Mathematics KCSE Mock Exams and Answers {Latest Best Collections}

MOCKS 1 2023

121/1MATHSPAPER 1MARKING SCHEME

Numerator:

Denominator:

Numerator

Denominator

N;(

Numerator

denominator

Expression

CAO

Comparing powers

Equation

Expression

No.	Log
2	0.3010
0.324	+
1.7642	0.2465 x 2 0.4930 –
5.42	0.7340 + 1.2270

0.4428

All logs correct

Attempt to divide

2y<x +4

4y ≥ – x- 4

x≤2

Midpoint (

LCM = 900 = 2² x 3² x 5²

36 = 2² x 3²

60 = 2² x 3 x 5

Least possible number = 2 x 3 x 5²= 150

GCD/LCM

36/60

10.

11.

Substitution

For both

12.

(a) 5 Tan θ =

(b) Cos (180 – θ) = –

13.

Complete net, well labelled

14.

15.

(i)

<BOD = 2 <DAB = 2 x 87 = 174⁰

(ii)

<AOB

Property

<ADT

Property

16.

(i)

(ii)

17.

(a) Distance after 30mins

Relative

= 20km/hr

(b)

(c)

For both distance

Relative distance

Relative speed

Relative time

18.

(a) OP = a + (b – a)

= a + b

BQ = a – b

(b) (i)OC = h

OC = b + k

= ka + (1 – k)b

h =

h = k

h = 1 – k

k = 1 – k

2k = 3 – 3k

5k = 3

k = h =

(ii) OC =

(iii)BC: CQ = :

BC:CQ = 3:2

19.

		fd
9.5	19	0.2
24.5	539	1.1
44.5	2225	2.5
69.5	1668	0.8
89.5	179	0.2

Total area = 2 + 22 + 50 + 24 + 2 = 100

50 – (2 + 22) = 26+26 = 2.5 x y, y = 10.4

Median = 34.5 + 10.4

= 44.9

f.d

scale

For median line

20.

x	-4	-3	-2	-1	0	1	2	3
y	30	13	2	-3	-2	5	18	37

c) i) x = + 0.4

x = -1.7 + 0.1

ii) y = 3x² + 4x – 2

0 = 3x² + 7x + 2

y = -3x – 4

x = -2 or x = -0.4 + 0.1

All ü

at least 6 ü

For equation of line

For ü line drawn

21.

(b) A¹(4,-4) B¹(7,-3) C¹(2,-1)

(d) A¹¹¹(4,-4) B¹¹¹(3,-7) C¹¹¹(1,-2)

For plotting

For ∆ABC

For ∆A¹B¹C¹

For construction or otherwise

For ∆A¹¹B¹¹C¹¹

For construction or otherwise

For ∆A¹¹¹B¹¹¹C¹¹¹

22.

b) 2.1 + 0.1cm

200km, 210km, 220km

c) i) Bearing of M from N = 010⁰+ 1⁰

ii) Bearing of N from M = 190⁰+ 1⁰

1cm rep.100km

<30⁰ at P

<45⁰ at Q

üpositions of PQM and N

ülabelling 540km, 360km, 500km allü

23.

(b)

(ii) Volume of frustum

(c)

24.

(i)

(ii)

(c)

Maximum speed,

For both

NAME:……………………………………………….. INDEX NO………………………………
SCHOOL:……………………………………..……… STREAM:…………… ADM:………….

CANDIDATE’S SIGN …………………………….… DATE …………………………………..

121/1

MATHEMATICS

Paper 1

FORM 4

JULY 2023
Time: 2 ½ Hours

MOCKS 1 2023

Kenya Certificate of Secondary Education (K.C.S.E)

INSTRUCTIONS TO CANDIDATES

Write your name, stream, admission number and index number in the spaces provided above.
The paper contains two sections, Section I and II
Answer all questions in section I and ONLY any FIVE questions from section II.
All answers and working must be shown on the question paper in the spaces below each question
Show all steps in your calculations, giving answers at each stage
Marks may be given for each correct working even if the answer is wrong
Non-programmable silent electronic calculators and KNEC mathematical tables may be used.

