**PAPER 1**

**SECTION A**

1. If 125_{n} = 85_{ten}, find n.

2. In a group of 29 girls, 22 liked Rice (**R**) and 18 liked Matoke (**M**). All girls liked at least one of the foods. How many liked both?

3. Solve the inequality: 10 - 3x < 4(x - 1).

4. Without using mathematical tables or a calculator, evaluate:

5. Two quantities **y** and **x** are related by the equation y = a + bx. When y = 4, x = 2 and when y = 6, x = 4. Find the values of **a** and **b**.

6. Given that sin a = ^{3/}_{5} and **a** is obtuse, without using mathematical tables or calculator, find the values of **cos** **a** and **tan** **a**.

7. A shop keeper bought an item at shs5,500 and sold it at 30% more than the buying price. Find the shopkeeper's.

a) Selling price

b) Profit

8. Given the matrix **P** = , find **P ^{2}**.

9. Use tables of logarithms to evaluate:

10. Solve the following pairs of simultaneous equations

5x - 9y = 1,

4y - 2 = x.

**SECTION B**

*Answer any five questions from this section, all questions carry equal marks.*

11. The following table shows marks obtained by 40 pupils in a mathematics test.

11 | 17 | 35 | 34 | 42 | 45 | 28 | 46 |

16 | 12 | 14 | 36 | 41 | 31 | 49 | 37 |

20 | 33 | 37 | 38 | 18 | 38 | 39 | 27 |

26 | 28 | 40 | 33 | 43 | 32 | 29 | 47 |

29 | 32 | 41 | 24 | 44 | 35 | 36 | 23 |

a) Draw a frequency distribution table for the marks, starting with a class of 10 - 14.

b) State the:

(i) Class interval,

(ii) Modal class.

c) Calculate the:

(i) Mean mark,

(ii) Median mark.

12. a) (i) Determine the range corresponding to the domain.

(-3, -2, 0, 1, 3, 4) for the mapping x →x^{2} + 1.

(ii) Represent the mapping in (i) on an arrow diagram.

b) Given the functions h(x) = x + 2, g(x) = x^{2} and f(x) = -x; find the values of **x** for which g [h(x)] = f(x).

13. A triangle with vertices **A **(2, 4), **B **(6, 4) and **C** (1, 6), undergoes two successive transformations **P _{1}** followed by

**P**The transformation

_{2}.**P**is represented by the matrix and

_{1 }**P**by the matrix

_{2}a) find the co-ordinates of the vertices of:

(i) Triangle **A'B'C'** the image of **ABC** under **P _{1}**

_{.}

(ii) Triangle **A"B"C"** the image of **A'B'C'** under **P _{2}**.

b) Show on the same axes the three triangles **ABC**, **A'B'C'** and **A"B"C".**

c) Use your graph in (b), to describe fully the transformations represented by

**(i) ****P _{1},**

**(ii) ****P _{2}.**

14. a) A student had his rectangular photograph of dimensions 30 cm by 20 cm framed with a uniform border. If the area of the border is 216 cm^{2}, how wide is the border?

b) A cone has a radius of 7 cm and a vertical height of 30 cm. find:

(i) Its volume,

(ii) The volume of another similar bigger cone which has a linear scale factor of 2.

15. a) Find the equation of a line passing through a point (2,0) and perpendicular to the line joining the points (-10,3) and (6, -9)

b) A triangle **PQR** has vertices with coordinates **P**(3,-1), **Q**(7,6) and **R**(0,2). Find the equation of its line of symmetry.

16. A hawker sells handkerchiefs at shs500 each. He sold 50 handkerchiefs in the first week. In the second week he sold 20% more than in the first week. In the third week he sold 10% more than in the second week. Each week he receives a commission of 8% on the price of the first 20 handkerchiefs sold and 12% for any handkerchiefs sold in excess of 20.

a) Express the number of handkerchief sold in the third week as a percentage of the number sold in the first week.

b) Calculate the commission he received in the third week.

c) If in the fourth week the hawker received a commission of 2,000/= calculate the number of handkerchiefs he sold in that week.

17. In the diagram below, **OA** = **a**, **OB** = **b**, =1:3, 3**OF **= 2**OA** and **E** divides **AC** in the ratio 3:2.

TRIANGLE

Express the following vectors in terms of **a** and **b**.

