ORDINARY LEVEL PAPER 2
REVISION OF MAJOR TOPICS IN MATHEMATICS
COMMIT YOUR WORK TO THE LORD AND YOUR PLANS WILL BE ESTABLISHED
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A DREAM DOESN’T BECOME REALITY THROUGH MAGIC IT
TAKES SWEAT DETERMINATION AND HARD WORK
A WINNER IS A DREAMER WHO DOES NOT GIVE UP
COMPILED AND SOLVED BY MR. MUNUNGA J
JM/2019
CONTACTS: 0762486410/0978934334
Email: justinmununga06@gmail.com
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TABLE OF CONTENTS
Topic
page
1. Algebra…………………………………………………………………………….…….………..…..1-2
2. Quadratic equations…………………………………………………..………..……….…….…….2
3. Matrices…………………………………………………………….………….………….……….…3-4
4. Geometric progressions……………………………………………..……………….………...5-6
5. Sets………………………………………………………………………...….……………….…….….7-9
6. Probability………………………………………………………...…………………….…….…10-11
7. Vectors…………………………………………………………...…………………….………….12-15
8. Pseudo codes and flowcharts………………………..…………………………….…….15-19
9. Mensuration……………………………………………….……………………………………20-23
10. Calculus…………………………………………………..……………………………….…….24-25
11. Locus……………………………………………………...………………….…………….…….25-27
12. Earth geometry……………………………………...……………………………………….27-30
13. Trigonometry……………………………………..………………………………………….31-34
14. Statistics………………………………………………………………………………………...35-39
15. Linear programming………………………...……………………………………….……40-42
16. functions………………………………………………………………………………………..42-45
TOPIC 1: ALGEBRA
1
Worked examples
1. Simplify the following
(a)
(b)
.(c)
(d)
2. Express the following as single fractions in their lowest terms.
(a)
(b)
Answers
1. (a)
(b)
.(c)
(d)
2.
(a)
Activity
1. Simplify the following
(a)
(e)
(m)
Compiled and solved by Mr. Mununga J
(b)
2
2. Express each of the following as a single fraction in simplest form.
(a)
(e)
TOPIC 2: QUADRATIC EQUATIONS
Worked examples
Solve the following equations, giving your answer correct to 2 decimal places
(1)
(2)
Answers
(1)
(2)
√
√
√
√
√
√
Activity
Solve the following equations, giving your answer correct to 2 decimal places
(a)
The foundation is the determiner-Justin Mununga
Compiled and solved by Mr. Mununga J
3
TOPIC 3: MATRICES
Worked examples
1. Given that matrix
(
), find
(i) The determinant of K
2. given that matrix
(
(ii) the inverse of K
)
(i) Find the value of x for which the determinant of M is 22.
(ii) Hence find the inverse of M
Answers
1. (i) | |
(ii)
(
)
= -20+22
=2
2. (i)
(ii)
(
)
(
Activity
1. given that matrix
(
) and
(
)
(i) Find the value of y for which the determinant of A and B are equal.
(ii) Hence find the inverse of B.
2. given that matrix
(
)
(i) Find the positive value of x for which the determinant of A is 12.
(ii) Hence or otherwise, write A-1.
3. The determinant of matrix
(
(i) The value of x,
(ii) The inverse of Q.
Compiled and solved by Mr. Mununga J
) is 8. Find
)
4
(
4. Given that matrix Q
) , find the
(i) Value of x, given that the determinant of Q is 2,
(ii) Inverse of Q.
(
5. Given that the determinant of matrix
) is 4,
(i) find the value of x,
(ii) Write the inverse of the matrix A.
6. Given that matrix
(
),
(i) Write an expression in terms of a for the determinant of Q,
(ii) Find the value of a, given that the determinant of Q is 2,
(iii) Write Q-1.
(
7. Given that
) , find
(b) A-1.
(a) The determinant of A,
(
8. The matrix M satisfies the equation,
form (
. Find M expressing it in the
)
9. Given that
(a) Find
(
(
)
(
(i) A-B,
(b) Write down
10.
)
)
)
(ii) the inverse of matrix A.
(
(
)
)
(a) Find AB.
(b) Given that B is the inverse of A, write down the value of x and the value of y.
You’ve been armed with strength for every battle. The forces that are for you
are greater than the forces against you.-Joel Osteen
Compiled and solved by Mr. Mununga J
5
TOPIC 4: GEOMETRIC PROGRESSION
Worked examples
1. In a geometric progression, the third term is and the fourth term is
. Find
(i) The first term and the common ratio,
(ii) The sum of the first 5 terms of the geometric progression,
(iii) The sum to infinity.
2. The first three terms of a geometric progression are 6+n, 10+n and 15+n. Find
(i) The value of n,
(ii)The common ratio,
(iii) The sum of the 6 terms of this sequence.
Answers
1. (i)
(ii)
( )
( )
(iii)
2. (i)
(ii) 6+n, 10+n, 15+n
6+10, 10+10, 15+10
=16, 20, 25…
(iii)
( )
Compiled and solved by Mr. Mununga J
6
Activity
1. If x+1, x+3 and x+8 are the first three terms of a GP. Find
(i) The value of x,
(ii) The first term and common ratio,
(iii) Sum of the first 5 terms.
