.

(ii) Height of pyramid **VABCD.**

(iii) Volume of the remaining part of pyramid **VABCD** when **VEFGH** is cut off.