FOR EXAMINERS USE ONLY

Section I

Total

Section II Grand Total

Total

This paper consists of 16 printed pages. Candidates should check the question paper to ensure that all pages are printed as indicated and no questions are missing.

SECTION I (50 MARKS)

Answer all questions in this Section

Evaluate : (3 mks)

Simplify completely (3 mks)

Use the exchange rates below to answer this question.

Buying Selling

1 US dollar 63.00 63.20

1 UK £ 125.30 125.95

Abwanja, a tourist arriving in Kenya from Britain had 9600 UK Sterling pounds (£). He converted the pounds to Kenya shillings at a commission of 5%. While in Kenya, he spent ¾ of this money. He changed the balance to US dollars after his stay. If he was not charged any commission for this last transaction, calculate to the nearest US dollars, the amount he received. (3 mks)

Solve for x in the following equation. (3mks)

4^x (8^{x – 1}) = tan 45^o

The sum of interior angles of two regular polygons of sides; n and n + 2 are in the ratio 3:4. Calculate the sum of the interior angles of the polygon with n sides. (3mks)

Use logarithms to evaluate the following correct to 4 decimal places.

(3mks)

By shading, show the region defined by the following linear inequalities (3mks)

2y < x + 4; 4y ≥ –x – 4; x ≤ 2

Find the equation of locus of points equidistant from points A (6, 5) and B (-2, 3) in the form

y = mx + c (3mks)

The GCD of three numbers is 6 and their LCM is 900. If two of the numbers are 36 and 60, find the least possible third number. (3mks)

Use the tables of squares, cube roots and reciprocals to evaluate (3mks)

Solve the following pair of simultaneous equations using substitution method (3mks)

Given that Sin q = 0.8 and q is an acute angle, find without using tables or calculators

(a) Tanq (2mks)

Cos (180 – q) (1mk)

The figure below is a triangular prism of uniform cross-section in which AF = FB =3cm,

AB = 4cm and BC = 5cm. Draw a clearly labeled net of the prism. (3mks)

The mass of two similar cans is 960g and 15000g. If the total surface area of the smaller can is 144cm², determine the surface area of the larger can. (3mks)

In the circle below, O is the centre, angle DAB = 87⁰ , minor Arc AB is twice minor arc AD. CD is a tangent to the circle at D.

Giving reasons, Calculate the size of;

(i) Angle AOB. (2mks)

(ii) Angle ADT (2mks)

A sector of a circle of radius 42cm subtends an angle of at the centre of the circle. The sector is folded into an inverted right cone. Calculate

(i) The radius of the cone (3mks)

(ii) To one decimal place the vertical height of the cone (1mk)

SECTION II: 50 MARKS

Answer any FIVE questions in this section

A bus and a Nissan left Nairobi for Eldoret, a distance of 340 km at 7.00 a.m. The bus travelled at 100km/h while the Nissan travelled at 120km/h. After 30 minutes, the Nissan had a puncture which took 30 minutes to mend.

Find how far from Nairobi did the Nissan caught up with the bus (5mks)

At what time of the day did the Nissan catch up with the bus? (2mks)

Find the time at which the bus reached Eldoret (3mks)

In the diagram below OA = a, OB = b the points P and Q are such that AP = ²/₃ AB, OQ = ¹/₃ OA

(a) Express OP and BQ in terms of a and b (2 mks)

(b) If OC = hOP and BC = kBQ, Express OC in two different way and hence

(i) Deduce the value of h and k. (5 mks)

(ii) Express vector OC in terms of a and b only. (2 mks)

(iii) State the ratio in which C divides BQ (1 mk)

The table below shows the marks scored in a Mathematics examination.

Calculate the mean mark (3mks)

Marks	Frequency
5 – 14	2
15 – 34	22
35 – 54	50
55 – 84	24
85 – 94	2

Draw a histogram to represent the above information (4mks)

Using the histogram, find the median mark (3mks)

Given the quadratic function y = 3x² + 4x – 2
a) Complete the table below for values of x ranging – 4 < x < (2mks)

x	-4	-3	-2	-1	0	1	2	3
y

b) Using the grid provided draw the graph of y = 3x² + 4x – 2 for -4 < x < 3 (3mks)

c) Using the graph, find the solution to the equations.
i) 3x² + 4x – 2 = 0 (2mks)

ii) 3x² + 7x + 2 = 0 (3mks)

A triangle ABC has vertices A(2,1), B(5,2) and C(0,4).

