DEPARTMENT OF STATISTICS
UNIVERSITY OF BOTSWANA
STA 101 : Mathematics for Business and Social Sciences 1
QUESTION BANK
CHAPTER 1
Real number system: Properties of Real numbers- Transitive, Commutative, Associative,
Distributive and Inverse. Binary Relations - Reflexivity, symmetry and transitivity. Laws of
indices and simple examples.
1. State the four categories of number system and give two examples for each.
2. State any four rules of algebra on number system and illustrate the rule by giving an
example each.
3. Identify the rules of algebra associated with the number system in each of the
following examples:
(i)
3+6 = 6+3
(ii)
10  3 = 3  10
(iii) (3+7)+2 = 3+(7+2)
(iv)
(5  3)  2 = 5  (3  2)
(v)
2  (4+6) = (2  4)+( 2  6)
(vi)
3+(-3) = (-3) + 3 = 0
1
(vii) 2  21  2   1
2
1
1 1
(viii)
0  0 
5
5 5
1
(ix)
4 1  1 41  1
4. Write the Cartesian product of the set A  1, 2,0, 2 and set B  1, 2,3 . Giving
reasons examine whether the following are relations from A to B .
(a) R1   1,1,  1,2,  2,2, 0,3, 2,1, 2,3
(b) R2   1,1 2,2,  2,30,3, 2,1, 2,3, 2,2
(c) R3   1,2 1,3, 0,10,3, 2,1, 2,3, 3,1
5.
If A  1,2,3,4,B  1,0,1  , give an example of a relation and represent it by an
arrow diagram.
6. Let A= The set of vowels, B= The set of distinct letters in ‘STATSTICS’ and C = The
set of vowels in ‘I OWE YOU’. Write down the Cartesian product of the sets (i) A
and B (ii) A and C (iii) A and A B.
7. Let the relation R be defined as R 
 x, y  y is the additive identity of x on the set
A  0,1, 2,3, 4 . Represent the relation by an arrow diagram and write the ordered
pairs of the relation. Is this a relation on A ? Is the relation reflexive? Give reason.
8. Let
R
R
be
the
relation
on
the
 x, y  y is the additive inverse of x
set A  2,  2,3,  3,1, 1 defined
on
the
as
set A  2,  2,3,  3,1, 1 .
Represent the relation by an arrow diagram and ordered pairs of the relation. Verify
whether R satisfies the property of symmetry?
be defined as ‘multiplicative inverse of ’on the set
1


A  2,  1, 0.5,3 to the set B    1, 0, 0.5,1, 2,3 . Represent the relation by an
3


arrow diagram and write the ordered pairs of R .
9. Let the relation
R
10. Let the relation R be “is a square of” on a set A = {-2,-1, 0, 1, 2, 3, 4}. Represent the
relation by an arrow diagram and ordered pairs. Determine, giving reasons, which of
the reflexive, symmetric and transitive properties are satisfied.
11. Let the relation R be “is a factor of” on the set A = A = {5, 10, 15, 20, 225,}.
Represent the relation by an arrow diagram and ordered pairs. Determine, giving
reasons, which of the reflexive, symmetric and transitive properties are satisfied.
12. Let the relation R be ‘is a factor of’ on the set A  3,6,9, 27 . Represent the
relation by an arrow diagram and write the ordered pairs of the relation. Determine,
giving reasons, which of the reflexive, symmetric and transitive properties are
satisfied by the relation.
1


13. Let the relation R be ‘cube root is’ on the set A  8, , 64  to the set
27


1


B  4,  2, , 3, 4  . Represent the relation by an arrow diagram and write the
3


ordered pairs of R .
14. Let the relation R be ‘greater than or equals to’ on the set A  2,4,6,8 .
(i)
Represent the relation by an arrow diagram and write the ordered pairs of the
relation.
(ii)
Determine, giving reasons, which of the reflexive, symmetric and transitive
properties are satisfied by the relation R .
(iii)
Show that R  A  A .
15. All roads linking different neighborhoods in a certain city are one-way. The following
arrow diagram shows how 4 such localities are linked.
(i) Write down the ordered pairs for the relations
(ii) Verify which of the properties of relations, namely: Reflexive, Symmetric,
Transitive and equivalence are satisfied.
16. Suppose that the set A  {2, 1, 0, 1, 2}, the set B  {0, 1, 2, 3, 4, 5} and R is
the relation ‘the square is’ on the set A to set B .
i)
Represent R by an arrow diagram or on a graph.
ii)
Write the ordered pairs of the relation.
17. Let A = { - 3,-2,-1,0,1,2,3 } and B = {-6,-5,-4,-2,0,2,3,4,5,6} and the Relation R be
defined by (i) R 1 = { (x, y) x = - y},(ii) R 2 = { (x, y) x 2 = y 2 } , (iii)
R 3 = { (x, y) xy is a multiple of 5} . Represent the relation by an arrow diagram and
ordered pairs.
18. Let A= {a,e,i,o,u}and the relation R on A be defined by
R= { (a, a) (a, e),(a, o),(e, a),(e, e) ,(e, u),(i, i),(i, a),(o, a)(o, o),(o, i),(u, u)}.
Check if reflexive, symmetric, transitive properties are satisfied by the relation R.
19. Let the relation R denote “ an even number ” on the set A = { 1,2,3,4,5,6} to set B =
{ 0,1,4, 8,9,10}. Represent the relation by an arrow diagram and ordered pairs.
20. Simplify the following algebraic expressions:
3
(i) 4 16 x 4 y 8
(ii)
 27 x
3
8y
6
3
(ii)
 3a b  2a
2
3 2
 2x4 y 2 
 2 3
 8x y 
1 2
b

4
1 1 1
  0
x y z
1 x
1 y
1 z
22. Find the value of
.


