**Kenya Certiﬁcate of Secondary Education**

2019 Mathematics Paper 1

Section I (50 marks)

**Answer all the questions in this section.**

1. Without using mathematical tables or a calculator, evaluate: 5.4∕ 0.025 x 3.6 (3 marks)

2. Express 1728 and 2025 in terms of their prime factors. Hence evaluate:(4 marks)

∛1728 √ 2025

3. Juma left his home at 8.30a.m. He drove a distance of l40km and arrived at his aunt’s home at 10.15 a.m.

Determine the average speed, in km/h, for Juma’s journey.(3 marks)

4. Expand and simplify:

4 (q – 6) + 7 (q — 3).(2 marks)

5. In the trapezium PQRS shown below, PQ = 8 cm and SR = 6 cm.

If the area of the trapezium is 28 cm², find the perpendicular distance between PQ and SR. (2 marks)

6.Given that = ∛ 9^{4}=3^{n} find the value of n.(3 marks)

7. Three villages A, B and C rife Such that B is 53 km on a bearing of 295° from A and C IS 75 kin east of B.

(a) Using a scolc of 1 cm to represent 10 km, draw a diagram to show the relative positions of villages A, B and C. (2 marks)

(b) Determine the distance, in km, of C from A. (2 marks)

8. A retailer bought a bag of tea leaves. If the retailer were to repack the tea leaves into smaller packets of either 40 g, 250g or 350 g, determine the least mass, in grams, of the tea leaves in the bag.(3 marks)

9. Given that sin 2x = cos (3z — 10°), find tan z, correct to 4 significant figures. (3 marks)

10. A tourist converted 5820 US dollars into Kenya Shillings at the rate of Ksh 102.10 per dollar.

While in Kenya, he spent Ksh450 000 and converted the balance into dollars at the rate of Ksh 103.00 per dollar.

Calculate the amount of money, to the nearest dollar, that remained.(3 marks)

11. Given that b =

(2) |

(4) |

C,=

(3) |

(2) |

and a = 3c — 2b, find the magnitude of a, correct to 2 decimal (4 marks)

12. Using a ruler and a pair of compass only, construct a rhombus PQRS such that PQ = 6cm and dSPQ = 75°.

Measure the length of PR.

13. Solve the inequality 2x — 1 ≤ 3x + 4 < 7 — x. (3 marks)

14. Given that A = ( 2 3 4 4 ) B, ( X 1 2 3 ) and that AB is a Singular matrix, find the value ofx.(3 marks)

15. A trader bought two types of bulbs A and B at Ksh 60 and Ksh 56 respectively.

She bought a total of 50 bulbs of both types ct a total of Ksh 2872.

Determine the number of type A bulbs that she bought. (3 marks)

16. A bus plies between two towns P and R via town Q daily. On each day it departs from P at 8.15 a.m. and stops for 40 minutes at Q before proceeding to R.

On a certain day, the bus took 5 hours 40 minutes to travel from P to Q and 3 hours 15 minutes to travel from Q to R.

Find, in 24 hour clock system, the time the bus arrived at R. (3 marks)

**SECTION II (50 marks)**

17. A rectangular water tank measures 2.4 m long, 2 m wide and 1.5 m high.

The tank contains some water up to a height of 0.45 m.

(a) Calculate the amount of water, in litres, needed to fill up the tank (3 marks)

(b) An inlet pipe was opened and water let to flow into the tank at a rate of 10 litres per minute.

After one hour, a drain pipe was opened and water allowed to flow out of the tank at a rate of 4 litres per minute.

Calculate:

(i) the height of water in the tank after 3 hours;(4 marks)

(ii) the total time taken to fill up the tank. (3 marks)

18. (a) A line, L, posies through tho points (3,3) and (5,7). Find the equation of L, in the form y = mx+c where m and c arc constonti. (3 marks)

(b) Another line L2 is perpendicular to L, and passes through (—2, 3). Find:

(i) the equation of L2;

(ii) the x-intercept of L2.

(c) Determine the point of intersection of L, and L2. (3 marks)

19. A triangle ABC wÎth Vertices A (—2,2),B (1,4)and C (-1,4) is mapped on to triangle A’B’C by a reflection in the line y=x+1.

(a) On the grid provided draw

(i) triangle ABC (3 marks)

(ii) the line y = x + 1; (2 marks)

(iii) triangle A’B’C’. (2 marks)

(b) Triangle A”B”C“ is the image of triangle A’B’ C’ under a negative quarter turn (0,0).

On the same grid, draw triangle A“B“C“. (3 marks)

20.The figure below is a right pyramid VEFGHI with a square base of 8cm and a slant edge of 20cm Points A B C and D lie on the slant edges or the pyramid such that VA = VB = VC = VD = I0 cm and plane ABCD is parallels to the base EFGH.

(a) Find the length of AB. (2 marks)

(b) Calculate to 2 decimal places

(i) The length of AC (2 marks)

(ii) The perpendicular height of the pyramid VABCD (2 marks)

(c) The pyramid VABCD was cut off. Find the volume of the frustrum ABCDEFGH correct to 2 decimal places (4 marks)

21. The heights of 40 athletes in a county athletics competition were as shown in the table below.

Height,cm | Frequency |
---|---|

150 -159 | 2 |

160 – 169 | 8 |

170 – 179 | 10 |

180 – 189 | x |

190 -199 | 6 |

200 – 209 | 2 |

(a) Find the value of X. (1 mark)

(b) State the modal class. (1 mark)

(c) Calculate:

(i) The mean height of the athletes;(4 marks)

(ii) The median height, correct to 1 decimal place,of the athletes (4 marks)

22. The figure below represents a triangular 8ower garden ABC in which AB = 4 in, BC = 5m and ∠BCA = 30°. Point D lies on AC such that BD = 4 m and ∠BDC is obtuse.

Find, correct to 2 decimal places: (a) ∠BDC; (3 marks)

(b) the length of AD; (3 marks)

(c) the length of DC; (2 marks)

(d) the area of the flower garden ABC. (2 marks)

23. The shaded region on the graph below shows a piece of land ABCD earmarked for building a sub-county hospital.

(a) Write down the ordinates of curves AB and DC for x = 0, 200, 400, 600, 800, 1000 and 1200. (2 marks)

(b) Use trapezium rule, with 6 strips to estimate the area of the piece of land ABCD, in hectares.(4 marks)

(c) Use mid-ordinate rule with 3 strips to estimate the area of the piece of land, in hectarea. (4 marks)

24. The equation of a curve is y=x^{3}+x^{2}-x-1

(i) Determine the stationary point of the curve (3 marks)

(Îi) the nature of the stationary points in (a) (i) above.(2 marks)

(b) Determine:

(i) the equation of the tangent to the curve at x = 1; (3 marks)

(ii) the equation of the normal to the curve at x = 1. (2 marks)