(a) On the grid provided plot the triangle ABC. (2 mks)

(b) A¹B¹C¹ is the image of ABC under a translation . Plot A¹B¹C¹ and state its coordinates. (2mks)

(c) Plot A¹¹B¹¹C¹¹ the image of A¹B¹C¹ under a rotation about the origin through a negative quarter turn. State its coordinates. (3 mks)

(d) A¹¹¹B¹¹¹C¹¹¹ is the image of A¹¹B¹¹C¹¹ under a reflection on the line y = 0. Plot A¹¹¹B¹¹¹C¹¹¹ and state its coordinates. (3 mks)

Two Airstrips P and Q are such that Q is 500km due East of P. Two warplanes M and N

Leave from P and Q respectively at the same time. Warplane M moves at 360km/h on a bearing of 030⁰. Warplane N moves at a speed of 240km/h on a bearing of 315⁰. The two warplanes landed at Police camps A and B respectively after 90 minutes. Using a scale of 1cm represent 100km

a) Show the relative positions of the two police camps A and B (6mks)

(b) Find the shortest distance between the police camps A and B. (2mks)

i) Police camp A from B (1mk)

ii) Police camp B from A (1mk)

The diagram below represents square based pyramid standing vertically. AB = 12cm, PQ = 4cm and the height of pyramid PQSV is 10cm.

If PQRSV is a solid, find the volume of material used to make it. (2mks)

Find the
height of the frustum ABCDPQRS (2mks)

Volume of the frustum (3mks)

The liquid from a hemisphere is poured into PQRS. Find radius correct to 4 significant figures of the hemisphere if the liquid from hemisphere filled the solid completely. (3mks)

The displacement h metres of a particle moving along a straight line after t seconds

is given by h = -2t³ + ³/₂ t² + 3t

(a) Find the initial acceleration. (3mks)

(b) Calculate

(i) The time when the particle was momentarily at rest. (3mks)

(ii) Its displacement by the time it comes to rest momentarily. (2mks)

(c ) Calculate the maximum speed attained. (2mks)

MOCKS 1 2023

121/2 MATHEMATICS PAPER 2 MARKING SCHEME

Workings

Marks

Comments

b² = 4ac

5² = c + 2

25 = c + 2

c = 23

Correct expression in C

Truncated = 0.777

Rounded off = 0.778

A.E = 0.778 – 0.777 = 0.001

% E = x 100

= 0.12870012870012870012870012870013

For both values correct

Expression for % Error

Allow 0.1287

P(-8.5, -20, -11)

Expression

Correct matrix

Co-ordinate form

Correct substitution in sine rule

Surd form for sin 60⁰

Correct attempt to rationalize

CAO

(x³)⁶ -6 (x³)⁵ + 15 (x³)⁴ ²– 20 (x³)³ ^{3 . . .}

– 20 (x³)³ ³

– 20 x 8 =

– 160

Expansion up to the 4^th term

Correct attempt to simplify

Constant term stated

Let log₃x = y

2y² – y – 3 = 0

(2y – 3)(y + 1) = 0

y = -1 or y = 1 ½

if log₃x = -1, x = 3^-1 = ¹/₃

if log₃x = 1 ½ , x = 3^1.5 = 5.196

Quadratic equation formed

For both correct

P = cp – d = 13800 – 2280 = 11520

I = 11520 x 20 x 2/100= 4608

A = P + I = 11520 + 4608 = 16128

MI= 16128 ÷ 24

= 672

M1 A1

Expression for simple interest

Expression for MI

2ax + x² = 3v

x² + 2ax – 3v = 0

x2 + 2ax +a² = 3v + a²

√(x + a)² = √(3v +a²)

x + a = ±√(3v +a²)

x = -a ± √(3v +a²)

Formation of quadratic equation

Completing the square

Correct attempt to solve

x	1.5	2.5	3.5	4.5	5.5
y	14.75	26.75	77.75	68.75	98.75

A= 1(14.75 +26.75+77.75+68.75+98.75)

= 253.75 square units

Correct values of mid-ordinates

Expression for area

10.

