**Instructions to candidates **

- Write your name, index and admission number in the spaces provided above.
- Sign and write the date of the examination in the spaces provided above.
- The paper contains TWO Sections: Section I and Section II.
- Answer ALL the questions in Section I and only five questions from Section II.
- All answers and working must be written on the question paper in the spaces provided below each question
- Show all the steps in your calculations, giving your answers at each stage in the spaces below each question.
- Marks may be given for correct working even if the answer is wrong.
- Non-programmable silent electronic calculators and KNEC Mathematical tables may be used except where stated otherwise,
- This paper consists of 16 printed pages.
- Candidates should check the question paper to ascertain that all the pages are printed as indicated and that no questions are missing.
- Answer all the questions in English.

FOR EXAMINER'S USE ONLY

SECTION I

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | Total |

SECTION II

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | TOTAL |

**QUESTIONS**

**SECTION I (50 marks)**

Answer all questions in this section in the spaces provided

1. Evaluate without using mathematical tables or calculators

(3 marks)

2. Solve the equation using completing the square method x^{2}+x-6=0 (2 marks)

3. Express:

in the form m+n√6 where m and n are whole numbers (3 marks)

4. Make h the subject of the formula

(3 marks)

5. Given the rectangle ABCD below, locate and shade a region within the rectangle in which a variable point P must lie given that P satisfies the following conditions:

- AP ≥BP
- ∠APB ≤ 90°
- P is closer to DC than BC
- AP > 7 cm (4 marks)

6. Expand and simplify

(2 marks)

Hence use the first four terms of the expansion to evaluate (1.98)^{5} to 5 significant figures (2 marks)

7. A coffee dealer mixes two types of coffee, type A and type B in the ratio 3:2. Type A costs sh. 72 while type B costs sh. 66 per kg. At what price should he sell the mixture per kg in order to make 5% profit? (3 marks)

8. State the Amplitude, the Period and the Phase angle of the function :

(3 marks)

9. A fair coin and a fair die are thrown together once. Find the probability of getting

- A tail and a 2 (1 mark)
- A head and a prime number (2 marks)

10. Two port cities Melboume in Australia and Santiago in Chile both lie 40° South of the Equator at 150°E and 75°W respectively.

- Calculate the shorter distance between them to the nearest nautical miles (2 marks)
- Hence, find the time in hours a ship sailing at 85 knots will take to sail from Melbourne to Santiago (2 marks)

11. A variable y varies as the square of x and inversely as the square root of z. Find the percentage change in y when x is increased by 10% and a decreased by 36% (3 marks)

12. The matrix below is a singular matrix. Find the value of x.

(3 marks)

13. The 2^{nd}, 4^{th} and the 7^{th} term of an Arithmetic Progression (AP) form the first 3 consecutive terms of a Geometric Progression. Find the common ratio of the G.P if the first term of the AP is 2. (3 marks)

14. Find the radius and the co-ordinates of the centre of the circle whose equation is

(3 marks)

15. The diagram below shows a sketch of the curve y = x^{2}(x - 2). Calculate the area enclosed by the curve, the x-axis and the lines x = 1 and x = 3.

(3 marks)

16. The length of two specimens in a research laboratory are stated as a = 0.12 cm and b = 0.5cm

Calculate the maximum possible value of a + b . (3 marks)

a**SECTION II (50 marks)**

Answer any five questions in this section in spaces provided.

17. The table below shows the Kenya tax rates in the year 2010.

Income (Ksh. per month) | Tax rate (per Ksh) |

0-9680 | 10% |

9681 - 18800 | 15% |

18801 - 27920 | 20% |

27921- 37040 | 25% |

Over 37040 | 30% |

In that year, Wandera earned a basic salary of Ksh 30,000 per month. In addition, he was entitled to 15% of the basic salary as house allowance, a medical allowance of Ksh 2,800 per month and a commuter allowance of Ksh 1800 per month. Wandera is entitled to a monthly tax relief of Ksh 1,162.

Calculate:

- Wandera's monthly taxable income. (2 marks)
- The tax paid by Wandera per month (5 marks)
- Other monthly deductions were union dues Ksh 445, WCPS Ksh 490, NHIF Ksh 320, COOP shares Ksh 1000 and risk fund Ksh 100. Calculate his net income per month (3 marks)

18. The figure below is of a right pyramid with a rectangular base ABCD and a vertex T.

TC TB = TA = TD. The area of the base is 60 cmand the volume of the pyramid is 280 cm^{3}. AB=10 cm

Calculate correct to 2 decimal places:

- The vertical height OT of the pyramid (2 marks)
- The length of the line
- AC (2 marks)
- TC (1 mark)

- The angle between the edge TC and the plane ABCD (2 marks)
- The angle between the planes TCB and TAD (3 marks)

19. (a) Complete the table below giving your values correct to 2 decimal places (2 marks)

xº | 0º | 30º | 60º | 90º | 120º | 150º | 180º |

2cos (2x-15) | 1.93 | -1.93 | -1.41 | ||||

3sin(x +30) | 1.50 | 3.00 | 2.60 | 0 |

b) Using the grid provided, draw on the same axis the graph of y-2cos (2x-15) and y = 3sin (x+30) for 0° ≤ X ≤ 180°

Scale: x-axis : 2 cm represents 30°

y-axis : 2 cm represents 1 unit

(5 marks)

c) Use the graph in (b) above to solve the equation 3 sin(x+30) = 2 cos (2x-15) (2 marks)

d) Find the range of values of x for which 3 sin(x+30) ≥ 2 (1 mark)

20. The table below shows the average marks scored by form two students in a test.

Marks | 35-39 | 40-44 | 45-49 | 50-54 | 55-59 | 60-64 | 65-69 | 70-74 |

Frequency | 3 | 2 | 5 | 10 | 12 | 11 | 5 | 2 |

Using an assumed mean of 54, calculate the :

- mean mark (5 marks)
- Variance (4 marks)
- Standard deviation (1 mark)

21. Triangle ABC has vertices A(3,-2) B(4,3) and C(-3, 3). On the grid provided below, draw triangle ABC. (1 mark)

- Point A is mapped on to A'(3, 4) by a shear y-axis invariant. On the grid above, draw triangle A'B'C' under the shear. (2 marks)
- Determine the matrix representing the shear (2 marks)
- Triangle A'B'C' is mapped onto triangle A"B"C" by the transformation matrix
- State the coordinates of triangle A"B"C" (2 marks)
- Draw triangle A"B"C" (1 mark)

- Find a single matrix that maps triangle A"B"C" onto triangle ABC (2 marks)

23. In the figure below, O is the centre of the circle. BC and DC are tangents to the circle.

∠BCO=25º. OEC is a straight line.

Find the value of the following angles, stating the reason in each case:

- ∠BOC(2 marks)
- ∠OED(2 marks)
- ∠CDE(2 marks)
- ∠BED(2 marks)
- Reflex ∠DAE(2 marks)

24. A large car park in Kikuyu town has an area of 1400m with space for x cars and y vans. Each car

requires 14m- of space and each van requires 35m- of space. There must also be space for at least 50 vehicles. The parking space for cars must be more than 25 and at least 20 for vans.

- Write all the inequalities to represent the above information (4 marks)
- On the grid provided, draw the inequalities in part (a) above and shade the unwanted region. (4 marks)
- The company charges sh. 300 for parking each car and sh. 500 for parking each van. Find the number of cars and the number of vans which would enable the owners of the car park to break even the minimum number of cars and Vans ) (1 mark)
- Hence find the possible minimum income. (1 mark)

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