1
1
1 x
1 y
1  z 1
21. If a x  b y  c z and abc  1 show that
x 1  y 1 y 1  z 1 z 1  x 1


x y
yz
zx
23. Find the value of
3
 13 3 
x y 
 at x  3 and y  2 .
24. Evaluate 
3 2 2
x y 
25. Let P 
1562  5 x3
3
184  2 x 4
. Find the value of P when x  2 .
26. Simplify completely:
(i)
3xyz 2 x  7 yz   3 y  2 xz   4 z  5xy  6 z 
(ii)
10uv  u  2v  3u 2v 2   3uv  2u  5  u  v  
(iii)
x 2  4 y 2 x 2  4 xy  4 y 2

4x  8 y
x  2y
(iv)
 x   x 2   x 2  x  1  x 
(v)
27. Factorize:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
9 y

2

 14 yz  8z 2   3z  4 y  5z   2 y  3 y  2 z 
16 x4  16 x3  4 x2
5 p3q  10 p2 q2  5 pq3
2ab4  4ab2c2  2ac4
25a2b2 100a2c2
100 x 2  36 y 2
2 x4  8x2 y 4
4 p2  5 p  2  8 p  5 p  2  3 5 p  2
28. Perform the indicated operation and simplify:
6
(i)
3
x2
  x 12 y 2 3
1
(iv)  1 3

3
4
 x y
6

 (v)


(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
1
4a  6b
 2
2a  3b 4a  9b2
2a 2  3ax  x 2 2a  x
 2
x y
x  xy
x2  4 x  4 4 x  8

2x2  8 x2  2x
 p 2  16 p  36 3 p 2  12
 2
3 p 2  27
3p  9 p
4 x2  y 2
x2  y 2

x 2  2 xy  y 2 4 x 2  4 xy  y 2
x x

2 5
3x
2x 
10
2
p q  4q 3
pq  2q 2

4 p  8q 12 p  4 pq
2
 1 2 1   1  x  

  2  3 
x   x 2 
x x
1
1

x 1 x
1
1

x x 1
29. If x2  y 2  45 and x  y  9 , find the value of x  y
1
1
1
1
30. If x   2, find the value of (i) x 2  2 (ii) x 3  3 (iii) x 4  4
x
x
x
x
1
1
1
1
31. If x   2, find the value of (i) x 2  3 (ii) x 3  3 (iii) x 4  4
x
x
x
x
Chapter 2
Equations & Inequalities
Equations & Linear equations and inequalities in one and two variables and their graphing.
Quadratic equations and inequalities and their solutions.
1. Solve the following equations:
(i) 7 x  3(2 x  4)  1  4 x  1  2(2  2 x)  3
2
1
1
(ii)
x  ( x  3)   (6  3x)  1
3
6
2
2. Solve the following inequalities and indicate the solution set on the number line.
(i) 2 x  3  3
(ii) 2 x  5  8
(iii) 15  31  x 
(iv)  4( x  2 y)  3( x  3)
(v) 3x  2 y  1  0
(vi) x  1  3 y  1  1
3. The product of two consecutive even integers is six more than three times their sum.
Find the integers.
4. A rectangular field has a perimeter of 1000 meters. The length of the field is twice the
breadth. Form the equation to the perimeter of the field and hence find the length and
breadth of the field.
5. Of the two numbers, the first number is three less than the second number. Two times
the first number is six more than half the second number. Find the numbers.
6. The sum of three consecutive numbers is 273. Find the numbers.
7. One side of the triangle is 4 m longer, and another 3 m longer than the shortest side.
If the perimeter of the triangle is 25 m, find the lengths of the three sides.
8. Among three numbers, the second number is six more than the first while the third
number is two more than the second. Thrice the third number is three less than four
times the first number. Find the numbers.
9. Bianca traveled 1200 km. If her average speed had been 5 more km per hour, she
could have made the trip in 12 hours less time. Find Bianca’s average speed.
10. Mr.Mogatwe is walking from his village to the nearby town which is 15 kilometers
away. If he increases his walking speed by 2kms per hour he can reach the town 2
hours earlier. Find the speed at which Mr. Mogatwe walked to reach the town.
11. Mmatalodi estimated she would need 2 hours to complete a math test if she did not
use a calculator. However, if she used a calculator, she would only need 45 minutes.
On the day of the test, Mmatalodi’s instructor indicated she could use a calculator but
she must complete the test in a one-hour period.
Halfway through the test
Mmatalodi’s calculator died.
(i)
How long will it take Mmatalodi to complete the test?
(ii)
Will she complete the test in the 1 hour period allowed?
12. A University cinema charges students P1.50 , staff’s children P3.00 and staff P5.00 .
For one show 2600 tickets were sold. There were ten times as many students’ tickets
as there were staff tickets and the children’s tickets were 200 more than the staff
tickets. What is the total collection from the cinema?
13. Solve the following quadratic equations:
(i) 4 x 2  5x  6  0
(ii) 2 x 2  4 x  3
(iii) 2 x 2  7 x  15  0
(iv) 2  6 2 y  9 y 2  0
(v) 4 x 2  1  0
14. Solve the following quadratic inequalities:
2
x
x
2
(ii)  x  1  4
(i) 3 
(iii) 2 x2  x  1  0
(iv) x2  x  6
(v) 2 x2  8x  10
(vi) x2  x  12
(vii) 2 x2  8x  10
(viii) 2 x 2  3x  2
.
15. (i)
Given x 
k
2
s

