THE KENYA NATIONAL MATHS GURUS EXAMINATIONS COUNCIL
Featuring Kenya Certificate Of Secondary Education (K.C.S.E.) 2024.
−
𝟏𝟐𝟏/𝟏
−
MATHEMATICS
ALT A
FORM FOUR
OPENER EXAM
𝟏
Jan. 2024−𝟐 𝟐 𝐡𝐨𝐮𝐫𝐬
𝐏𝐚𝐩𝐞𝐫 𝟏
Name……………………………….……………. Index Number:…….……………....
Candidate’s Signature……………………… Date………………………..…………..
Instructions to candidates
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Write your name and admission number in the spaces provided above.
Sign and write the date of examination in the spaces provided.
This paper consists of two sections: Section I and Section II.
Answer all questions in section I and only five questions from section II.
Show all the steps in your calculations, giving the answers at each stage in the spaces provided 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.
Candidates should answer the questions in English.
For Examiner’s Use Only
Section I
𝟏
𝟐
𝟑
𝟒
𝟓
𝟔
𝟕
𝟖
𝟏𝟗
𝟐𝟎
𝟐𝟏
𝟗
𝟏𝟎
𝟏𝟏
𝟏𝟐
𝟏𝟑
𝟏𝟒
𝟏𝟓
𝟏𝟔
𝐓𝐨𝐭𝐚𝐥
Section II
𝟏𝟕
𝟏𝟖
𝟐𝟐
𝟐𝟑
𝟐𝟒
𝐓𝐨𝐭𝐚𝐥
Grand Total
© 2024 The Kenya National Maths Gurus Examinations Council
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Turn over
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SECTION I (50 marks)
Answer all the questions in this section in the spaces provided.
1. A number N is formed by writing the first three composite numbers between 0 and 10 in
descending order.
(a) Express N as a product of its prime factors.
(2 marks)
(b) Find the smallest number Q which when multiplied by N, makes it a perfect square.
(1 mark)
2. Given that tan(90−∝)° = √5 find the value of sin ∝ without using a calculator or a
mathematical table.
(3 marks)
a
3. The image of a point Q (1, 2) after a translation represented by T = ( ) is Q1 (−1, 2).
b
Determine the coordinate of a point R whose image is R1 (−3, −3) after undergoing the
same translation.
(4 marks)
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4. Use square roots and reciprocal tables to evaluate the expression below correct to 4
significant figures.
(3 marks)
1
2
(0.06458)2 + (
)
0.4327
5. Three bells are programmed to ring at intervals of 0.2 of an hours, 0.25 of an hour and 0.4 of
an hour respectively. If the bells will next ring together at 10.00 a. m, find the time the bells
had last rang together.
(4 marks)
6.
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An old cylindrical container whose mass is 343 g is similar to a new cylindrical container of
mass 1,331 g. Given that the old container has a cross-sectional area of 196 cm2 , find the
cross-sectional area of the new container.
(3 marks)
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2x − 4
−1
−
2
12 − 3 x
3x + 6
7.
Simplify the expression,
8.
A Point P(−4, 1) is mapped onto P1 (2, 11) by an enlargement, scale factor 3 centre (a, b)
Determine the centre (a, b) of the enlargement.
(3 marks)
9.
Consider a line AB below. Without any measurement, divide the line AB into 6 equal parts
and hence locate a point X such that AX: XB = 2: 4.
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(3 marks)
(2 marks)
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10. A Kenyan bank buys and sells foreign currencies as shown below
Currency
Buying (Ksh. ) Selling (Ksh. )
1 Hong Kong dollar
9.74
9.77
1 South African rand
12.35
12.50
A tourist arrived in Kenya with 𝐊 Hong Kong dollars and changed the whole amount to
Kenyan shillings. While in Kenya, she pent Ksh. 383,975 and changed the balance to South
African rand before leaving for South Africa. If she received Ksh. 51,098 South African rand,
determine the value of 𝐊.
(3 marks)
11.
The size of one interior angle of a regular polygon of n sides is 140°. Calculate the number
of sides of the polygon and hence state its name.
(3 marks)
12.
Simplify: (8m)3 × m3 + 6 ÷ 2m−1.
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(3 marks)
6
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13.
A bicycle has two wheels of different sizes. The bigger wheel has a radius of 36.75 cm
while the smaller one has a radius of 26.25 cm. Before the bicycle is moved in a straight
line on a horizontal ground, the points on the tires in contact with the ground are marked.
Calculate the distance the bicycle travels when the marked points are next in contact with
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the ground at the same time. (Take π as 7 ).
(3 marks)
14.
Hassan has 10 mangoes less than Khadija who has x mangoes. If Hassan adds Khadija 6
more mangoes, then Khadija will have three times as many mangoes as Hassan. Find the
value of x.
(3 marks)
15.