, ….., Find
2. For the geometric progression 20, 5,
(i) The common ratio,
(ii) the nth term,
(iii) the sum of the first 8 terms.
3. The first three terms of a geometric progression are x+1, x-3 and x-1. Find
(i) the value of x,
(ii) the first term,
(iii) the sum to infinity.
4. In a geometric progression, the third term is 16 and the fifth term is 4. Calculate
(i) the first term and the common ratio,
(ii) the tenth term,
(iii) the sum to infinity.
5. Given the GP 2, -6, 18, …
(a) Find the formula for the nth term of the GP.
(b) Calculate the value of the 9th term.
(c) Which term is equal to 1458?
6. The first term of a GP is 16 and the 5th term is 1.
(a) Find the eighth term of the GP.
7.
(b) Will this GP converge? Give a reason for your answer.
8, x and 18 are three consecutive terms in a GP. Find two possible values of x.
8. T2, T3, and T4 of a GP are n-2, n and n+3 respectively. Find:
(a) The value of n,
(b) the common ratio,
(c) S10.
S5 of a GP is 121 and the common ratio is . Find
(a) T1,
(b) the sum to infinity.
.
Compiled and solved by Mr. Mununga J
7
TOPIC FIVE (5): SETS
Worked examples
1. The Venn diagram below shows tourist attractions visited by students in a certain week.
(i)
Find the value of y if 7 students
visited mambilima falls only?
(ii) How many students visited
(a) Victoria falls but not Gonya falls,
(b) two tourist attractions only,
(c) One tourist attraction only.
2. A survey carried out at Kamulima farming Block showed that 44 farmers planted maize, 32
planted sweet potatoes, 37 planted cassava, 14 planted both maize and sweet potatoes, 24 planted
both sweet potatoes and cassava; 20 planted both maize and cassava, 9 planted all the three crops
and 6 did not plant any of these crops.
(i) Illustrate this information on a Venn diagram.
(ii) How many farmers
(a) Were at this farming block,
(b) Planted maize only,
(c) Planted 2 different crops.
Answers
1. (i)
(ii) (a)
2. (i)
(iii)
Compiled and solved by Mr. Mununga J
(b)
(ii)
(iv)
(c)
8
Activity
1. The diagram below shows how learners in a Grade 12 class at Twaenda School travel to
school. The learners use either buses (B), cars (C) or walk (W) to school.
E
B
C
2
14
4
X
7
3
7
W
(i) If 22 learners walk to school, find the value of x.
(ii) How many learners use
(a) Only one mode of transport?
(b) Two different modes of transport?
2. At Sambilileni college, 20 students study at least one of the three subjects; Mathematics
(M),Chemistry (C) and Physics(P). All those who study chemistry also study mathematics, 3
students study all the three subjects, 4 students study mathematics only, 8 students study
chemistry and 14 students study mathematics.
(i) Draw a Venn diagram to illustrate this information.
(ii) How many students study;
(a) Physics only, (b) Two subjects only (c) Mathematics and physics but not chemistry
3. The Venn diagram below shows the optional subjects that all the Grade 10 learners at
Kusambilila secondary school took, in a particular year.
(i) Given that 12 learners took music, find the value of x.
(ii) How many learners were in Grade 10 this particular year?
(iii) Find the number of learners who took
(a) One optional subject only,
(b) Two optional subjects only.
Compiled and solved by Mr. Mununga J
9
4. Of the 50 villagers who can tune in to Kambani Radio station, 29 listen to news, 25
listen tosports, 22 listen to music, 11 listen to both news and sports, 9 listen to both sports
and music, 12 listen to both news and music, 4 listen to all the three programs, and 2 do not
listen to any programme.
(i) Draw a Venn diagram to illustrate this information
(ii) How many villagers
(a) Listen to music only,
(b) Listen to one type of programme only,
(c) Listen to two types of programmes only.
5. The results of a survey of 31 students are shown in the Venn diagram.
(i) Write down the value of
(a) x,
(b)
.
(ii) Write down a description, in words, of he set that has 16 members.
6. In a class of 36 students, 25 study History, 20 study Geography and 4 study neither History
nor Geography. How many students study
(a) Both History and Geography,
(b) One subject only.
If you want a positive life surround yourself with positive people.
Compiled and solved by Mr. Mununga J
10
TOPIC 6: PROBABILITY
Worked examples
1. A box contains identical buttons of different colours. There are 20 black, 12 red and 4 white
buttons in the box. Two buttons are picked at random one after another and not replaced in
the box.
(i) Draw a tree diagram to show all the possible outcomes.
(ii) What is the probability that both buttons are white?
2. A box of chalk contains, 5 white, 4 blue and 3 yellow pieces of chalk. A piece of chalk is
selected at random from the box and not replaced. A second piece of chalk is then selected
(i) Draw a tree diagram to show all the possible outcomes.
(ii) Find the probability of selecting pieces of chalk of the same colour.