 2 y   , make s the subject of the formula.
k

(ii)
Solve the literal equation p 
(iii)
Solve 6q  2 z 
(iv)
rs  pq
by making q as the subject of interest
ps  rq
3q
 2 x  3 y  for q .
4
r

Solve the following for r : G   B   R t
s

Solve for x : z 
(v)
(vi)
(vii)
5k
 x  i
4
k
Solve the following for x : n  [2l  (k  1) x]
2
3q
Solve 6q  2 z 
2 x  3 y) for q .
4
16. Find the solution to the following:
(i) A straight line that is parallel to the line  x  2 y  2  0 passes through the points
 a ,1
and  2, 2  . Find the value of a .
(ii) Two straight lines A and B are perpendicular. The equation of B is
2 x  3 y  6  0 . Find the slope of A .
(iii)Find the equation of a straight line passing through the points 12,13 and
14,16  .
(iv) Find the equation of a straight line passing through the point ( -3,-2 ) and parallel
to the straight line 3x  2 y  1  0.
(v) Find the equation of a straight line passing through the point
 2,  3
and is
parallel to the straight line 3 y  2 x  3  0 .
(vi) Find the slope and the y -intercept in each of the following cases: (a) y  1  0
and (b) x  5  0 .
(vii) Given that 2 x  3 y  1  0 and 2 px  4 y  5  0 are the equations of two
parallel lines, find the value of p .
(viii) Find the equation of a straight line passing through the point ( -3,-2 ) and
parallel to the straight line 3x  2 y  1  0.
(ix) Find the equation of the straight line that passes through the point  1,1 and is
parallel to the line 2 x  6 y  4 .
(x) Find the equation of a straight line passing through the point ( -2,3) and
perpendicular to the straight line 2 x  3 y  2  0.
(xi) Find the equation of a straight line passing through the point 1,  2  and is
parallel to the straight line 2 y  4 x  3  0 .
(xii)
Find the equation of the straight line passing through the point 1,3 and is
perpendicular to the line 3x  2 y  2  0 .
(xiii) Find the equation of a straight line passing through the point ( -1, 2) and
perpendicular to the straight line 3x  2 y  1  0.
(xiv)
Find the equation of a straight line passing through the point ( -2,3) and
perpendicular to the straight line x  3 y  1  0.
(xv)
Find the equation of the straight line that passes through the point  3,5 and
has slope
(xvi)
1
.
3
Find the equation of the straight line passing through the points 1,5 and
 3,15
17. Mr T buys a brand new house today for P 200000 . He expects it to be worth
P 800000 in ten years’ time.
(i)
Write a linear equation that relates the value of the house  y  to its age  x  .
(ii)
What would be the value of the house five years from today?
18. A server purchased at a cost of P 60000 in 2002 has a scrap value of P 12000 at the
end of 4 years. If the straight line method of depreciation is used,
(i) Find the rate of depreciation.
(ii) Find the linear equation expressing the server’s book value at the end of t
years.
(iii)Find the server’s book value at the end of third year.
19. John buys an apartment building for P150 000. He expects it to be worth P30 000
after 30 years. (i) What is the linear equation that states the value of the building in
Pula given its age in years? and (ii) What is the value of the building after 20 years?
20. Modise invests in property by buying a new house worth P170 000. He expects to sell
the house in 10 years for P300 000.
(i) What is the linear equation that states the value of the building in Pula given
its age in years?
(ii) It Modise was to sell the house after 15 years, how much would be the selling
price of the house?
(iii)What would be the age of the house if the value of the house triples?
21. Assume that a firm can sell as many units of its product as it can manufacture in a
month at P 20 each. It has to pay out P 300 fixed costs in addition to a marginal cost
of P 15 for each unit produced. How many units must be produced to break even?
22. The market value of a machine is known to be linearly related to its age in years. The
value of the machine was known to be P250, 000 at the end of the first year and P
150,000 at the end of the third year. Express the linear relationship between the
market value and the age in years. When will the machine be a scrap?
23. Sales of Golden Garments were 5.1 million Pula in 1992 and 7.50 million Pula in
1993. Suppose the sales grow linearly with time in years since 1992.
(i)
Find an equation relating sales to time.
(ii)
Predict sales for 1995, 1998, 2001 and 2010.
(iii)
When will annual sales reach 15 million Pula?
24. A retail merchant pays a flat weekly rental fee of P 225 . She is considering adding
an additional product line. She can sell each unit of the new line for P 2.50 profit.
However, if she adds the new line, she must rent additional space at an added cost of
P125.00 weekly (because of the new space). How many units of the new product
must she sell each week to break even?
25. Arabang has decided to manufacture ‘Genius’ computer systems at a cost of P1800
each and sell at P3000 each. She must spend P150 000 to start the manufacture in
terms of license fees and other initial expenses. How many computer systems must
she sell in order to break even?
26. A manufacturer of Christmas tree ornaments knows that the total cost of making x
thousand ornaments of a certain kind is given by C  600  60 x where C is in Pula
and that the corresponding sales revenue is given by R  200 x  4 x2 which is also in
Pula. How many thousands of ornaments will the manufacturer have to produce and
sell to break even?
27. A furniture rental shop has set apart an amount of P5000 for buying new chairs. The
shop manager learnt from the advertiser that Funmart is offering a discount of P10 on
each chair the shop wants to buy. If the discount offer is availed, the manager figures
out that he could buy 25 more chairs than what P5000 could otherwise fetch. Find the
cost of each chair and the number of chairs purchased.
28. A farmer was offered a flock of sheep for P 8000 . But he made a tough bargain and
reduced the price of each sheep by P 40 . As a result he could buy 10 more sheep with
the same amount of money. How many sheep did the farmer buy and how much did
he pay for each sheep?
CHAPTER 3
Linear Programming Problem
Graphical method of its solution, Application to real life problems
1. Solve the L.P.P
(i) Maximize z  3x  4 y subject to: 4 x  2 y  80 ; 2 x  5 y  180 ; x  0 , y  0 .