OA = OP = 5 units

AM = 5 – 2 = 3 units

OM = √(5² – 3²) = 4 units

C(5,4) , r = 5

(x – 5)² + (y – 4)² = 5²

x² – 10x + 25 + y² – 8y + 16 = 25

x² + y² -10x – 8y + 16 = 0

Expression for midpoint

Radius, r

Expression for OM

Correct substitution

Correct expanded form

11.

= 1.736k

% change =

Correct substitution

Expression for percentage change

12.

3sin²x – sin x – 2 = 0

Let sin x = y

3y² – y -2 = 0

(3y + 2)(y – 1 ) = 0

y = 1 or y = ^-2/₃

sin^-1(1) = 90⁰

sin^-1(^-2/₃) = 221.8⁰, 317.8⁰

x = 90⁰, 221.8⁰, 317.8⁰

Quadratic equation formed

Correct attempt to solve

For both

All values correct

13.

i) k + 2k + 3k + 4k + 5k + 6k = 1

21k = 1

ii) P(5&6) 0r P(6&5)

(

Addition of probabilities (allow for any correct)

Allow

14.

(a) Let VU = x

8(8 + x) = 12²

8x = 144 – 64 =80

x = 10

b) VX =

XU =

XT = 6 + 8 = 14

SX = √(14² – 12²) = 7.211

x = 10

Expression for XT

15.

h	10-19	20-29	30-39	40-49	50-59	60-69	70-79
f	9	16	19	26	20	10	4
cf	9	25	44	70	90	100	104

Q₁ =

Q₃ =

Quartile deviation =

Q1 and Q3

Expression for quartile deviation

Allow 16.47

16.

= 4:5

Correct substitution

17.

(a) (i) x-intercept

x²(2x + 3) = 0

(ii) y-intercept

When x =0, y = 0

(b) (i) Stationary points of curve

= 0

6x(x + 1) = 0

x = 0 or x = -1

stationary points (0,0) and (-1,1)

(ii)

x	-2	-1	-0.5	0	1
	12	0	-1.5	0	12
sketch

maximum point (-1, 1), minimum point (0,0)

iii)

Factorized form

Both correct

Derivative equated to zero

Attempt to solve

For both

Checking points

For both

Points plotted (-1.5,0), (-1,1), (0,0)

Smooth curve

18.

a) (i)

ii)

r = 5.2cm ± 0.1

iii)

h = 5cm ± 0.1

b) area of circle – area of triangle

= 84.98 – 21.25

= 63.73cm²

Construction of 30⁰

Construction of 105⁰

Complete triangle, well labeled

Line bisectors

Complete Circle drawn

radius

height dropped

follow through for r and h ± 0.1

x	0	30	60	90	120	150	180	210	240	270	300	330	360
2cosx-1	1	1.73		-1		-2.73		-2.73				0.73	1
sinx		0.50	0.87		0.87			-0.50	-0.87		-.087	0.50	0

Scale

Plotting for both

Smooth curve

20.

a) i)

ii)

b)i)

ii)

Tree diagram draw with probabilities indicates

ü1 probability

Addition of the probability

ü probability

Addition

21.

(a)(i)

Distance =

=longitude difference

=40+140=180⁰

=17,337.8Km

b) =60Î2

=1200

Distance =

=13,346.7km

B(30⁰W,140⁰E)

Difference in longitude=140+40

=180⁰

1⁰=4min

180=?

180Î4=720minutes

8.00+12.00=20.00

=12.00hrs/8.00pm

For 180^o

22.

