# Mathematics Paper 1

September 19, 2011 KCSE Exams

**KSCE Past Exam – Mathematics Paper 1 **

**SECTION 1** (50 marks)

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

1 Without using a calculator, evaluate

2 simplify

3 simplify the expression

4. Simon traveled by train from Butere to Nairobi. The train left Butere on Sunday at 23.50 hours and traveled for 7 hours 15 minutes to reach Nakuru. After a 45 minutes stop in Nakuru, the train took 5 hours 40 minutes to reach Nairobi. Find the time in 12 hours clock system and the day Simon arrived in Nairobi. (2 marks)

5 The figure below shows a net of a solid.

Below is the part of sketch of the solid whose net is shown above.

complete the sketch of solid, showing the hidden edges with broken lines. (3 marks)

6. A fuel dealer makes a profit of Ksh 520 for every 1000 liters of petrol sold and Ksh 480 for every 1000 liters of diesel sold. In a certain month the dealer sold twice as much diesel as petrol. If the total fuel sold that month was 900000 liters, find the dealer’s profit for the month. (3 marks)

7 A liquid spray of mass 384 g is packed in a cylindrical container of internal radius 302cm. Given that the density of the liquid is 0.6g/cm3 , calculate to 2 decimal places the height of liquid in the container. (3 marks)

8 Line BC below is a side of triangle ABC and also a side of a parallelogram BCDE.

Using a ruler and a pair of compasses only, construct:

(i) the triangle ABC given the angle ABC= 120 degree and AB= 6cm (1 mark)

(ii) the parallelogram BCDE whose area is equal to that of the triangle ABC and point E is online AB. (3 marks)

9 A solid metal sphere of radius 4.2cm was melted and molten material used to make a cube. find to 3 significant figures the length of the side of the cube. (3 marks)

10 An angle of 1.8 radians at the center of the circle subtends an arc of length 23 .4cm.

Find:

(a) the radius of the circle (2 marks)

(b) The area of the sector enclosed by the arc and the radii. (2 marks)

11 Three vertices of a rhombus ABCD are : A(-3, -4) and C(3,4).

(a) Draw the rhombus on the grid provided below. (2 marks)

(b) Find the equation of the line AD in the form y=mx+c, where m and c are constant. (2 marks)

(12) Two matrices A and B are such that

Given that the determinant of AB=4, find the value of K. (3 marks)

(13) A rectangular and two circular cut-outs of metal sheet of negligible thickness are used to make a closed cylinder.the rectangular cut-outs has a height of 18cm.each circular cut-out has a radius of 5.2cm.calculate in terms of ?, the surface area of the cylinder. (3 marks)

14 Given that log 4=0.6021 and log 6 and log 6=0.7782, without using mathematical tables or a calculator, evaluate log 0.096. (3 marks)

15 The equation of line L1 is 2y -5x -8 = 0 and line L2 passes through the points (-5,0) and (5, -4). Without drawing the lines L1 and L2, show that the two lines are perpendicular to each other. (3 marks)

16 Solve the equation:

2cos 2? = 1 for 0? ? ? ? 360? (4 marks)

**SECTION II (50 marks)**

*Answer any five questions in this section.*

17 (a) The ratio of Juma’s and Akinyi’s earnings was 5:3. Juma’s earnings rose to Ksh 8400 after an increase of 12%. Calculate the percentage increase in Akinyi’s earning given that the sum of their new earnings was Ksh 14100. (6 marks)

(b) Juma and Akinyi contribute all the new earnings to bye maize at Ksh 1175 per bag. The maize was then sold at Ksh 1762.50 per bag. The two shared all the money from the sales of the maize in the ratio of their contributions. calculate the amount that Akinyi got. (4 marks)

18 The figure below is a sketch of the curve whose equation is

it cuts the line y=11 at points P and Q.

(a) Find the area bounded by the curve

and the line y=11 using the trapezium rule with 5 strips. (5 marks)

(b) Calculate the difference in the area if the mid-ordinate rule with 5 ordinates was used instead of the trapezium rule. (5 marks).

19 In the figure **AB=p, AD=q, DE=1/2AB and Bc=2/3BD**

(a) Find the terms of p and q the vectors:

(i) **BD**; (1 mark)

(ii) **BC;** (1 mark)

(iii) **CD;** (1 mark)

(iv) **AC.** (2 mark)

(b) Given that AC =KCE, where K is a scalar, find:

(i) The value of K; (4 marks)

(ii) The ratio in which C divides AE. (1 mark)

20 The diagram below represent two vertical watch-towers AB and CD on a level ground. P and Q are two points on a straight road BD. he height of the tower AB is 20m and road BD is 200m.

(a) A car moves from B towards D.at point P the angle of depression of the car from point A is 11.3?. Calculate distance BP to 4 significant figures. (2 marks)

(b) If the car take 5 seconds to move from P to Q at an average speed of 36Km/h, calculate the angle of depression of Q from A to 2 decimal places. (3 marks)

(c) Given that QC=50.9m, calculate

(i) The height of CD in meters to 2 decimal places (2 marks)

(ii) The angle of elevation of A from C to the nearest degree. (3 marks)

21 The diagram below shows a triangle ABC with A(3,4), B(1,3) and C(2,1).

(a) Draw ?A’B’C, the image of ?ABC under a rotation of +90? about (0,0). (2 marks)

(b) Draw ?A”B”C”, the image of ?A”B”C” under a reflection in the line y=x. (2 marks)

(c) Draw ?A”B”C”, the image of ?A”B”C” under a rotation of -90? about (0,0). (2 marks)

(d) Describe a single transformation that maps ?ABC onto ?A”B”C”, (2 marks)

(e) Write down the equation of the lines of symmetry of the quadrilateral BB”A”A. (2 marks)

22 The diagram below represents a conical vessel which stands vertically.the vessel contains water to a depth of 30cm.the radius of the water surface in the vessel is 21cm. (take ?=22/7).

(a) Calculate the volume of the water in the vessel in cm3. (2 marks)

(b) When a metal sphere is completely submerged in the water, the level of the water in vessel rises by 6cm.

Calculate:

(i) The radius of the new water surface in the vessel. (2 marks)

(ii) The volume of metal sphere in cm3; (3 marks)

(iii) The radius of the sphere. (3 marks)

23 A group of people planned to contribute equally towards a water project which needed Ksh 2000000 to complete. However 40 members of the group withdrew from the project. As a result, each of the remaining members were to contribute Ksh 2500 more.

(a) Find the original numbers of members in the group. (5 marks)

(b) Forty five percent of the the value of the project was funded by constituency development fund (CDF). Calculate the amount of contribution that would be made by each of the remaining members of the group. (3 marks)

(c) Member’s contributions were in terms of labour provided and money contributed.if the ratio of the value of labour to the money contributed was 6:19, calculate the total amount of money contributed by the members. (2 marks)

24 The distance S meters from a fixed point O, covered by a particle after t seconds is given by the equation ;

(a) Calculate the gradient of the curve at t=0.5 seconds. (3 marks)

(b) Determine the values of s at the maximum and minimum turning points of the curve. (4 marks)

(c) on the space provided, sketch the curve of

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