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2016 KCSE 4MCK Joint Exam

Mathematics Paper 2

SECTION I (50 Marks)

Answer all the questions in this section in the spaces provided.

The base and height of a triangle were measured to be 10cm and 4â‹…0cm respectively.
Calculate the percentage error in calculating its area.

4 marks


Make y the subject of the formula

3 marks


a) Simplify: express it in the form a + b√c; where a, b, and c are rational numbers. (2 marks)
b) Hence find the values of a, b and c. (1 mark)

3 marks


In the figure below (not drawn to scale), points A, B, C and D lie on the circumference of
the circle and PAQ is a tangent of the circle at A. Given that angles BAQ and ABD are
40° and 30° respectively; and CB = CD;

3 marks


a) A rectangle whose area is 6 square units undergoes a transformation represented by
matrix . Determine the area of its image. (2 marks)
b) A point P undergoes a transformation by matrix M above in (a) and gets mapped onto P′(6,4).
Find the co-ordinates of P. (2 marks)

4 marks


The position of points A and B are a = 4i + 4j – 6k and b = 10i + 4j + 12k respectively.
A point D divides AB in ratio 2:1. Find the magnitude of the position vector of D.

4 marks


An arc of length 24cm subtends an angle of 1â‹…6 radians at the center of the circle.
Calculate the area of the circle. (Take π = 3⋅142).

3 marks


Form a quadratic equation whose roots are -1/4 and 2/3 expressing your answer in the
form ax2 + bx + c = 0 where a, b, and c are integers.

2 marks


Solve the equation for 0° ≤ x ≤ 180°
-3 Cos (2x + 10)° = 0⋅6

3 marks


a) Expand and simplify the first four terms of the binomial expression (2 – 1/4x)6 (2 marks)

b) Use your expansion in (a) above to evaluate (1â‹…975)6 correct to 2 decimal places. (2 marks)

4 marks


A curve whose gradient function is 3x2 – 4x + 5 passes through point (-1, 1). Determine
the equation of the curve.

3 marks


Without using tables or calculator, find the value of x if;
Log(20x + 35) – log2x = 1 + 2log3 (3 marks)

3 marks


The probability of Joyce passing her exams is 2/3. If she fails in her exams, she repeats
and the probability of her passing increases by 10%. Find the probability she will pass her
exams after repeating.

2 marks


Tangent PQ and chords AB and CD intersect at point Q outside the circle in the figure
below. If AB = 5cm, DQ = 4cm and BQ = 6cm;

a) The length of chord CD (2 marks)
b) The length of tangent PQ (2 marks)

4 marks


Find the co-ordinates of the center and the diameter of the circle whose equation is
2x2 + 2y2 + 16x -4y = -2

3 marks


A right pyramid ABCDV below (not drawn to scale) has a square base ABCD whose AB
= 10cm and its vertex V is such that AV = BV = CV = DV = 13cm

Calculate the angle between planes BCV and ABCD.

2 marks

SECTION II (50 Marks)

Answer only five question in this section in the spaces provided.

The table below shows the income tax rates:

Monthly Income (K£) Rate (%)
1 – 2000
2001 – 4000
4001 – 6000
Above 6000
Rate (%)

Mr. Mutua earns a basic salary of Kshs 64, 000 per month. He is entitled to allowances
amounting to Kshs 45,600 per month; and a personal relief of Kshs 1056 per month.
a) His total taxable income in K£ p.m (2 marks)
b) The total P.A.Y.E tax he pays in Kshs p.m (8 marks)

10 marks


The figure below shows a scale diagram of a rectangular piece of land ABCD in which
AB = 10cm and AD = 6cm.

a) Using a pair of compasses and a ruler only; construct on the diagram above
i) The locus of a point P equidistant from lines AB and AD (2 marks)
ii) The locus of a point Q such that AQ = 6cm (2 marks)
iii) The locus of a point R such that <ARB = 90° (2 marks)
b) A bore hole H lies inside the piece of land ABCD above such that its nearer to AD than
AB, and AH ≤ 6cm and also angle AHB greater than 90°. Show by shading out the
unwanted regions, and labeling H on the diagram above, the region where the borehole H
lies. (4 marks)

10 marks


A ship sails non stop from port P(20°N, 30°E) northwards to a port Q(45°N, 30°E) then
due west to port R a distance of 4500nm
a) Find the total distance in nautical miles travelled by the ship. (2 marks)
b) Determine the position of port R, to nearest degree. (3 marks)
c) Find the average speed of the ship in knots if it took 8 hours. (2 marks)
d) Determine the local time at R when the ship arrived there if it set off at P at 9:00am
local time. (3 marks)

10 marks


The sketch below shows the graphs of curve y = 4x – x2 and line y = x, which intersect at
points O and B

a) Determine the coordinates of points O, A, B and C. (5marks)
b) Use integration to calculate the area of the region enclosed by the curve y = 4x – x2 and
the line y = x (shaded) (5 marks)

10 marks


a) The nth term of a sequence is given by Tn = n2 –n + 3
i) the 10th term of the sequence (1 mark)
ii) the difference between 10th and 25th terms of the sequence (2 marks)
iii) the value of n if Tn = 243 (3 marks)
c) In a research station, it was found that the number of bacteria doubles in every one hour.
If the number of bacteria started with 200; how long does it take for the bacteria to lit 1
million. (4 marks)

10 marks


The marks obtained by 10 students in a maths test were; 25,24,22,23,x,26,21,23,22,and 27. The sum of the squares of the marks (Σx2) is 5154.
a) Calculate
i) The value of x (2 marks)
ii) The mean (2 marks)
iii) The standard deviation ` (4 marks)
b) If each mark was increased by 3 and the doubled; determine
i) The new mean (2 marks)
ii) The new standard deviation (2 marks)

10 marks


The volume Vcm3 of a solid varies jointly as the square of the radius r cm of its base and
its height h cm. Given that V = 180cm3, when r = 3 cm and h = 10 cm;
a) Determine the value of constant of proportionality (2 marks)
b) Find the diameter of the base when V = 480 cm3 and h = 15 cm (3 marks)
c) Calculate the percentage change in volume V when r is increased in ratio 5:4 and h is
decreased by 5%. Correct to 4 significant figures. (5 marks)

10 marks


A tailoring company makes two types of garments A and B. Garment A requires 3 meters
of material to make while B requires 2 meters of material. The company uses not more
than 600 meters of material in making the garments. It must make at most 100 of type A
and at least 80 of type B.
a) By letting the number of type A garments made be x and type B garment made be y; form
four inequalities for the above information. (4 marks)
b) Represent the inequalities in a (a) above in the squared grid provided. (4 marks)
c) If the company makes a profit of Kshs 100 on garment A and 60 on garment B; find the
number of garments of each type that should be made to maximize the profit. Hence
determine the maximum profit. (2 marks)

10 marks

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