Answers
B
1. (i)
(ii)
R
B
W
B
R
R
W
B
W
R
W
G
1. (i)
(ii)
B
G
Y
G
B
B
Y
G
Y
B
Y
Compiled and solved by Mr. Mununga J
11
Activity
1. A small bag contains 6 black and 9 green pens of the same type. Two pens are taken at
random one after the other from the bag without replacement. Calculate the probability that
both pens
(i) Are black,
(ii) are of different colours.
2. In a box of 10 bulbs, 3 are faulty. If two bulbs are drawn at random one after the other, find
the probability that
(i) Both are good,
(ii) one is faulty and the other is good.
3. A bag contains 3 white, 8 green and 4 blue beads. A bead is selected and not replaced. A
second bead is selected. Draw a tree diagram to show all the possible outcomes. Find the
probability that
(a) Both beads are white,
(b) one bead is white and the other bead green,
(c) First bead is white and the second bead blue,
(d) The two beads are of different colours.
4. Thirteen cubes of the same size numbered 1 to 13 are placed in a bag. If two cubes are drawn
at random from the bag one after the other and not replaced, what is the probability that
(i) Both cubes are odd numbered,
(ii) only one is even numbered.
5. A survey was carried out at certain hospital indicated that the probability that patient tested
positive for malaria is 0.6. What is the probability that two patients selected at random
(a) One tested negative while the other positive,
(b) Both patients tested negative.
6. Two pupils are to present a school at a human rights conference. If the two are chosen at
random from a group of 8 girls and 6 boys, calculate the probability that the two pupils picked
(a) Are both girls,
(b) at least one is a boy.
7. A bag contains 3 black and 2 white balls. Two balls are taken from the bag at random, without
replacement. By drawing a tree diagram, or otherwise, calculate the probability that
(a) Both balls are black,
(b) at least one ball is white,
(c) The two balls are of the same colour.
8. A bag contains five balls, numbered 1, 2, 3, 4, and 5, another bag contains six balls, numbered
1, 2, 3, 4, 5 and 6. One ball is drawn at random from each bag, find the probability that
(i) One ball is numbered 1 and the other 6, (ii) both balls have an odd number,
(iii) Both balls have the same number,
(iv) the sum of the number on the ball is 9.
9. A bag contains 1 red, 1 blue and 3 green balls. Two balls are taken from the bag, at random,
without replacement. Calculate the probability that
(a) Both balls are blue,
(b) neither ball is green.
Compiled and solved by Mr. Mununga J
TOPIC 7: VECTORS
12
Worked examples
1. (a) Show that the points L (-2, -10), M (2, 2) and N(5, 11) are collinear
(b) In the diagram below, ⃗⃗⃗⃗⃗ =2p, ⃗⃗⃗⃗⃗⃗ =4q and PX: XQ=1: 2
(i) Express in terms of
and /or
(a) ⃗⃗⃗⃗⃗
(b) ⃗⃗⃗⃗⃗⃗
(ii) Given that ⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗ show that ⃗⃗⃗⃗⃗ = (
(c) ⃗⃗⃗⃗⃗⃗⃗
)
2. In the diagram below, OABC is a parallelogram in which ⃗⃗⃗⃗⃗ = and ⃗⃗⃗⃗⃗ =b. AC and OB
meet at D such that OD=DB. OC is produced to E, such that
Express each of the following in terms of a and / or b
(i) ⃗⃗⃗⃗⃗⃗⃗
(ii) ⃗⃗⃗⃗⃗⃗
(iii) ⃗⃗⃗⃗⃗
Answers
1. (a) ⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗
(
)
(
( )
)
(
( ) ⃗⃗⃗⃗⃗⃗⃗
(b) (i) (a) ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ =⃗⃗⃗⃗⃗ +⃗⃗⃗⃗⃗⃗
(
)
)
( )
( ) and M is common, then, L, M and N are collinear.
(b) ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
=-2 +4
Compiled and solved by Mr. Mununga J
(c) ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ =⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
13
(ii)
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
2. (i) ⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗ =
⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
(ii)⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ =⃗⃗⃗⃗⃗
(
)
(iii) ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
=
Activity
1. In the diagram below, ⃗⃗⃗⃗⃗ =a and ⃗⃗⃗⃗⃗ =b. C on AB is such that AC: CB=2:1 And B on OD
is such that OB: OD=1:2
Express as simply as possible in terms of a and/or b
(i) ⃗⃗⃗⃗⃗⃗
(ii) ⃗⃗⃗⃗⃗⃗⃗
(iii) ⃗⃗⃗⃗⃗
2. In the quadrilateral ABCD below, ⃗⃗⃗⃗⃗ =a, ⃗⃗⃗⃗⃗ =b, ⃗⃗⃗⃗⃗ =2b and AE: AC=1: 3
(i) Find in terms of a and /or b
(a) ⃗⃗⃗⃗⃗
(b) ⃗⃗⃗⃗⃗⃗
Compiled and solved by Mr. Mununga J
(c) ⃗⃗⃗⃗⃗⃗
14
(ii) Hence or otherwise, show that the points, B, E and D are collinear.