Length in cm	9.5-12.5	12.5-15.5	15.5-18.5	18.5-21.5	21.5-24.5
No. of leaves	3	16	36	31	14
cf	3	19	55	86	100

c) i) Q₃ = 19.25, Q₁= 17.15

½ (Q₃ – Q₁) = ½ (19.25 – 17.15)

= 1.05

ii) 13cm – – 15.2, 17cm – – 15.8

15.8 – 15.2 = 0.3

All values correct

At least 4 values correct

Q₃ & Q₁ correct

Correct cf values

23.

x	0	1	2	3	4	5	6
y	3	3.5	5	7.5	11	15.5	21

Error: 54.5-54=0.5

24.

300

250

200

150

100

300

250

200

150

100

c) Objective function

For each correct inequality

For each correct line drawn

CANDIDATE’S SIGN …………………………….… DATE …………………………………..

121/2

MATHEMATICS

Paper 2

FORM 4
Time: 2 ½ Hours

MOCKS 1 2023

Kenya Certificate of Secondary Education (K.C.S.E)

INSTRUCTIONS TO CANDIDATES

Write your name, stream, admission number and index number in the spaces provided above.
The paper contains two sections, Section I and II
Answer all questions in section I and ONLY any FIVE questions from section II.
All answers and working must be shown on the question paper in the spaces below each question
Show all steps in your calculations, giving answers at each stage
Marks may be given for each correct working even if the answer is wrong
Non-programmable silent electronic calculators and KNEC mathematical tables may be used.

FOR EXAMINERS USE ONLY

Section I

Total

Section II Grand Total

Total

This paper consists of 18 printed pages. Candidates should check the question paper to ensure that all pages are printed as indicated and no questions are missing.

SECTION I

Answer all the questions in the spaces provided (50marks)

The expression x² + 10x + c + 2 = 0is a perfect square. Find the value of c if it is a scalar. (2mks)

Muya was asked to truncate ⁷/₉ to 3 significant figures. He rounded it off instead to 3 decimal places. Calculate the percentage error resulting from his rounding off. (3mks)

The co-ordinates of a point A is (2, 8, 3) and B is (-4, -8, -5). A point P divides AB externally in the ratio 7: -3.

Find the co-ordinates of P (3mks)

In a triangle XYZ, XY = 2cm, YZ (2√3-1) cm, and angle YXZ = 60⁰. Determine Sine XZY giving your answer in the form m + √3,where M and N are integers (4mks)

Find the independent term of x in the expansion of (x³ – ²/_X³) ⁶ (3mks)

Solve for x: (log₃x)² – ½ log₃x= ³/₂ (3mks)

The cash price of a T.V set is Ksh.13,800. Walter opts to buy the set on hire purchase terms by paying deposit of Ksh.2,280. If simple interest of 20% p.a is charged on the balance and the customer is required to pay by monthly installments for 2 years, calculate the amount of each installment. (3mks)

Make x the subject of the formula ax = 3r – x² (3mks)

2 2

Calculate the area under the curvey = 3x² + 8 and bounded by lines;y = 0, x = 1 and x = 6, using the mid-ordinate rule with 5strips. (3mks)

A circle is tangent to the y – axis and intersects the x- axis at (2,0) and (8,0). Obtain the equation of the circle in the form x² + y² +ax +by +c = 0, where a, b and c are integers (4mks)

A variable y varies as the square of x and inversely as the square root of z. Find the percentage change in y when x is changed in the ratio 5:4 and z reduced by 19% (3mks)

Solve for X in the equation:

2 Sin²x – 1 = Cos²x + Sin x, for 0⁰ ≤ x ≤ 360⁰ (3mks)

A die is biased so that when tossed, the probability of a narrator of a number n showing up, is given by p(n) = kn where k is a constant and n = 1, 2, 3, 4, 5, 6 (the numbers of the faces of the die)
Find the value of k (1mk)

If the die is tossed twice, calculate the probability that the total score is 11 (2mks)

In the figure below, the tangent ST meets chord VU produced at T. Chord SW passes through the Centre, O of the circle and intersects chord VU at X. Line ST = 12cm and UT = 8cm.