(ii) Maximize P  3x  5 y subject to: x  4 ; y  6; x  2 y  10 ; x  0 , y  0 .
2. Solve the L.P.P
(i) Minimize C  2 x  y subject to: 5x  10 y  50 ; x  y  1 ; y  4 ; x  0 ; y  0 .
(ii) Minimize Z  x  y subject to: 2 x  y  12 ; 5x  8 y  74 ; x  6 y  24 ; x  0 ;
y  0.
3. Soundex produces two models of clock radios. Model A requires 12 minutes of work
on assembly line I and 15 minutes of work on assembly line II. Model B requires 10
minutes of work on assembly line I and 15 minutes of work on assembly line II. At
most, 30 labour hours of assembly time on line I and 25 labour hours of assembly
time on line II are available each day. The market research has indicated that demand
for the two types of clock models is limited 50. It i s anticipated that Soundex will
realize a profit of P10 on model A and P15 on model B.
(i) Formulate the Soundex’s production scheduling problem as a L.P.P.
(ii) Indicate the feasible and the non-feasible regions graphically.
(iii)How many clock radios of each model should be produced and what is the
maximum profit?
4. Mr.Modise has 20 hectares for planting black-eye beans and millet. The farmer has to
decide how much of each to grow. The cost per hectare for black-eye beans is P30
and for millet is P20. The farmer has budgeted for P480. Black-eye beans require 1
man-day per hectare and millet requires 2 man-days per hectare. There are 36 mandays available. The profit on black-eye beans is P100 per hectare and on millet is
P120 per hectare.
(i) Formulate the farmer’s crop production problem as a linear programming
problem.
(ii) Draw a graph showing the feasible region and shade the unwanted regions.
(iii)Using the graph, find the number of hectares of each crop the farmer should
sow to maximize profit. What is the profit at this production level?
5. A farmer has at most 500 acres of on which to plant two crops –tomatoes and potato.
Producing tomatoes requires 2 hours of labour per acre and potato requires 3 hours of
labour per acre. The farmer has 1200 hours of labour available. The profit per acre is
P80 for tomatoes and P 100 for potatos.
a) Formulate the farmer’s production scheduling problem as a L.P.P.
b) Indicate the feasible and the non-feasible regions graphically.
c) How many acres of each crop should be planted to maximize profit?
6. A weaver makes two kinds of cloth, both of which contain red and white wool. Type
A uses 0.2 kg of red wool and 0.3 kg of white wool per metre of cloth. Type B uses
0.4 kg of red wool and 0.1 kg of white wool. He has 200 kg of red wool and 150 kg of
white wool available. Type A sells for 15 Pula per metre and type B sells for 25 Pula
per meter.
(i) State the equations/inequalities which describe the production conditions.
(ii) Draw a graph of these equations/inequalities and hence find how much of each
kind of cloth should he manufacture to maximize his revenue.
(iii)What is his maximum revenue?
7. A manufacturer produces tables and desks. Each table requires 5 hours for
assembling, 6 hours for buffing and 2 hours for crafting. Each desk requires 2 hours
for assembling, 6 hours for buffing and 4 hours for crafting. The firm can use no more
than 40 hours for assembling, 60 hours for buffing and 32 hours for crafting each
week. The profit on a table is P 80 and that on a desk is P 50 and the firm expects to
sell all its products.
(a) State the equations/inequalities which describe the production conditions.
(b) Draw a graph to represent this problem shading any unwanted regions.
(c) Find the number of tables and desks the firm should manufacture weekly in order
to maximize profit and determine the maximum profit.
8.
A firm makes two food products A and B and the contribution to profit is P 2 per
unit of A and P 3 per unit of B . There are three stages in the production process:
cleaning, mixing and canning. The number of hours of each process required for each
product and the total number of hours available for each process are given in the
following table. The firm wishes to maximize its profit.
Unit of A
requires
Unit of B
requires
Total hours
available
Cleaning
3
Mixing
6
Canning
2
6
2
1.5
210
120
60
(i)
Formulate the firm’s problem as a linear programming problem.
(ii)
Draw a graph of the feasible region.
(iii)
What combination of A and B should the firm produce so as to maximize its
profit?
(iv)
What is the maximum possible profit of the firm?
9. Pearl’s company has just received an urgent order for its bathroom cabinets, which it
makes in two styles, Standard and De Luxe. The order is for ‘at least 100 bathroom
cabinets of either variety, including at least thirty of the De Luxe style.’ The Standard
model takes two hours of assembly time and has variable costs of P40 , whereas the
De Luxe model takes five hours of assembly time and has variable costs of P 60 .
There are 400 hours in total available for assembly. The equipment can be used to
assemble either style of cabinet in any combination. Other reasons dictate that at least
as many Standard cabinets as De Luxe cabinets must be made. The company wishes
to minimize its variable costs of production on this special order.
(a) Formulate this problem as a linear programming problem.
(b) Graph the constraints, shading the unwanted region.
Recommend the best product mix for the company. What is the variable cost incurred?
10.
A furniture manufacturer makes two types of chairs: reclining and non-reclining
chairs. The reclining chair requires 2 hours of assembly time and 1 hour of packing
time. The non-reclining chair requires 1 hour of assembly and 1 hour of packing time.
Each month, the manufacturer has 420 work-hours available for assembly and 300
work-hours for packing. The profit on each reclining chair is P140 and on each nonreclining chair is P100. How many chairs of each type should be produced per month
in order to maximize profit?
CHAPTER 4
MATRIX ALGEBRA
Addition and Multiplication of two matrices. Square, Identity and Diagonal Matrices. Transpose
of a Matrix. Inverse of a matrix using augmented matrix and Cofactors.
1. Define the following terms in one or two sentences. Give an example in each case: (i)
Diagonal matrix, (ii) Transpose of a matrix, (iii) Identity matrix,(iv) square matrix .
2. The matrices A to D are defined as follows:
 2  3 9  4
 3  1 2
1 
 11 2