3. In the triangle below, ⃗⃗⃗⃗⃗ =a, ⃗⃗⃗⃗⃗ =b and
(i) Express in terms of a and/or b
(a) ⃗⃗⃗⃗⃗⃗
(b) ⃗⃗⃗⃗⃗⃗
(c) ⃗⃗⃗⃗⃗⃗
(ii) Given that M is the midpoint of OC, show that ⃗⃗⃗⃗⃗⃗
.
4. In the diagram below, OAB is a triangle in which ⃗⃗⃗⃗⃗ =3a and ⃗⃗⃗⃗⃗ =6b. OC: CA= 2: 3 and
AD: DB =1: 2. OD meets CB at E.
(i) Express each of the following in terms of a and /or b
(a)
⃗⃗⃗⃗⃗
ii) Given that ⃗⃗⃗⃗⃗
(b) ⃗⃗⃗⃗⃗⃗
(c) ⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ , express ⃗⃗⃗⃗⃗ in terms of h, a and b.
Compiled and solved by Mr. Mununga J
15
5. In triangle OPR below, the line OX produced, meets PR at Y. PQ and OY meet at X such
that XQ
. ⃗⃗⃗⃗⃗ =p, ⃗⃗⃗⃗⃗⃗ =q and OQ=QR.
(i) Express each of the following as simply as possible in terms of p and /or q
(a) ⃗⃗⃗⃗⃗⃗⃗
(b) ⃗⃗⃗⃗⃗⃗⃗
(ii) Given that ⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗ , show that ⃗⃗⃗⃗⃗ =(
)
.
TOPIC 8: PSEUDO CODES AND FLOWCHARTS
Worked examples
1. study the pseudo code below
Start
Enter a, r, n
R=1-r
IF R=0 THEN
Print ‘’ the value of r is not valid’’
Else
End if
Print Sn
Stop
Construct a flow chart corresponding to the pseudo code above.
Compiled and solved by Mr. Mununga J
16
2. Study the flow chart below
Start
Enter r
Is
R < 0?
YES
Error ‘’ r must be positive’’
NO
Display Area
Stop
Write a pseudo code corresponding to the flowchart program above.
Happiness is within. It has nothing to do with how much applause you get
or how many people praise you. Happiness comes when you believe that
you have done something truly meaningful.-Martin Yan
Compiled and solved by Mr. Mununga J
17
Answers
1.
Start
Enter a, r, n
R=1-r
is
R=0?
YES
Print ‘’ the value of r is not
valid
NO
Print Sn
Sn
Stop
2.
Start
Enter r
IF
THEN
Display error message ‘’ r must be positive’’
Else
End if
Display Area
Stop
Compiled and solved by Mr. Mununga J
18
Activity
1. The program below is given in pseudo code.
Start
Enter x, y
Let M=square root(x squared + y squared)
IF M < 0
THEN display error message ‘’ M must be positive’’
ELSE
END IF
Display M
Stop
Draw the corresponding flowchart for the information given above.
2. The program below is given in the form of a pseudo code
Start
Enter radius
IF radius < 0
Then display ‘’error message’’ and re-enter positive radius
Else enter height
IF height < 0
Then display ‘’error message’’ and re-enter positive height
Else
End if
Display volume
Stop
3. The program below is given in the form of a flowchart.
Start
Enter a, r
Is
I r I< 1?
NO
YES
Display sum to infinity
Stop
Write a pseudo code corresponding to the flowchart Program above.
Compiled and solved by Mr. Mununga J
19
4 The flowchart below shows the steps in calculating the volume of a solid given the base area
(A) and height (h)
Start
Enter A
YES
Is A<0
Error message ‘’A must be
positive
NO
Enter h
YES
Is h<0
Error message ‘’h must be
positive
NO
V=A*h
Display V
Stop
Write the corresponding pseudo code for the flowchart given above.
5. Study the flowchart below
Start
Enter a, b, c
D=b*b-4*a*c
Is
D < 0?
YES
Print ‘’no real
solutions’’
NO
X1= (-square root D)/2*a
X2= (-b-square root D)/2*a
Print x1, x2
stop
Write a pseudo code corresponding to the flow chart program above
Compiled and solved by Mr. Mununga J
20
TOPIC 9: MENSURATION (FRUSTUM)
Worked examples
1. The figure below is a cone ABC from which BCXY remained after the small cone AXY
was cut off (Take
Given that
,
and
, calculate
(i) The height AE, of the smaller cone AXY,
(ii) The volume of XBCY, the shape that remained.
2. The figure below is a frustum of a cone. The base diameter and top diameter are 42cm and
14cm respectively while the height is 20cm. (
Calculate its volume.
Compiled and solved by Mr. Mununga J
21
3. The diagram below shows a bin in the form of a frustum with square ends of sides 4cm and
10cm respectively. The height of the bin is 9cm
Find the volume of the bin.
4. The diagram below is a frustum of a rectangular pyramid with a base 14cm long and 10cm wide. The
top of the frustum is 8cm long and 4cm wide.
Calculate its volume.
Answers
1. (i)
(ii)
2.
3.
Volume
Compiled and solved by Mr. Mununga J
22
√
4.