**a) ****BC.**

**b) ****CA.**

**c) ****BE.**

**d) ****FE.**

**PAPER 2**

**SECTION A**

1. Express 0.341666... in the form ^{p}_{/q,}_{ }where **q** ≠0.

2. Solve for **x **in 32^{3/}_{5} ÷ **x**_{}^{1/}_{2} = 2.

3. Given two points **P** (4, 5) and **Q** (-2, 9), find the equation of the line through **P** and **q.**

4. Simplify . Give your answer in the form where **a** and **b **are constants.

5. A rectangle 6cm long and 5cm wide is enlarged so that its area becomes 270cm^{2}. Find the linear scale factor of the enlargement.

6. In the figure below, **O** is the centre of the circle, angle **JKQ** = 40^{0} and **KOQ** is a straight line.

CIRCLE

Find the angles marked **n** and **p**.

7. Given that **a** = , **b** = and **m** = **a + 2b**, find the magnitude of **m**.

8. If **n = x **, express m in terms of **n** and **x**.

9. A function f(x) = . Find the values of **x** for which f(x) = 4.

10. Three girls, Auma, Assimwe and Nakato shared shs10, 500. Nakato got twice as much as Assimwe and Assimwe got twice as much as Auma. Find how much money Assimwe got.

**SECTION B**

*Answer any five questions from this section. All questions carry equal marks.*

11. A speed boat sets off from an island **M** on a bearing of 080^{0} to an island **X **at an average speed of 150kmh^{-1}, island **X** is 450 km from island **M**. at **X** it alters its course to a bearing of 200^{0} and maintains the average speed of 150kmh^{-1} for 3 hours until it reaches island **Y**. it then moves to island **P** is 400 km from island **M**.

a) Using a scale of 1 cm to represent 50 km, construct a scale drawing to show the route of the speed boat.

b) Use the scale drawing in (a) to find the distance **PY**.

c) Calculate the

(i) Total time taken for the speed boat to move from **M **to** P**.

(ii) Speed boat's average speed for the whole journey.

12. The Venn diagram below shows the members of a district council who sit on three different committees of works (**W**), production (**P**) and Finance (**F**).

VENN DIAGRAM

a) Determine the value of **x, y** and **z**.

b) Find the total number of members who

(i) Make up the district council.

(ii) Belong to more than one committee.

c) Given that a member is selected at random from the district council, find the probability that the member belongs to only two committees.

13. a) express in the form

b) A mini bus travels from migyera to Kampala, a distance of 156km, at a certain average speed of **V**km/hr. on the return journey, it increases the average speed by 4 km/hr and takes 15minutes less. Find the average speed **V** from Migyera to Kampala.

14. The table below shows time (**t**) in seconds and velocity (**V**) in m/s of an object.

t(s) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

V(m/s) | 0.0 | 1.0 | 1.7 | 2.0 | 1.7 | 1.0 | 0.0 |

a) Using a scale of 2cm to represent one second on the horizontal axis and 4cm to represent 0.5 m/s on the vertical axis, plot the values of **t** and **V **and join the points with a smooth curve.

b) Use your graph in (a), to find the

(i) Times at which the speed of the object is 0.8 m/s.

(ii) Acceleration of the object when the time is 2 seconds.

c) If the total distance covered by the object was 7.5m, what was its average speed?

15. a) without using mathematical tables or a calculator, find the value of

2 log _{10} 50 + log _{10} 80 - log _{10} 2.

b) (i) find the prime factors of 150.

(ii) Using your result in (i), find log _{10}150, given that

Log _{10} 5 = 0.6990, log _{10} 3 = 0.4771 and log _{10} 2 = 0.3010.

16. a) solve the following simultaneous equations using the matrix method

5x + 2y = 5,

3x - 0.2y = 10.

b) Given that **P**= , **Q**= and **R** = ;

Find:

**(i) ****QR - P,**

(ii) The determinant of **QR - P.**

17. Mr. Oketcho's monthly gross salary is shs900,000 which includes the following allowances:

Shs

Water and electricity 20,000

Relief and insurance 30,000

Housing allowances 50,000

Medical allowance 25,000

Transport allowance 28,000

Marriage allowance 20,000

Family allowance

(for only 4 children):

- From 0 to 9 years 20,000 per child

- Between 9 and 16 years 15,000 per child

- Over 16 years 10,000 per child

Mr. Oketcho has five children; two of whom are aged between 0 and 9 years, one aged 14 years and the other two are over 16 years.

The income tax structure is shown in the table below:

Taxable income per month in shillings. | Tax rate % |

01 - 50,000 50,001 - 110,000 110,001 - 200,000 200,001 - 350,000 350,001 - 600,000 Above 600,000 |
10.0 20.0 24.5 35.0 40.0 49.0 |

a) Calculate Mr. Oketcho's

(i) taxable income

(ii) Income tax

b) Express the income tax as a percentage of his monthly gross salary.