KCSE MATHEMATICS QUESTIONS AND SOLUTIONS

​Mr. Kimutai a teacher from Tuiyotich Secondary School earns K₤ 12000 per annum and lives in a house provided by the employer at a minimum rent of Ksh.2000 per month.  He gets a family relief of K₤ 1320p.a. and is entitled to a relief of 10% of his insurance of K₤ 800p.a.

​(a) Calculate his annual tax bill based on the table below. (6mks)

​(b) Kimutai other deductions include.
- W.C.P.S = sh 600.00pm
- NHIF = sh 500.00pm
Calculate Kimutai’s net salary monthly. (4mks)

​A particle P moves in a straight line such that t seconds after passing a fixed point Q, its velocity V m/s, is given by the equation V = t² - 5t – 12

​Find; 
(a) The value of t when P is instantaneously at rest.     (2mks)

(b) An expression for the distance, S metres, moved by P after t seconds given that when t = 0, s = 0 (2mks)

​(c) The total distance traveled by P in the first 3 seconds after passing point Q.     (3mks)

(d) The distance of P from Q when the acceleration is zero. (3mks)

​The diagram below shows a triangle OPQ in which M and N are points on OQ and PQ respectively such that OM = ⅔ OQ and PN = ¼ PQ.  Lines PM and ON meets at X.

(a) Given that OP = P and OQ = q express in term of P and q  the vectors.
(i) PQ

​(ii) PM.                    (2mks)

​(iii) ON.                     (2mks)

​(b) You are further given that OX = KON and PX = hPM.
(i) Express OX in terms of P and q in two different ways.      (2mks)

(ii) Find the value of h and K.   (2mks)

(iii) Find the ratio PX : XM.               (1mk)

(a) Complete the table for y = Sin x + 2cos x. (2mks)

​(b) Draw the graph of y= sin x + 2 Cos x using a scale of 1cm to represent 30o on x-axis and 2cm to represent 1 unit on y-axis. (3mks)

​(c) Solve sin x + 2 cos x = 0 using the graph. (2mks)

(d) Find the range of values of x for which y≤-0.5. (3mks)

The diagram below shows a square based pyramid V vertically above the middle of the base.  PQ = 10cm and VR = 13cm.  M is the midpoint of VR.

​Find
(a) (i) the length of vector PR.                 (2mks)

​(ii) The height of the pyramid.                 (2mks)

(b) (i) the angle between VR and the base PQRS. (2mks)

(ii) The angle between MR and the base PQRS.   (2mks)

(iii) The angle between the planes QVR and PQRS. (2mks)

(ii) If it is 1200hrs at P, what is the local time at Q.      (3mks)

​A dam containing 4158m-cubed of water is to be drained. A pump is connected to a pipe of radius 3.5cm and the machine operates for 8 hours per day. Water flows through the pipe at the rate of 1.5m per second. Find the number of days it takes to drain the dam. (3mks)

​Atieno is now four times as old as her daughter and six times as old as her son. Twelve years from now, the sum of the ages of her daughter and son will differ from her age by 9 years. What is Atieno’s present age? (3mks)

​(i) Expand

​to the term in x cubed.

​(ii) Use your expansion to estimate the value of

​Correct to one decimal place.         (2mks)

​Chords AB and CD of a circle meet at X.

​If AB = 8cm, BX = 5cm and DX = 6.  Calculate the length of chord CD. (3mks)

WORKED ANSWER:

2X² + 2y² - 6X + 10y + 9 = 0 is the equation of a circle.  Find the radius and the centre of the circle. (4mks)

​Given that P = QXⁿ make n the subject of the formula and simplify your answer.                     (3mks)

Differentiate with respect to x    ​(3mks)

Worked Answer:

​T is a transformation represented by the matrix

Under T, a square of area ​10cm square is mapped onto a square of area 110cm square find the value of x. (3mks)

​Worked Answer:

​Solve for θ in the equation sin (3 θ + 120)degrees

Worked Answer:

​Two grades of coffee one costing sh.42 per kilogram and the other costing sh.47 per kilogram are to be mixed in order to produce a blend worth sh.46 per kilogram in what proportion should they be mixed.     (3mks)

Worked Answer:

In the figure below, EAF and EDG are tangents to the circle. AB is parallel to DC, angle FAB = 46° and angle AED = 110°.

​Find the values of angles P and q. (3mks)

Worked Answer:

​A map has a scale of 1 : 2500 and on it is square plot of land which has an area of 4cm². Calculate the actual area in hectares of the land. (3mks)

​​​Worked Answer:

​The initial salary of Mr. Lutta is sh. 42, 000 per annum. His salary increases by 13% each year. Determine his total earnings after 15 years. Give your answer to the nearest thousands.  (3mks)

​​Worked Answer:

Find the value of a and b where a and b are ​rational numbers.   (3mks)

​Worked Answer:

Solve the equation below.  (3 marks)

Worked Answer:

Use logarithms correct to 4 decimal places to evaluate. (3mks)

Worked Answer:

(a) A small field was surveyed and the measurements recorded in a surveyor’s field book as in the table below. ​

​(i) Using a scale of 1cm to 10m make an accurate drawing of the map of the field.         (4mks)

​(ii) Find the area of the field. (3mks)

(iii) Assuming that the baseline in (a) runs in a northern direction give the position of D relative to A using bearing and distance.    (3mks)

(a) Using the scale: 1cm represents 10km, construct a diagram showing the positions of B, C, Q and D. (5mks)

(b) Determine the 
(i) Distance between B and C           (1mk)

(ii) Bearing of D from B                      (2mks)

(c) Find the distance and bearing of D from C.  (2mks)

​(a) The diagram represents a solid frustum with base radius of 35cm and top radius of 21cm.The frustum is 22.5cm high and is made of a metal whose density is 4g/cm³.  
Taking  = 22/7, calculate

​(i) The volume of the metal that makes the frustum. (6mks)

​(ii) The mass of the frustum.                      (2mks)

​(b) The frustum is melted in down and recast into a solid cube in the process 20% of the metal is lost.  Calculate to 2d.p the length of each side of the cube.                                                                    (2mks)