**PAPER 1**

**SECTION A**

1. Simplify:

2. Solve for **x** in the inequality:

3. Express: in the form . hence evaluate correct to 3 significant figure if

4. A trader made a 35% profit after selling a goat at shs45, 900. How much profit did the trader get?

5. Simplify: log 75 + 2 log 2 - log 3.

6. Find the value of **a** and **b** such that: =

7. A line of gradient ^{7/}_{9} passing through the point **Q** (3, 4), cuts the **y** - axis at a point **P**. find the co-ordinates of **P.**

8. The height of a small box is 2cm and its volume 10cm^{3}. If the height of a small box is 6cm, what is its volume?

9. The points; **A, B, C** and **D** are on the circumference, if a circle of centre **O** and <**ADC **= 30^{0}. Find the values of the marked angles **a **and **b**.

CIRCLE

10. Solve the equation:

**SECTION B**

11. Shown below are marks obtained by 50 candidates in a certain **S4** mathematics mock examination.

25 | 30 | 29 | 60 | 72 | 59 | 40 | 40 | 62 | 70 |

40 | 39 | 62 | 65 | 40 | 59 | 39 | 43 | 80 | 21 |

58 | 29 | 19 | 25 | 30 | 32 | 56 | 59 | 40 | 55 |

69 | 90 | 81 | 50 | 31 | 45 | 60 | 20 | 51 | 49 |

31 | 30 | 56 | 58 | 50 | 50 | 50 | 60 | 40 | 70 |

a) (i) construct a grouped frequency table having class intervals of 10 marks, beginning with the 15 - 24 class group.

(ii) use your grouped frequency table to calculate the mean mark of the mock examination.

b) Represent the above mock results on a histogram and use it to estimate the mode.

12. a) (i) Plot on a graph the triangle **ABC** whose vertices are (1, 1), (3, 2) and (2, 4) respectively.

(ii) one the same graph, enlarge the triangle **ABC** using (-1, -1) as the centre of enlargement and a scale factor of 2 to obtain its image **A'B'C'**.

(iii) state the coordinates of **A'B'C'** the image of triangle **ABC.**

b) using your graph, find the area of the triangle **ABC**. Hence determine the area of the triangle **A'B'C'.**

13. using a pair of compasses and ruler only:

a) (i)construct triangle **ABC**, such that = 10.0cm, = 9.2cm, <**ABC** = 105^{0},

(ii) measure length

b) (i)construct an inscribed circle of triangle **ABC **with center **O**,

(ii) measure the radius of the circle.

14. A poultry farm has three poultry units; **A, B** and **C**. unit **A** produces 30 trays of eggs and 20 broilers every month. Unit **B** produces 40 trays of eggs and 15 broilers and unit **C**, 35 trays of eggs and 10 broilers during the same period if a tray of eggs costs shs3,000 and a broiler shs4,000.

a) (i) represent the above information in matrix form of order 3 x 2 for the eggs and broilers.

(ii) form a 2x1 cost matrix produced on the farm for the eggs and broilers.

(iii) find the sales of the farm if all eggs and broilers were sold.

b) If the farm charges a 17% VAT, find the total income from the sales of the farm every month.

15. a) An FM radio commercial section charges fees for any radio announcements as follows:

The first ten words shs5,000 and any additional word shs100 each . Find the total charge for the announcement below:

"Mr. John Musoke, the chairman organizing committee of the wedding preparatory meetings of Mr. James Lima and Miss Vanessa Tukko announces the cancellation of the wedding meetings which were scheduled to begin on Wednesday, 11^{th} August, 2004 at Kalori Gardens, National theatre Kampala. Any inconveniences caused are highly regretted. A new date and venue for the meetings will be announced later."

b) Mr. Ronald Anguyo bought a car at shs4,500,000. The car depreciates at a rate of 12% per annum. After 2 years, Ronald decided to sell the car to his friend at 25% less of the value of the car then. Find the price at which his friend bought the car.

16. At a graduation party, the guests are to be served with beer and soda. At least twice as many crates of beer as crates of soda are needed. A crate of beer contains 25 bottles and a crate of soda contains 24 bottles. More than 200 bottles of beer and soda are needed. A maximum of shs500,000 may be spent on beer and soda. Assume a crate of beer costs shs40, 000 and that of the soda costs shs15,000.

a) (i) form inequalities to represent the above statements.

(ii) represent the above inequalities on the same axes.

(iii) by shading the unwanted region, represent the region satisfying the inequalities in (a)(i) above.

b) From your graph, find the number of crates of beer and soda that should be bought if the cost is to be as low as possible. Find the amount that was paid for these crates of beer and soda.

17. TRAPEZIUM

The figure **ABCD** shows a plot of land in form of a trapezium. Lengths = 6m, =21m and = 10m.

a) Find the:

(i) Length of the plot,

(ii) Area of the plot.

b) The diagram below shows road **AO** intersecting road **OB** at 90^{0} at point **O**. the two roads are also connected to **A** and **B** by an arc - like shaped road measuring a quarter of a circle 70m in radius.

