**PAPER 1**

**SECTION A**

1. If **a * b**= , evaluate

2. Make c the subject from the expression: **a = b** -

3. The point **R**( 10, 7) is reflected in the line **y** = **x** to give point **S**. given that **M** is the mid point of **RS**, find the co-ordinates of M.

4. Find the area of a triangle whose sides are 13 cm, 24 cm and 13cm.

5. Given the sets:

**A** = {all natural numbers less than 30}

**B =** {all prime numbers between 10 and 30}

Find:

a) n(A∩B')

b) n(A'∩B)

Where **B' **stands for the complement of the set **B.**

6. If , find the values of k and n.

7. Use the prime factor method to find the cube root of 3375.

8. In a revenue authority department, the tax earned income is calculated as follows:

The first shs120,000 is tax free and the remaining income is taxed at 25%. Find the tax payable on an earned income of:

a) Shs100,000

b) Shs440,000

**SECTION B**

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

9. a) Given that , solve for **x.**

b) Solve the simultaneous equations x^{2} + 4y^{2} = 4

y = x - 1

10. Using a pencil, a ruler and a pair of compasses only, construct a triangle **ABC** in which =9.2cm, angle **CAB **= 45^{0} and angle **ABC** = 75^{0}.

a) Measure the length of

b) Draw a circumscribing circle through the points **A, B** and **C**.

c) Measure the radius of the circle.

11. a) in the figure below, vectors **OA** = **a** and **OC** = **c,** =^{1/3 }and = 3

FIGURE

(i) by expressing vectors in terms of **a** and **c**, find **OD,** **AB** and **OB**

(ii) Show that points **O, D** and **B** are collinear.

b) Points **A** and **B** have co-ordinates (0, -1) and (-6, 7) respectively.

Find:

(i) **AB**,

(ii) The magnitude of **AB**

12. In a certain school, a sample of 100 students was picked randomly. In this sample, it was found out that 78 students play Netball (**N**), 82 play volley ball (**V**) 53 play tennis (**T)** and 2 do not play any of the three games. All those that play Tennis also play volleyball. 48 play all the three games.

a) Represent the given information on a Venn diagram.

b) How many students play both netball and volleyball but not tennis?

c) If a student is picked at random from sample, what is the probability that the student plays two games only?

13. a) Draw a table showing the values of sin 2**θ** for 0^{0}≤**θ**≤90^{0}, using values of **θ** at intervals of 15^{0}.

b)use the table in (a) above, a horizontal scale of 2cm for 15^{0} and a vertical scale of 2cm for 0.5 units to draw a graph of sin 2 **θ**,

c) From the graph, find the values of θ for which sin 2 **θ** = 0.6.

14. A manager of an industry earns a gross salary of shs2,000,000 per month, which includes an allowances of shs500,000 tax free. The rest of her income is subjected to an income tax which is calculated as follows:

7.5% on the first shs800,000

12.5% on the next shs500,000

20% on the next shs100,000

30% on the next shs60,000

35% on the remainder.

a) Find her taxable income

b) Calculate her monthly income tax

c) Express her monthly tax as a percentage of monthly gross salary.

15. A school constructed an office which required 34 tones of sand the school hired a lorry and a tipper truck with capacities of 7 tones and 5 tones respectively to transport the sand. The cost per trip either by lorry or by tipper truck was shs30, 000. The money available for transportation was shs180, 000. The trips made by the lorry did not exceed those made by the tipper truck.

a) If x and y represent the number of trips made by the lorry and the tipper truck respectively;

(i) Write down five inequalities to represent the given information.

(ii) Plot these inequalities on the same axes, shading the unwanted regions.

b) (i) From your graph in (a)(ii) above, list all the possible numbers of trips, that each vehicle can make so as to maximize the total tonnage of sand transported.

(ii) Find the number of trips by each vehicle that made the greatest total tonnage.

16. The figure below shows a cuboid **ABCDEFGH** in which **BC** = 8cm. **BF** = 6 and **CD** = 5cm. **K** is the mid point of **AB.**

CUBOID

Find the:

a) (i) length **AG**

(ii) Angle which **AG** makes with the plane **ABCD.**

b) Angle between planes **KGH **and **FGHE**.

