BOQ Tutorial

1. For each statement below, state whether the statement is true or false.
i. The union of two all more sets is never an empty set.

ii. Dividing a rational number by an integer will give an irrational number.
iii. {2, 5, 9} ⊆ {0, 2, 4, 5, 7, 8, 9, 10, 13}.
iv. {2, 5, 9} = {5, 9, 2}
2. Consider the universal set ψ that contains even numbers. ψ contains the sets A and B.
Set A is the set A = {x|x > 0}, and set B is the set A = {y|y < 15}. Using roster
method, give the set A∩B.
3. Simplify the following expressions:
i. (4x)3 × (2x)1 −1 .
ii. log3y 2y 3 − log3y 6y 4 .
4. Suppose demand of a product, which depends on price, is given by a linear equation. When
the price is 4 the demand is 20, and when the price is 8 the demand is 12. Determine the
demand equation.
5. Solve the equations below.
i. 25x = 22x ×26

e3x
ii. ex−3
= 30
iii. log[2x − 50]3 = 30, where 2x − 50 > 0.
6. A bank gives an interest rate of 7%. How long, in years, will it take to double one’s
investment?
7. Estimate the following using the binomial theorem:
i. (1.3)15
ii. (0.8)16
8. Solve the inequalities below.
i. 4 − 3x≤ − 5
ii. |2x − 4|≤6
iii. |2x + 4|≥6
iv. 2x2 + 7x > 4
y−3
v. y 2 −4
≤0
9. For each case provided below, determine the values that must be excluded from the set
of inputs so that the case becomes a function. In each case the input values are real
numbers.
1
i. v(x) = x
+ 20
4x2 +3x
ii. p(x) = x2 +2x+1
+ 20
10. Express each expression below using signma notation.
1
i. 2 + 3x + 4x2 + 5x3 + 6x4
ii. 0 + x2 − 2x3 + 3x4 − 4x5 + 5x6
11. Consider the polynomial function f (x) = x4 + 6x3 − 32x.
i. Determine the roots of f (x) = 0. Hence

ii. sketch the graph of f (x).
3
12. Consider the functions f (x) = x
− 2 and g(x) = 2x − 1 where x > 0. Determine
i. (f −1 ◦ g)(x).
ii. (g ◦ f )−1 (2).
13. Sketch the graphs of the functions provided.
i. f (x) = 2|x + 3| − 2
ii. g(y) = −|y − 1| − 1
iii. h(x) = 2(x + 1)2 − 2
iv. k(x) = −(x − 1)2 + 1
v. y = log0.5 [x]
vi. m(x) = 0.5x
vii. y = 2x−2 + 2
viii. y = −2x+2 + 2
14. The prices of a good in three different markets are interrelated in a manner given by the
system of simultaneous linear equations below.
5x1 + 3x2 − 7x3 = 0

7x1 + 5x2 − 3x3 = 16
3x1 − 5x2 + 2x3 = −8
x1 , x2 , and x3 are prices in markets 1, 2, and 3 respectively.
i. Rewrite the system of simultaneous linear equations in the matrix form
Ax = d.
ii. Use Cramer’s rule to solve the system of simultaneous linear equations.
iii. Check the solution obtained above using the inverse method (solve the system of
simultaneous linear equations using the inverse method- this is for the sake of prac-
tice).
15. An entrepreneur is repaying a loan of MK853, 600. S/he is repaying the loan in monthly
installments. S/he pays MK1, 000 in the first month, then increases the payment by
MK200 in every installment. In how many installments is s/he going to clear the loan?
16. Chisomo’s contract with VHC Company has come to an end. S/he is given a gratuity of
MK50, 000, 000.00. Every months s/he uses 0.5% of the gratuity money s/he had at the
beginning of the month. Determine the amount of the gratuity money that s/he remains
with after 20 months.
2
17. Chikondi borrowed MK1, 000 and agreed to pay a maturity value of MK1, 450 after 3
months. If simple interest was charged, determine the interest rate (annual rate).
18. On 12th March 2015 George Phiri borrowed MK1, 200, 000 from a bank. If he agreed to
pay 9% annual rate of interest (simple interest), calculate the total amount of money he
was supposed to repay if the loan was for 125 days. Assume the bank considers a 365-day
year.
19. What is the future value of MK13, 310, 000 if invested for a period of three years in an
account that pays interest at a rate of 12% compunded semi-annually?
20. For how long should MK10, 000, 000 be left in an account that pays interest at a rate of
12% compunded semi-annually to grow to MK14, 000, 000?
21. What is the present value of MK13, 310, 000 received in three years’ time at 10% interest
compunded quarterly?
22. An organisation wants to make a decision on two possible projects (project A and
project B) to take for a period of four years. Both projects require an initial invest-
ment of MK23, 000, 000. Investment on project A is expected to generate a cash inflow
of MK7, 500, 000 each year for a period of four years. The salvage value of project A is
zero. Investment on project B is expected to generate cash inflows of MK4, 000, 000 at
the end of the first year, MK5, 000, 000 at the end of second year, MK6, 700, 000 at the
end of the third year, and MK8, 400, 000 at the end of the fourth year. For project B, at
the end of the fourth year all the equipment will be sold for MK6, 500, 000.
i. Given that the target (dscount) rate of return for project A is 14% per annum, and
the discount rate for project B is 18%
(a) For each project, calculate the net present value (NPV). Give your answer to
the nearest Kwacha (MK).
(b) Using the NPVs obtained above, in terms of the project in which to invest
MK23, 000, 000, what advice would you give the organisation?
ii. Suppose the target rate of return for both projects is 10%
(a) For each project, calculate the internal rate of return (IRR).
(b) Using the IRRs obtained above, in terms of the project in which to invest
MK23, 000, 000, what advice would you give the organisation?
23. An organisation is considering to undertake a project that requires an initial investment

of MK10, 000, 000. The project is expected to generate MK2, 050, 000 per year for 10
years. Determine the payback period of this project. What advice would you give the
organisation?
24. An institution is considering of undertaking a project that requires an initial investment

of MK520, 000, 000. The institution expects the project to generate a cash inflow of
MK200, 000, 000 in the first year; a cash inflow of MK200, 000, 000 in the second year; a
cash inflow of MK190, 000, 000 in the third year; a cash inflow of MK170, 000, 000 in the
fourth year; and a cash inflow of 165, 000, 000 in the fifth year. Determine the payback
period for this project.

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1. For each statement below, state whether the statement is true or false.

i. The union of two all more sets is never an empty set.

3. Simplify the following expressions:

5. Solve the equations below.

i. 25x = 22x ×26

7. Estimate the following using the binomial theorem:

8. Solve the inequalities below.

10. Express each expression below using signma notation.

11. Consider the polynomial function f (x) = x4 + 6x3 − 32x.

i. Determine the roots of f (x) = 0. Hence

13. Sketch the graphs of the functions provided.

5x1 + 3x2 − 7x3 = 0

x1 , x2 , and x3 are prices in markets 1, 2, and 3 respectively.

i. Rewrite the system of simultaneous linear equations in the matrix form

23. An organisation is considering to undertake a project that requires an initial investment

24. An institution is considering of undertaking a project that requires an initial investment

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