)
√
Activity
1. The frustum of a cone has height 6cm and the radius of its ends is 9cm and 15cm. find its
volume
2. The diagram shows a waste paper basket in a form of a frustum with rectangular ends
with sides 20cm by 25cm and 24cm by 30cm respectively. Its height is 35cm. find the
volume of the basket
Work for a clause, not for applause, live life to express, not to impress.
Compiled and solved by Mr. Mununga J
23
3. The diagram below shows a frustum TQRS of a cone. (Take
Given that US=3cm, UV=10cm and RV=8cm,
calculate Its volume.
4 The diagram below shows a prism with base ABCD. ABCD is a trapezium with AB II DC.
AB=7cm, AD=5cm, DC=3cm and AF=20cm. Calculate the volume of the prism
5. The diagram below shows a pyramid with square base with sides 10cm and perpendicular
height of 12cm
(a) Calculate the slant height l,
(b) Calculate the total surface area of the pyramid.
Compiled and solved by Mr. Mununga J
24
TOPIC 10: CALCULUS
Worked examples
1. (a) Determine the equation of the normal to the
the point (3, 7).
(b) Evaluate ∫
that passes through
.
2. (a) Find the equation of the tangent to the curve
at the point where x=2.
(b) The equation of the curve is
Find the coordinates of the
stationary points.
Answers
1. (a)
(b) ∫
*
+
(
2. (a)
)
(b) At stationary points,
(
Compiled and solved by Mr. Mununga J
)
25
Activity
1. The equation of a curve is
Find
(a) The equation of the normal where x=2,
(b) The coordinates of the stationary points.
2. (a) Find the coordinates of the points on the curve
gradient is zero.
(b) Evaluate ∫
where the
.
3. (a) Evaluate ∫
(b) Find the equation of the normal to the curve
4. The gradient function of a curve is
the point (1, 2).
at the point where x=4.
Find the equation of the curve passing through
TOPIC 11: LOCUS
Worked examples
1. (a) (i) Construct triangle PQR in which PQ=10cm, QR=8cm and
=50°.
(ii) Measure and write the length of PR.
(b) On your diagram, within triangle PQR, construct the locus of points which are
(i) Equidistant from P and Q,
(ii) Equidistant from PR and PQ,
(iv) 5cm from R.
(c) A point T within triangle PQR is such that it is 5cm from R and equidistant from P and Q.
label the point T.
(d) Another point X is such that it is less than or equal to 5cm from R, nearer to Q than P and
nearer to PQ than PR. Indicate clearly, by shading, the region in which X must lie.
2. (a) Construct a quadrilateral ABCD in which AB=10cm, angle, ABC=120°, angle
BAD=60°, BC=7cm and AD=11cm.
(b) Measure and write the length of CD.
(c) Within the quadrilateral ABCD, draw the locus of points which are
(i) 8cm from A,
(ii) equidistant from BC and CD.
(d) A point P, within the quadrilateral ABCD is such that it is 8cm from A and equidistant
from BC and CD. Label point P.
(e) Another point Q, within the quadrilateral ABCD, is such that it is nearer to CD than BC
and greater than or equal to 8cm from A. indicate, by shading, the region in which Q must
lie.
Compiled and solved by Mr. Mununga J
26
Answers
1. (a) (i)
2. (a)
(ii) 7.9cm
(b) CD=8.6cm
Activity
1. (a) (i) Construct a triangle ABC where AB=BC=CA=7cm.
(ii) Measure and write the size of
.
(b) Within the triangle ABC, construct the locus of points
(i) Equidistant from AB and BC,
(ii) 4cm from B,
(iv) 3cm from AB.
(c) A point R, within triangle ABC, is such that it is nearer to BC than AB, less than 3cm from
AB and less than 4cm from B. shade the region in which R must lie.
2. (a) (i) Construct a triangle JKL in which KL=8cm, KJ=6cm and JL=10cm.
(ii) Measure and write angle JLK.
(b) Within the triangle JKL, draw the locus of points which are
(i) 5cm from J,
(ii) 3cm from JL,
Compiled and solved by Mr. Mununga J
27
(iii) Equidistant from JK and JL
(c) A point Q, within triangle JKL, is such that it is greater than or equal to 5cm from J, less
than or equal to 3cm JL and nearer to JK than to JL. Indicate by shading the region in which
Q must lie.
3. (a) (i) Construct a parallelogram ABCD in which AB=8cm, BC=5.3cm and
C=60°.
(ii) Measure and write the length of AC.
(b) On your diagram draw the locus of points within the parallelogram ABCD which are
(i) 2.5cm from AB,
(ii) 3cm from C,
(iii) Equidistant from BC and CD.
(c) P is a point inside parallelogram ABCD such that P is nearer to BC than CD, less than or
equal to 3cm from C, less than or equal to 2.5cm form AB. Indicate clearly by shading, the
region in which P must lie.
TOPIC 12: EARTH GEOMETRY
Worked examples
1. In the diagram below, A and B are points on latitude 60°N while C and D are points on
].
latitude 60°S [
(i) Calculate the distance BC along the longitude 60°E in nautical miles.
(ii) A ship sails from C to D in 12 hours. Find its speed in Knots.