**PAPER 2**

**SECTION A**

1. Express as an improper fraction in its simplest form.

2. If a = 14, b = 8 and , find the value of **c**.

3. A line is given by the equation 45- 15x + 3y = 0. Find the co-ordinates of its **x** - intercept.

4. Given that **f(x)** = 2x + 4 and **g(x**) = x + 5, find **fg(x).** Hence evaluates **fg(4).**

5. Expand the expression;

6. A butcher sells 5kg of meat at shs10, 000, if the cost of meat is increased by 20%, find the weight of meat which can be bought at shs3, 600.

7. The data given below represents the ages in years of 30 senior four students of a certain school:

Age class | 15 - 17 | 19 - 20 | 21 - 23 | 24 - 26 |

Number of students | 7 | 11 | 9 | 3 |

Use the table above to draw a histogram and state the modal class.

8. Triangle **ABC** with vertices **A** (0, 0), **B** (1, 0) and **C** (1, 1) underwent two transformations represented by **T _{2}T_{1}**. If

**T**is a translation represented by and

_{1}**T**

_{2}is a reflection in the

**x**- axis, find the co-ordinates of the final image of the triangle.

9. Given **A** = and **B** = , evaluate (A +B)^{2}

10. Study the diagram below:

TRIANGLE

If **AD** = 12cm. find the area of the shaded region.

11. Given that **V** is inversely proportional to **t ^{2}** and

**V**= 25 when

**t**= 2, find

**V**when

**t**= 5.

12. The figure below shows a net of a cone which can be folded to form a right circular cone.

FIGURE

Calculate the radius of the cone formed.

**SECTION B**

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

13. a) Given that 212_{n} = 25_{nine}, find the base that n represents.

b) A positive integer **r** is such that pr^{2} = 168, where p is such that 3 ≤ p ≤ 5. Find the integral values of **r**.

14. a) Find the length marked **x** in the diagram below correct to two significant figures.

FIGURE

b) A dog tied by a silk rope 4.5 m long is tethered to a tree stamp 2.5 from a straight path. For what distance along the path is one in danger of being of being bitten by the dog?

15. By shading the unwanted regions, show the region which satisfies the inequalities:

X + y ≤ 3

Y > x - 4

Y + 7x ≥ - 4

Find the area of the wanted region.

16. The table below shows the weight in kilograms of 28 children sampled in a primary school:

**Weight (kg) number of children**

15 - 19 2

20 - 24 4

25 - 29 7

30 - 34 3

35 - 39 5

40 - 44 6

45 - 49 1

a) State the modal class.

b) Calculate the cumulative frequency and

(i) Hence, estimate the median weight correct to one decimal place,

(ii) Calculate the mean weight of the children,

(iii) Find the probability that a child selected at random from the school weighs 40kg and above.

17. Musa is a businessman who deals in an agricultural produce business, he visited four markets in a certain week:

In market **A** he bought 3 bags of beans, 5 bags of maize, 10 bags of potatoes and 3 bags of millet,

In market **B**, he bought 1 bag of beans, 4 bags of potatoes and 2 bags of millet,

In market **C** he bought 5 bags of beans, 1 bag of maize,

In market **D** he bought 4 bags of beans, 3 bags of maize, 6 bags of potatoes and 1 bag of millet.

He bought each bag of beans at shs45, 000, a bag of maize at shs30,000, a bag of potatoes at shs15,000 and a bag of millet at shs50,000. He later sold all the produce he had bought at shs50,000 per bag of beans; shs35,000 per bag of maize, shs18,000 per bag of potatoes and shs55,000 per bag of millet.

a) Form a 4x4 matrix to show the produce musa bought from the four markets.

b) (i)form a cost matrix for the price of the produce,

(ii) By matrix multiplication, find the amount of money spent on the produce in each market.

c) Find also the amount of money he got from the sale of the produce

d) Find Musa's profit.

18. Town A is 170km from town **B**. a Tata lorry left town **B** for town **A** at 8.25 am. And travelled at a steady speed of 40 kmh^{-1}_{. }A saloon car left town **A** for town **B** at 8.25am and travelled at a steady of 80kmh^{-1}.

a) Calculate the:

(i) Distance from town **A** to the point at which the two vehicles met.

(ii) Time at which the two vehicles met.

b) Just as they met, the Tata lorry driver increased the speed by 10kmh^{-1}. Find the difference in their times of arrival at their destinations.

19. The figure **QRSTUV** below is a plan of Mr. Rukidi's farm. The area marked **A** is in form of an equilateral triangle, area **B** is rectangular and **C** is a semi circle. =14cm and =100m.

FIGURE

Find the:

a) Length which divides the farm into two equal parts,

b) Area of the farm,

c) Length of barbed wire required to fence Rukidi's farm.