 3 
6
7 
0
1 4
A =
,B= 
, C = 1 0 3 4 5 , D =   .
 6
3
 2
0
2
9 
2 1




 
1 5 8 
 5
 1 0 8 
0 
(i)
Write down the size of each matrix A to D.
(ii)
Write down the elements a14 , a23 , a34 , a42 , b21, b32 , b42 , c13 , d 21 .
(iii)
Identify the row matrix and write down its transpose.
(iv)
Identify the column matrix and write down its transpose.
(v)
Identify square matrix and write down its transpose.
4 2
1
T
3. Given A  
, find (i) A and (ii) A A .

9 5 
4. Solve the following matrix equation for x, y and z.
 x 3  2  y z   3 7 
 z 2   2  z  x    2 0 

 
 

 1 1
1 2
5. Given A  
and
B

 2 3 ,



3 4 
(i) Find D such that 2 A  3D   B .
(ii) Find A B .
 3 2
1 2
1
6. Given A  
and
, find D such that A  D   B .
B




2
 1 5
3 4 
3
3 5
 1 1
2
7. Given A  
B
C


 , and k  2 . Find matrix D given that
5 1 
 1 - 1
 1  3
A  kB  C  D .
3 1
 2 6
 2 1
B

C

8. Given A  
,
and
1 2 

 3 3 , find D if A  2  B  D   2C .


 2 2


3 1 
9. Let A = 
 and B =
0 2 
 4  2
2.
2 1  , Compute (A+B)


 1 0 
 1 2 
10. Given A  
show that AT BT
,B  


 1 2
 0 2

show that AT

T
 2 1 1
 BA. If A =  0 2 1 ,
1 0 1
T
2
A2 .
11. Find the product matrix AB given A
1
4
1
0
2 1
1
2 and B
1
2
1
1
3 .
2
1
7 0
12. Let B  
 . (i) What entries constitute the main diagonal of B ? (ii) Is B a
 3 4
1
diagonal matrix? (iii) Find B (the inverse of B ).
1 2
 2 1
1
1
1
13. Given A  
and B  
, find (i)  AB  and (ii) A .