Compiled and solved by Mr. Mununga J
28
2. W, X, Y and Z are four points on the surface of the earth as shown in the diagram below.
(i) Calculate the difference in latitude between W and Y.
(ii) Calculate the distance in nautical miles between
(a) X and Z along the longitude 105°E,
(b) Y and Z along the circle of latitude 30°.
3. P (80°N, 10°E), Q (80°N, 70°E) and R (85°S, 70°E) and S (85°S, 10°E) are four points
on the surface of the earth
(i) Show these points on a clearly labeled sketch of the surface of the earth.
(ii) Find in nautical miles
(a) The distance QR along the longitude,
(b) The circumference of the circle of latitude (85°S
Answers
1. (i)
2. (i)
(ii)
°
(b)
Compiled and solved by Mr. Mununga J
(ii) (a)
.
29
3. (i)
(ii) (a)
(b)
Activity
1. The diagram below shows a model of the earth, the points C and D are on the same
longitude. The latitudes of C and C are 45°N and 55°S respectively. (Take =3.142 and
R=3437nm).
(i) Write the position of the point C.
(ii) Calculate the difference in latitude between C and D.
(iii) Find the distance CD in nautical miles.
(iv) Calculate the circumference of the latitude 45°N in nautical miles.
2. The points A (15°N, 40°E), B (35°S, 70°E) and C (35°S, 40°E) are on the surface of the earth.
(use
.
(i) Calculate the distance AC in kilometers.
(ii) An aeroplane takes off from point B and flies due west on the same latitude covering a
distance of 900km to a point Q
(a) Calculate the difference in longitude between B and Q,
(b) Find the position of Q.
Compiled and solved by Mr. Mununga J
30
3. In the diagram below, A(65°N, 5°E), B(65°N, 45°W) and C are three points on the surface of
the model of the earth and O is the centre of the model. The point C, due south of A, is such
]
that angle AOC=82°. [
(i)
(ii)
(iii)
State the longitude of A.
Calculate the latitude of C.
Calculate, in nautical miles, the shortest distance
(a) Between A and C measured along the common longitude,
(b) Between A and B measured along the circle of latitude.
4. The points K, L and M are on the surface of the earth as shown in the diagram below
(
(i) Find the difference in longitude between points K and L.
(ii) Find, in kilometers the distance
(a) LM,
(b) KL.
Compiled and solved by Mr. Mununga J
31
TOPIC 13: TRIGONOMETRY
1. (a) Three villages A, B and C are connected by straight paths as shown in the diagram
below
Given that AB=15km, angle ABC=79° and angle ACB=40°, calculate the
(i) Distance AC,
(ii) Area of triangle ABC,
(iii) Shortest distance from B to AC.
(b) Solve the equation
.
(c) Sketch the graph of
.
2. (a) The diagram below shows the location of houses for a village Headman (H), his
secretary (S) and a trustee (T). H is 1.3km from S, T is 1.9km from H and angle
THS=130°
Calculate
(i) The area of triangle THS,
(ii) The distance TS,
(iii) The shortest distance from H to TS.
(b) Find the angle between 0° and 90° which satisfies the equation
As long as God is involved in your situation, victory is
guaranteed! Prophet Shepherd Bushiri
Compiled and solved by Mr. Mununga J
.
32
Answers
1. (a) (i) Distance AC
(ii)
̂
(iii)
(b)
(c)
y
90°
1
180°
0
270°
-1
360°
0
1
0
90
180
1
2. (a) (i)
(iii)
(b)
Compiled and solved by Mr. Mununga J
360
270
3
3
3
3
(ii)
3
33
Activity
1. (a) in the diagram below, K,N, B and R are places on horizontal surface KN=80m,
NB=50m,
=60° and
=52°.
(i) Calculate
(a) KR,
.(b) The area of triangle KNB.
(ii) Given that the area of triangle KNR is equal to 3260m2, calculate the shortest
distance from R to KN.
(b) Sketch the graph of
2. (a) In triangle PQR below, QR=36.5m, angle PQR=36° and angle QPR=46°
Calculate
(i) PQ,
(ii) The area of triangle PQR,
(iii) The shortest distance from R to PQ,
(b) Solve the equation
Compiled and solved by Mr. Mununga J
.
34
3. (a) In Zambezi constituency, A, B and C are polling stations as shown on the diagram below
Calculate
(i) Angle BAC,
(ii) Angle ACB,
(iii) The area of triangle ABC correct to 1 decimal place,
(iv)The shortest distance from C to AB.
(b) Find the angle between 0° and 90° which satisfies the equation
.
4. (a) in triangle ABC below AC=275km, angle BAC=125° and angle ACB=40°.
Calculate
(i) The distance BC,
(ii) the area of triangle ABC,
(iii) The shortest distance from A to BC.
(b) Solve the equation
.
Compiled and solved by Mr. Mununga J
35
TOPIC 14: STATISTICS
Worked examples
1. A farmer planted 60 fruit trees, in a certain month, the number of fruits per tree was
recorded and the results were as shown in the table below.