Calculate the length of chord VU (1mk)

If VX : XU = 2 : 3, Find SX (2mks)

Dota measured the heights in centimeters of 104 trees seedlings are shown in the table below

Height	10 – 19	20 – 29	30 – 39	40 – 49	50 – 59	60 – 69	70 – 79
No. of Seedlings	9	16	19	26	20	10	4

Calculate the quartile deviation (4mks)

Given that the ratio x:y = 2:3, find the ratio (5x – 2y) : (x +y) (2mks)

SECTION II

Answer ONLY five questions in this section (50marks)

A curve is represented by the function, y = 2x³ + 3x²
Find:(i) the x-intercept of the curve (2mks)

(ii) the y-intercept of the curve (1mk)

(i) Determine the stationary points of the curve of the curve (3mks)

(ii) For each point in b(i) above, determine if it is maximum or minimum (2mks)

Sketch the curve in the space below (2mks)

Use ruler and a pair of compasses only in this question
Construct; (i) triangle ABC in which AB = 8.5cm, BC = 7.5cm and <BAC = 30⁰and <ABC = 105⁰

(3mks)

ii) a circle that passes through the vertices of triangle ABC. Measure the radius (3mks)

the height of triangle ABC with line AB as the base. Measure the height. (2mks)

Determine area of the circle that lies outside the triangle (2mks)

a) Complete the table below, giving your values to 2 decimal places (2mks)

x	0	30	60	90	120	150	180	210	240	270	300	330	360
(2cos x) -1			0		-2		-3		-2	-1	0		1
Sin x	0			1		0.50	0			-1			0

b) Draw the graph of y= (2 co x) – 1 and y=sin x on the grid provided below. Use the scale 1cm represent 30⁰ horizontal 2 cm represent 1 unit vertically and 2cm for 1 unit on the y-axis (4 mks)

c) Use the graph to solve:
i) (2cos x) – 1 = -1.5 (1mk)

ii) 2 cos x – sin x =1 (2mks)

d) State the amplitude of the wave y=2cos x – 1 (1mk)

A bag contains blue, green and red pens of the same type in the ratio 8:2:5 respectively. A pen is picked at random without replacement and its colour noted.
a) Determine the probability that the first pen picked is
i) Blue (1mk)

ii) Either green or red. (2mks)

b) Using a tree diagram, determine the probability that
i) The first two pens picked are both green (4mks)

ii) Only one of the first two pens picked is red. (3mks)

A and B are two points on the earth’s surface and on latitude 30⁰N.The two points are on the longitude 40⁰W and 104⁰E respectively.

Calculate

(a) (i) The distance from A to B along a parallel of latitude in kilometres. (3mks)

(ii) The shortest distance from A to B along a great circle in kilometre (4mks)

(Take =and radius of the earth =6370km)

(b) If the local time at B is 8.00am, calculate the local time at A (3mks)

Lengths of 100 mango leaves from a certain mango tree were measured t the nearest centimeter and recorded as per the table below,

Length in cm	9.5-12.5	12.5-15.5	15.5-18.5	18.5-21.5	21.5-24.5
No. of Leaves	3	16	36	31	14
Cumulative frequency

Fill in the table above. (2 mks)
Draw a cumulative frequency curve from the above data. (3 mks)

b) Use your graph to estimate
i) The quartile deviation of the leaves (3mks)

ii) The number of leaves whose lengths lie between 13cm and 17cm. (2mks)

a) Use the trapezium rule with 7 ordinates to estimate the area enclosed by the curve and the lines x = 0, x = 6 and the x-axis. (4 mks)

b) Determine the exact area bounded the curve and the lines in section a) above (3 mks)

c) Calculate the percentage error from the trapezoidal rule (3 mks)

A manufacturer sells two types of books X and Y. Book X requires 3 rolls of paper while Book Y requires 2¹/₂ rolls of paper. The manufacturer uses not more than 600 rolls of paper daily in making both books. He must make not more than 100 books of type X and not less than 80 of type Y each day
Write down four inequalities from this information (4mks)

On the grid provided, draw a graph to show inequalities in (a) above (3mks)

If the manufacturer makes a profit of sh 80 on book X and a profit of sh 60 on book Y, how many books of each type must it make in order to maximize the profit. (3mks)

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