0 4 
 1 1
1
1
14. Given A  
0
2
15. Given A   4
 2
16. Given that A
2
 2 1
1
1
and B  
, find (i)  AB  and (ii) A .


4
 1 1
3 2 
2 1 find the determinant of AT .
1 1 
2
1
3
2
find the inverse of A2 .
17. Find the determinant of the following matrices by Siruss’ rule and direct expansion
method and check whether they give the same answer.
2 6 2
 1 2 2 
0 1 1




(i) A   2 4 2 
(ii) A   2 3 1 
(iii) A  1 2 3
 6 10 1 
 2 1 1 
1 1 1
1
18. Find A ( the inverse of A ) using the cofactor method given
(i)
0 1 1
A  1 2 3
1 1 1
1 1 0 
(ii) A  0 1 1 
0 0 1 
(iii)
 1 2 3
A  1 3 5
 2 5 9 
19. Find the inverse of the following matrices using the elementary row operations:
1
(i) A
4
1
0
2
2
1 0 2 
 1 2 3
1 1 1 




1 (ii) B  1 3 5 (iii) B  0 1 1  (iv) B   0 1 2 
1 1 4 
 2 1 1 
 2 5 9 
10
2
CHAPTER 5
Systems of Linear Equations
Solutions of systems of linear equations in two and three variables – Elementary row operations,
Elimination method, Substitution method, Gauss-Jordan method and Matrix inversion method.
1. Solve the system of equations by elimination method and Gauss-Jordan method.
3x  y  1, x  2 y  5
(i)
(ii) 3x  2y  2  0,3x  2y  2  0
2. Solve the system using the method of separation and matrix inversion method.
4 x  y  1, 2 x  2 y  5 (ii) 3x  y  2, x  y  4 (iii) 2x  3y  8, 4x  y  2 .
(i)
1 1  x  1 
3. A system of linear equations in two variables is expressed as 
     .
1 2  y   3 
1 1 
A
 and hence
1 2 
(ii) obtain the solution of the system by pre-multiplying both sides of the equation by
A1 .
(i) Find the inverse,
 A  of
1
4. Solve the following system of linear equations by method of elimination and method
of substitution.
(i)
x1  x2  x3  14 ; x2  x3  10 ; 2 x1  x2  x3  16
(ii)
u  v  w  2 ; 3u  3v  w  2 ; u  w  0
2x  3y  z  3
(iii)
3x  y  z  7
3x  y  z  1
2 x  2 y  z  2
(iv)
2 x  y  3z  1
x  2 y  3z  2
5. Solve the following system of linear equation by Gauss Jordan method and matrix
inversion method.
3x  3 y  z  6
x  y  2 z  12
(i) x  y  2 z  5
2x  2 y  z  3
(ii) y  z  10
2 x  y  z  16
3x  3 y  z  6
2 x1  x2  3x3  5
(iv) 2 x2  x3  2
 x1  4 x2  1
(iv)
x  y  2z  5
2x  2 y  z  3
4 x  z  13
(iii)
2 x  y  3z  10
x  2 y  4z  5
6. Pula 100 can buy five liters of milk and 4 liters of ice cream. Pula 123 can buy 6 litres
of milk and five liters of ice cream of same brands. Find the per liter cost of milk and
ice cream.
7. The total cost of 5 chairs and 6 tables is P 3700; while the total cost of similar 3 chairs
and 4 tables is P2460. Find the cost of each chair and table.
8. Dunlop wishes to produce three types of tires: types A, B, and C. To manufacture a
type-A tire requires 2 minutes on machine I, 1 minute on machine II, and 2 minutes
on machine III. A type-B tire requires 1 minute on machine I, 3 minutes on machine
II, and 1 minute on machine III. A type-C tire requires 1 minute on machine I and 2
minutes each on machines II and III. There are 3 hours available on machine I, 5
hours available on machine II, and 4 hours available on machine III for processing the
order. Dunlop needs to determine how many tires of each type they should make in
order to use all of the available time.
Formulate, but do not solve the problem.
9. Cindy regularly makes long distance phone calls to three foreign cities- London,
Tokyo and Hong Kong. The matrices A and B give the length (in minutes) of her calls
during peak and non peak periods respectively, to each of these three cities during the
month of June.
 London Tokyo HongKong 
 London Tokyo HongKong 
A= 
and B = 


60
40
150
250
 80

 300

The costs for the calls (in dollars per minute) for the peak and non peak periods for
the month in question are given below, respectively by the matrices
London 0.34 
London 0.24 