Fruits per tree 2
3
4
5
6
7
8
Frequency
1
5
4
6
10
16
18
(a) Calculate the standard deviation
(b) Answer this part of the question on a sheet of graph paper.
(i) Using the table above, copy and complete the relative cumulative frequency table
below
Fruits per tree
2
3
4
5
6
7
8
Cumulative frequency
1
6
10
16
26
42
60
Relative cumulative
0.02 0.1
0.17 0.27
frequency
(ii) Using a scale of 1cm to represent 1 unit on the x-axis for
and 2cm to
represent 0.1 units on the y-axis for
, draw a smooth relative cumulative
frequency curve
(iii) Showing your method clearly, use your graph to estimate the 70th percentile.
2. The ages of people living at Pamodzi village are recorded in the frequency table below.
Ages
Number of
people
7
22
28
23
15
5
(a) Calculate the standard deviation.
(b) Answer this part of the question on a sheet of graph paper.
(i) Using the information in the table above, copy and complete the cumulative frequency
table below
Age
Number of people 7
29
100
(ii) Using a scale of 2cm to represent 10 units on both axes, draw a smooth cumulative
frequency curve where
(iii) Showing your method clearly, use your graph to estimate the semi-inter quartile range.
Compiled and solved by Mr. Mununga J
36
Answers
1. (a)
Midpoint (x) Frequency
2
1
3
5
4
4
5
6
6
10
7
16
8
18
60
∑
∑
Fx
2
15
16
30
60
112
144
379
4
45
64
150
360
784
1152
2559
̅
√
√
(b) (i)
Fruits per tree
Cumulative frequency
Relative cumulative
frequency
2
1
0.02
3
6
0.1
4
10
0.17
5
16
0.27
6
26
0.43
7
42
0.7
A true test of character isn’t how you are on your best days but how you
act on your worst days.
Compiled and solved by Mr. Mununga J
8
60
1
37
(ii)
1. (a)
Midpoints(x)
5
15
25
35
45
55
Frequency (f)
7
22
28
23
15
5
100
Fx
35
330
700
805
675
275
2820
175
4950
17500
28175
30375
15125
96300
∑
∑
∑
∑
̅
√
(b) (i)
Age
Number of people
7
Compiled and solved by Mr. Mununga J
29
57
80
95
100
38
(ii)
Activity
1. The frequency table below shows the distribution of marks obtained by 90 learners on a test.
Mark (x)
Frequency
2
10
15
23
30
(a) Calculate the standard deviation
(b) Answer this part on the question on a sheet of graph paper.
(i) Copy and complete the relative cumulative frequency table below
Mark (x)
Cumulative frequency 0
2
12
27
50
80
Relative cumulative
0
0.02
0.13
0.3
frequency
10
90
(ii) Using a scale of 2cm to represent 10 units on the x-axis for
and 2cm to
represent 0.1 units on the y-axis for
, draw a smooth relative cumulative
frequency curve.
(iii) Showing your method clearly, use your graph to estimate the 65th percentile.
Compiled and solved by Mr. Mununga J
39
2. The table below shows the amount of money spent by 100 learners at school on a particular
day.
Amount in kwacha
Frequency
13
27
35
16
7
2
(a) Calculate the standard deviation
(b) Answer this part of the question on a sheet of graph paper.
(i) Using the table above, copy and complete the cumulative frequency table below
Amount in kwacha
Frequency
0
13
40
100
(ii) Using a scale of 2cm to represent 5 units on the horizontal axis and 2cm to represent 10
units on the vertical axis, draw a smooth cumulative frequency curve.
(iii) Showing your method clearly, use your graph to estimate the semi-inter quartile range.
3. (a) The table below shows the distribution of the ages of 30 players at a school
Age (x) years
Frequency
10
0
11
2
12
5
13
7
15
8
16
6
17
2
Calculate the standard deviation
(b) (i) Using the table above, copy and complete the relative cumulative frequency table below.
Age (x) years
Cumulative
frequency
Relative cumulative
frequency
0
2
7
0.00
0.07
0.23
14
22
28
30
1.00
(ii) Using a scale of 22cm to represent 1 unit on the x-axis for
and a scale of
2cm to represents 0.1 units on the y-axis for
draw a smooth relative
cumulative frequency curve
(iii) Showing your method clearly, use your graph to estimate the 90th percentile.
You are because iam and iam because you are. We are one
another’s strength.-T. B. Joshua
Compiled and solved by Mr. Mununga J
TOPIC 15: LINEAR PROGRAMMING
1. A tailor at a certain market intends to make dresses and suits for sale.
(a) Let x represent the number of dresses and y the number of suits. Write the inequalities
which represent each of the conditions below.
(i) The number of dresses should not exceed 50
(ii) The number of dresses should not be less than the number of suits
(iii) The cost of making a dress is K140.00 and that of a suit is k210.00. The total cost should
be at least k10500.00
(b) Using a scale of 2cm to represent 10 units on both axes, draw x and y axes for
and
shade the unwanted region to indicate clearly the region where (x, y) must
lie.
(c) (i) The profit on a dress is k160.00 and on a suit it is k270.00. Find the number of dresses
and suits the tailor must make for maximum profit.