Water flows from a point A to a point B 10 km away at a speed of 𝐱 km/hr. A boat cruises
from A to B downstream and then returns back to A upstream. Given that the speed of the
boat in still water is 4 km/hr and takes three and a third hours more on its return journey
from B to A than from A to B, form an equation in x and hence find the speed of water.
(4 marks)
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16.
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Write down the three inequalities which define the region R.
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(3 marks)
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SECTION II (50 marks)
Answer only five questions from this section in the spaces provided.
17. Four ports A, B, C and D are situated in such a way that port B is 140 km on a compass
bearing of N65°E from port A. Port C is 200 km on a compass bearing of S32°E from B. Port
D is due south of B and directly west of C.
(a) Using a scale of 1cm to represent 20 km, draw a diagram to show the position of ports
A, B, C and D.
(4 marks)
(b) Using your diagram, find
(i) The distance between ports A and D.
(ii) The compass bearing of port A from B.
(c) Determine the area of quadrilateral ABCD in km2
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(2 marks)
(1 mark)
(3 marks)
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18. A car initially moving at 20 m/s accelerates uniformly at 3 m/s 2 for 5 seconds. It maintains
this velocity for 6 seconds before accelerating uniformly again at 2.5 m/s 2 for 2 seconds.
The car finally decelerates uniformly in 2 seconds.
(a) Draw a velocity time graph to represent the above information.
(3 marks)
(b) State the maximum speed attained by the car.
(1 mark)
(c) Calculate the total distance travelled by the car.
(3 marks)
(d) Determine the time taken by the car to cover the first half of the journey. (3 marks)
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19. Three solids, a sphere, a cylinder and a cone are such that their radii and surface areas are
equal. Given that the volume of the sphere is
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500
3
π cm3 . Determine:
(a) The radius of the solids.
(2 marks)
(b) The height of the cylinder.
(3 marks)
(c) The height of the cone, correct to 1 decimal place.
(3 marks)
(d) The volume of the cone, correct to 2 significant figures.
(2 marks)
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20. (a) A straight line L1 passes through points P(−6, −1) and Q(−2, 3). By expressing its
x
y
equation in the form m + n = 1; where m and n are constants, find the values of m and n.
(4 marks)
(b) A second line L2 is equidistant from the points P and Q above. Determine the equation of
L2 in the form of y = mx + c, where a, b and c are constants.
(3 marks)
(c) Given that L1 intersects with a third line L3 whose equation is given by 2x + 3y – 20 =
0 at point T, find the coordinates of point T.
(3 marks)
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21. The table below shows marks scored by 40 students in a mathematics contest paper.
Marks
10 – 13
Cumulative frequency
6
14 – 17 18 – 21 22 – 25 26 – 29 30 – 33
10
20
32
37
40
Frequency
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(a) Complete the table above to show the corresponding frequencies.
(2 marks)
(b) Determine:
(i)
The median class.
(1 mark)
(ii)
The mean mark.
(4 marks)
(iii)
The median mark.
(3 marks)
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2 1
3 𝑡
) and 𝐁 = (
) where t is a constant, find the value of
1 3
1 0
t for which the determinant of 𝐀2 𝐁 = −𝟏𝟎𝟓.
(4 marks)
22. (a) Given two matrices 𝐀 = (
(b) On a certain day, two buses P and Q were hired to ferry students for a county
competition. Given that bus P uses petrol while bus Q uses diesel, and that 8 litres of
petrol and 12 litres of diesel were pumped into the buses for a total cost of Ksh. 4,444.
On their return journey, 10 litres and 4 litres were added to bus P and Q respectively, for
a total cost of Ksh. 3,146. Taking Ksh. p and Ksh. d to represent the cost per litre of petrol
and diesel respectively on this particular day;
(i) Write a simplified matrix equation to represent the above information. (2 marks)
(ii) Use matrix method to determine the cost per litre of petrol and diesel. (4 marks)
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23. During a closing day, ENA coach buses were brought to carry students from a certain school
to Nairobi. Given that the buses charged Ksh. 1500 per seat as fare and an agent was paid a
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commission of 11 9 % of the total fare collected for every fully booked 45 −seater bus.
(a) On this day, the agent was paid a total commission of Ksh. 45,000. Determine:
(i) How much fare he collected on that day?
(2 marks)
(ii) The total number of buses which were fully booked on that day.
(3 marks)
(b) On another closing day, the students were given a 10% discount per seat and as a result,
the agent’s rate of commission changed in the ratio 3: 4. The agent earned a total of
sh. 20,250 as commission on this day. Calculate:
(i) The total amount he collected as fare.
(3 marks)
(ii) The total number of students who booked ENA coach on this day.
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(2 marks)
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24. The figure below shows a circle centre O; not drawn to scale, circumscribing a triangle ABC
in which the sides AB = 4.1 cm, BC = 7.8 cm and AC = 8 cm.
Calculate correct to 1 decimal place;
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(a) The size of angle ACB.
(3 marks)
(b) The radius of the circle.
(2 marks)
(c) The area of the shaded region.
(5 marks)
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BLANK PAGE.
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