Tokyo 0.42  and
C=
D = Tokyo  0.31
HongKong  0.48
HongKong  0.35
Compute the matrix AC + BD and explain what it represents.
10. The following table gives the number of shares of certain corporations held by Leslie
and Tom in their respective IRA accounts at the beginning of the year.
Leslie
Tom
IBM
500
400
GE
350
450
FORD
200
300
WAL-MART
400
200
Over the years they added more shares to their accounts, as shown in the following
table:
Leslie
Tom
IBM
50
0
GE
50
80
FORD
0
100
WAL-MART
100
50
(i)
Write a matrix A giving the holdings of Leslie and Tom at the beginning of the
year and a matrix B giving the shares they have added to their portfolios.
(ii)
Find a matrix C giving their total holdings at the end of the year
11. Matrix A gives the eligible percentage of voters in the city of Newton, classified
according to party affiliation and age group.
Dem Re p Ind
Under 30 0.50 0.30 0.20
30  50  0.45 0.40 0.15
Over 50 0.40 0.50 0.10
The population of eligible voters in the city by age group is given by the matrix B:
A=
Under30 30  50 Over50
B = 30,000 40,000 20,000
Find a matrix giving the total number of eligible voters in the city who will vote
Democratic, Republic and Independent.
CHAPTER 6
Introduction to the concepts of logarithms: Common and Natural Logarithms.
1. Find the value of the following using the definition of logarithm.
(i) log 4 1024
 1 
(ii) log3  
 27 
(iii) log 2 128
(iv) log 5 3125
( ).
(v)
2.
Find the value of x in each of the following:
(i) log x 9  2
(ii) log5 125  x
(iii) log9 3  x
(iv) log10 x  2
3
(v) log x 8 
4
(vi) log5 x  5
8
(vii) log x
3
27
1
(viii) log x
 2
16
(ix) log x 32  5
3. Simplify the following :
 1 
(i) log3 27  log3 81  log 4 64  log 2   .
 32 
(ii) log3 27  log3 81  log 4 64.
(iii) log 4 32  log9 81  log 2 2.
(iv) log2  log4  log8  log16  log32  log64
4. Show that log5  log25  log125  log625  log3125  15 log5
5.
Expand and simplify the following expressions:
(i)
log3 x 2 y3
(ii)
x2  1
log 2 x
2
(iii)
ln
(iv)
x2 x2 1
ex
(
)
6. Solve for x in each of the following:
(i) 2e x 2  5
(ii) 5ln x  3  0
(iii) 32 x1  4x2
)
(iv) (
(v)
(vi)
(
)
(vii) loge x  loge 25  0.2t
7. Solve for x, logx  1  logx  2  log 2  log 3
3
2
8. Given logb x  logb 4  logb 8  2logb 2 , find x .
2
3
9. Find the value of x , if log  x  2   log  x  3  log 2  log3 .
10. Given that log 4= 0.6020 and log 3= 0.4771, use the laws of logarithms to find
1
(i) log 12
(ii) log 0.75 (iii) log 81
(iv) log 48
(v) log
.
300
11. Given log 2  0.3010,log3  0.4771 , find the value of
(i) log 48
(ii) log 144
(iii) log0.06)
Chapter 7
Sequences and Series
Arithmetic and Geometric Sequences and progressions.
1. Write down the first five terms of the following sequences given the respective n th :
  1n 
(i) an   
 (ii) tn   n  n  2 n  4
 n ! 
(iii) t1  1, t2  1 and for n  3, tn  tn1  tn2 , (iv) t1  3, t2  5 , and for
k  3, tk  tk 1  tk 2
2. Find a formula (either explicit or recursive) for the nth term of the following
sequences and write down the next three terms.
(i)
-2, 4, -8, 16, -32, 64, -128,
1 1 1 1
(ii)
1, , , , ,
3 5 7 9
15
25
35
(iii)
5, , 10, , 15, ,
2
2
2
(iv)
0,1, 3, 4, 6, 7, 9,
3. Which term of the sequence 4, 9,14,19, is 99?
4. Suppose an arithmetic sequence has an initial term of 10 and a common difference of
3. (i) Write down the first 5 terms of the arithmetic sequence and (ii) Find the sum of
the first 50 terms.
5. Find the sum of the first 50 terms of the following sequence: 3, 7,11,15,19, .
6. The fourth and seventh terms of an A.P are respectively 25 and 34. Find the tenth
term of the A.P.
7. What is the least number of the terms of the Arithmetic Progression
3,10,17, 24, 31, that must be added to give a total sum greater than 1000?
8. The fourth and seventh terms of an A.P are respectively 25 and 34. Find the tenth
term of the A.P.
9. There are 20 terms in an AP. The sum of the first 10 terms is 55 and the sum of the
last 10 terms is 355. Find (i) the first term, a1 and (ii) the common difference, d .
10. Given that the first term and the common difference of an arithmetic sequence are 10
and 3 , respectively, find:
(i)
The first four terms of the sequence.
(ii)
The sum of the first 50 terms of the sequence.
11. Given that the first term and the common difference of an arithmetic sequence are 15
and 2 , respectively, find:
(iii)
The first four terms of the sequence.
(iv)
The sum of the first 25 terms of the sequence.
12. What is the first term of an A.P whose sum to 20 terms is 610 and the common
difference is 3?
13. A line which is 140 cm is divided into 20 pieces whose lengths form an AP. If the
longest piece is 11 cm, find the length of the shortest piece.
1 1 1 3
, , , ,    , find:
14. For the sequence
18 6 2 2
The 10th term.
The Sum of the first 10 terms.
1 1 1
15. Find the sum of the first 10 terms of , , ,
8 4 2
(i)
(ii)
.
16. The sum to infinity of a certain geometric sequence is 27. If the first term is 36, find
the common ratio.
17. The second and sixth terms of a geometric sequence are 1.6 and 25.6 respectively.
Find the tenth term.
18.
(a) Evaluate the following sums:
100
(i)
(ii)
k 1