SQUARE

Find the distance saved by a motorist who goes through the arc-shaped road instead of going through **AO** and **OB.**

**PAPER 2**

**SECTION A**

1. Simplify

(i)

(ii)

2. Solve for **x** in

3. An examination is marked out of 130 marks. If Rita obtained 60% in the examination, how many marks did she get out of 130?

4. Given that **P** = { (x, y): 2x - 3y ≤ 6} and **Q** = { (x, y): x + y < 0}, show by shading the unwanted region, the region representing **P****∩Q**.

5. Use logarithm tables to evaluate correct to 2 decimal places.

6. Evaluate 5600 + 80,000, leaving your answer in the form **a** x 10^{n} where 1 ≤ **a** < 10 and **n** is an integer.

7. In a homework marked out of 20, a group of pupils obtained the following marks: 15, 20, 18, 17, 8, 18, 16, 20, 18, 17, 12 and 19. Find the mode and median marks.

8. Under an enlargement of scale factor 3, the image of the point **P** (0, 3) is **P'** (4, 5). Find the coordinates of the centre of enlargement.

9. Express as a fraction in its simplest form.

10. A fair die is tossed only once and the number which appears on its top face noted. what is the probability of a top face showing

(i) A number greater than 4?

(ii) An odd number or prime number?

**SECTION B**

11. Draw graphs y = 2x^{2} + 3x - 3 and y - 7x + 3 = 0 for -3 ≤ x ≤ 3 using a scale of 1cm: 2 units for the vertical axis and 1 cm: 0.5 units for the horizontal axis. Using your graph find the:

a) Point of intersection of the line and the curve,

b) Gradient of the curve between the points of intersection of the line and the curve.

12. a) At a certain point on the level ground the angle of elevation of the top of a tower, **T** is 28^{0}. At another point 100 meters away from the first point, the angle of elevation is 35^{0}. Find the two expressions for the height of the tower hence find the height of the tower and give your answer to the nearest meter.

b) If cos x = -0.634 for 90^{0} < x < 270^{0}, find the two possible values of **x.**

13. A helicopter left Kampala at 0600hours and flew on a bearing of 090^{0}, at a velocity of 300 km per hour. It landed at Nairobi airport at 0830 hours. At exactly 0900 hours, it left Nairobi airport and flew on a bearing of 340^{0}, at the same original velocity. It then landed at Kitgum Airstrip at 1200 hours. Using graphical construction and a scale of 1cm: 100km, find the:

a) distance of kitgum from Kampala,

b) bearing of Kampala from kitgum.

14. Two cars **A **and **B** start off from rest at the same time moving in the same direction on a straight road. The speeds of the two cars in ms^{-1} are shown in the table below:

t (s) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |

speed of car A (ms^{-1}) |
0 | 4.5 | 9.0 | 13.5 | 18.0 | 22.5 | 27.0 |

speed of car B (ms^{-1}) |
0 | 2.0 | 5.0 | 10.5 | 23.0 | 27.0 | 28.5 |

Using a suitable scale, draw on the same axes the velocity time graphs of cars **A ** and **B**.

From your graph find the:

a) Time when the two cars have equal speed and the magnitude of that speed.

b) Difference in speed after a period of 9 seconds,

c) Distance covered by car **A** by way of estimating the area under the curve described by car **A **for the 12 seconds.

15. **OAB** is a triangle, **OA** = **a**, and **OB** = **b**. points **C** and **E** are points on lines and such that they divide the lines and in the ratios 1:2 and respectively. Point **D** lies on **OE** such that

TRIANGLE

a) Find the vectors **AB, OE** and **CB** in terms of vectors **a** and **b**.

b) Show that the points **B, D** and **C** lie on a straight line.

16. A man earns a gross annual income of shs10,500,000. He is entitled to the following monthly allowances:

Children | shs15,000 for each child aged 12 and below. shs12,000 for each child between age 13 and 18 inclusive |

Lunc | shs60,000 |

Transport | shs110,000 |

Medical | ^{1/10th} of gross monthly income |

Marriage | ^{1/25th} of gross monthly income. |

Housing | ^{1/}_{100}^{th}of gross annual income |

The man is married with five children of whom two are aged 12 and below, the other two aged 21 and 24 and the other aged 17.

The following tax structure is applicable on the taxable income in excess of shs30,000.

Taxable income (shs) | Rate (% ages) |

00001 - 30,000 | Free |

30,001 - 130,000 | 8.0 |

130,001 - 260,000 | 14.5 |

260,001 - 380,000 | 23.0 |

380,001 - 490,000 | 28.5 |

490,001 - 590,000 | 35.0 |

590,001 and above | 42.5 |

(NB: A month has 30 days and a year 360 days)

Calculate:

a) The man's

(i) Total monthly allowances,

(ii) Monthly taxable income,

(iii) Monthly income tax.

b) The percentage of his gross annual that goes to tax.

17. FIGURE

The figure above (in thick, heavy lines) shows a lamp shed **ABCD** bounded by circles of radii 15 cm and 25cm. the slanting side **AB** is 30 cm.

If the lamp shed was cut from an original figure **OABCD**, of a complete cone, calculate the:

a) (i) slanting length of the cone **OAB**,

(ii)the angle formed by producing **CD** and **BA** to **O**.

b) (i) vertical height of the lamp shed,

(ii) volume of the lamp shed.