(ii) Calculate this maximum profit.
Answers
1. (a) (i)
(ii)
(b)
Compiled and solved by Mr. Mununga J
(iii)
40
41
(c) (i) 50, 50
50 suits and 50 dresses
(ii)
K21500.00
Activity
1. A hired bus is used to take learners and teachers on a trip. The number of learners and teachers
must not be more than 60. There must be at least 35 people on the trip. There must be at least
6 teachers on the trip. The number of teachers on the trip should not be more than 14.
(a) Write four inequalities which represent the information above
(b) Using a scale of 2cm to represent 10 units on both axes, draw x and y axes for
and
respectively and shade the unwanted region to indicate clearly the region
where the solution of the inequalities lie.
(c) (i) If the group has 25 learners, what is the minimum number of teachers that must
accompany them?
(ii) If 8 teachers go on this trip, what is the maximum number of learners that can be
accommodated on the bus?
(d) If T is the amount in kwacha paid by the whole group, what is the cost per learner if
2. Makwebo prepares two types of sausages, Hungarian and beef, daily for sale. She
prepares at least 40 Hungarian and at least 10 beef sausages. She prepares not more than
160 sausages altogether. The number of beef sausages prepared are not more than the
number of Hungarian sausages.
(a) Given that x represents the number of Hungarian sausages and y the number of beef
sausages, write four inequalities which represents these conditions.
(b) Using a scale of 2cm to represent 20 sausages on both axes, draw the x and y axes for
and
respectively and shade the unwanted region to show
clearly the region where the solution of the inequalities lie
(c) The profit on the sale of each Hungarian sausage is k3.00 and on each beef sausage is
k2.00. How many of each type of sausages are required to be prepared to make
maximum profit?
(d) Calculate this maximum profit.
Compiled and solved by Mr. Mununga J
42
3. Mipando makes two types of chairs for sale: dining and garden. He intends to make at
least 10 dining chairs and at least 20 garden chairs. He wants to make not more than 80
chairs altogether. The number of garden chairs must not be more than three times the
number of dining chairs.
(a) Let x be the number of dining chairs and y the number of garden chairs. Write four
inequalities to represent the information above.
(b) Using a scale of 2cm to represent 10 chairs on each axis, draw x and y axes for
and
respectively and shade the unwanted region to indicate clearly the
region where the solution of the inequalities lie.
(c) Given that the profit on the sale of a dining chair is k80.00 and profit on a garden chair
is k50.00, how many chairs of each type should Mipando make in order to maximize the
profit?
(c) What is this maximum profit?
TOPIC 16: FUNCTIONS
Worked example
1. The diagram below shows the graph of
Compiled and solved by Mr. Mununga J
43
(i) Use the graph to solve the equations
(a)
(b)
(ii) Calculate an estimate of the
(a) Gradient of the curve at the point where x=-3,
(b) Area bounded by the curve, x=-3, x=-1 and y=-10.
Answers
1. (i) (a)
(b)
(ii) (a)
X -4
Y 6
-2
8
2 4
12 14
(b)
56
Activity
1. (a) The values of x and y are connected by the equation
corresponding values of x and y are given in the table below.
X
Y
-2
P
-1.5
-8.5
-1
0
-0.5
4
0
5
0.5
4.5
1.5
5
. Some
2
9
(i) Calculate the value of p.
(ii) Using a scale of 4cm to represent 1 unit on the x-axis for
and 2cm to
represent 5 units on y-axis for
, draw the graph of
(iii) Use your graph to solve the equation
.
(iv) Calculate an estimate of the gradient of the curve at the point where x=1.5.
Believe in yourself, take on your challenges, dig deep within yourself to conquer fears.
Never let anyone bring you down. You got to keep going.-Chantal Sutherland
Compiled and solved by Mr. Mununga J
44
2. The diagram below shows the graph of
(a) Use the
graph to find the solutions of the equations
(i) 3+3 2− −3=0
(ii) 3+3 2− −3=5
(b) Calculate an estimate of
(i) the gradient of the curve at the point (−3,0)
(ii) the area bounded by the curve, = 0, = 0 and =20
A meaningful silence is always better than meaningless words.
Compiled and solved by Mr. Mununga J
45
3. The diagram below shows the graph of = 3+ 2−5 +3
Use the graph
(a) to calculate an estimate of the gradient of the curve at the point (2,5).
(b) to solve the equations
(i)
,
(ii)
.
(c) to calculate an estimate of the area bounded by the curve =0, =0 and =−2.
4. The values of x and y are connected by the equation =( −2)( +2). Some corresponding
values of x and y are given in the table below
X
Y
-3
-15
-2
0
-1
3
0
0
1
-3
2
0
3
k
(a) Calculate value of
(b) Using a scale of 2cm to represent 1 unit on the x – axis for −3≤ ≤3 and 2cm to represent
5 units on the y- axis for−16≤ ≤16. Draw the graph of =( −2)( +2)
(c) Use your graph to solve the equations
(i) ( −2)( +2)=0
(ii) ( −2)( +2)= +2
The end of examples and questions,
answers to the questions next.
Compiled and solved by Mr. Mununga J