1
(vi)   
r 4  2 

20
k 2
 2n
2n  n
n 6
19
(vii)   2n  200  (viii)
n10
5
2  3k  1 (x)   
i 6  2 
k
(iii)
n 3
r

15
22
 2k

15
i

  4n  5n 
n 3
k
12
1
2     5k

k 5  3 
k 3
10

12
(iv)
(ix)
(v)
5
3
k 1
k
Chapter 8
Mathematics of Finance
Simple and Compound interest formulas and applications. Banker’s discount and Effective rate
of interest. Daily, Monthly, Quarterly, Yearly and Continuous Compounding. Annuity, Future
value of an ordinary annuity. Present value of an ordinary annuity.
1. Find the simple interest on a P2500 investment made for 5 years at an interest rate of
12% per year. What is the accumulated amount?
2. A deposit of P 22,000 in First National Bank Botswana becomes P 30250 at a simple
interest of 12.5%. How long the deposit was held in the bank?
3. How long does it take P 2000 to double at 10% compounded annually?
4. Find the time necessary for P 230 000 invested at 19.5% simple interest to become
P 400 000 .
5. A bank deposit paying simple interest grew from an initial sum of P1000 to a sum of
P1075 in 9 months. Find the interest rate.
6. A certain sum of money was invested at 14.75% simple interest per annum. If, in
three years, this earned an interest of P 546 , find the sum of money invested.
7. Find the accumulated amount and the total interest received at the end of 3 years on a
P1200 bank deposit paying compound interest at a rate of 8% year.
8. Find the effective rate of interest corresponding to the given nominal rate of interest
10% per year compounded quarterly.
9. Find the present value of P150, 000 due in 5 years at 8% per year compounded
monthly.
10. Find the future value of an ordinary annuity for P1800 per quarter for 6 years at 8%
per year compounded quarterly.
11. Find the present value of P150, 000 due in 5 years at 8% per year compounded
continuously.
12. The amount of P2000 is deposited in the bank at the rate of 12% interest annually.
What will be the total amount after 4 years if compounded daily?
13. Compute the present value of an ordinary annuity in which P600 is deposited every
month for 5 years. The interest rate is 10% p.a. compounded monthly.
14. Find the future value of an ordinary annuity after 8 years in which, P200 is deposited
every month with interest rate of 15% p.a. compounded monthly.
15. How much must a firm invest today in an account which pays a 14% annual interest
rate continuously compounded to assure that P1000000 will be obtained in 6 years?
16. An individual deposits P200 at the end of every quarter in to an account which yields
12% per year compounded quarterly. What is the value of the account in 3 years?
17. Suppose you want to pay off a balance of P3600 in 36 months with monthly
installments at interest rate 20% per year. How much total money and total interest
will you pay?
18. An individual wants to determine the present value of an account which will worth
P3000 at the end of 4 years with interest compounded annually at 5% per years. In
other words, how much should the individual deposit now to yield P3000 at the end of
4 years?
19. Modise invested in a money market mutual fund that pays interest on a daily basis.
Over a 2-year period in which no deposits or withdrawals were made, her account
grew from P4500 to P5268.24. Find the effective interest rate at which Modise’s
account was earning interest over that period,(assuming 365 days in a year).
20. What amount of money deposited today at a 12% annual rate compounded monthly
will provide exactly enough to pay a lobola of P5000.00 4 years from now?
21. An account earning 13.75% annual interest compounded quarterly was established
17 years ago. Today the account is worth P 285000 . What was the initial amount in
the account?
22. TC is planning to build his new house. To meet the building expenses, he wants to
deposit enough money in an account to be able to withdraw P10000 each month for
the next 6 months. The account pays 12% interest per year compounded monthly.
How much should TC deposit in the account today to have enough for these 6
withdrawals?
23. A firm wants to set aside an amount each quarter for 5 years which will result in an
accumulated amount sufficient to pay off a P600, 000 loan at the end of the period.
Payments are to be paid into an account yielding a 16% annual interest compounded
quarterly. Find the quarterly payments.
24. A recently married couple wants to establish a savings account to provide enough for
them to make a P15, 000 down payments on a home in 6 years. The account pays
12% interest per year compounded semi-annually. How much should they deposit at
the end of every 6 months to assure an exactly sufficient amount in this account?
25. Bruno estimates that he will need P 5000 in cash to pay for the lobola to marry his
girlfriend in four years from now. How much should he deposit at the end of each
month into an account which pays an interest of 13% per annum compounded
monthly so that there will be enough money in the account at the end of four years?
26. A recently married couple wants to establish a savings account to provide enough for
them to make a P15,000 down payment on a home in 6 years. The account pays 12%
interest per year compounded semi-annually. How much should they deposit at the
end of every 6 months to assure an exactly sufficient amount in this account?