TECHNICAL UNIVERSITY OF KENYA
FACULTY OF APPLIED SCIENCES AND TECHNOLOGY
SCHOOL OF BUSINESS AND MANAGEMENT STUDIES
MAIN EXAM FOR DECEMBER 2021 EXAMINATION SERIES
FIRST TERM EXAMINATIONS 2021/2022
FIRST YEAR EXAMINATION FOR THE DIPLOMA IN
BUSINESS STUDIES
ABBE 1122: BASIC MATHEMATICS
TIME: 2 Hours
DECEMBER 2021
Instructions to candidates:
This paper consists of FIVE Questions.
Answer Question ONE [30 Marks] and any other TWO Questions [20 Marks Each].
Write your college/admission number on the answer sheet.
This paper consists of 3 printed pages
Candidates should check the question paper to ascertain that all the pages are printed as
indicated and that no questions are missing.
© April 2021 The Technical University of Kenya Examinations
QUESTION ONE (30 MARKS: COMPULSORY)
a) Find the value of x using logarithm
[4mks]
𝑥
5 = 2(3)
b) Rationalize the following expression
𝑥
2
√5−√7
[2mks]
c) Use the rules of indices to simplify each of the following and where possible evaluation;
i)
15𝑥 6
3𝑥 4 5𝑥 2
and ii)
54 .6−2
[4mks]
52
d) Use the remainder theorem to determine the remainder when 3𝑥 3 − 2𝑥 2 + 𝑥 − 5
is divided by (𝑥 + 2).
[3mks]
2
e) Solve 𝑥 + 2𝑥 − 8 = 0 using quadratic formula.
[2mks]
f) Transpose the formula 𝑣 = 𝑢 +
𝑓𝑡
𝑚
to make 𝑓 the subject.
[2mks]
g) Without fully expanding the binomial series, (3 + 𝑥)7 determine the sixth term. [4mks]
h) Simplify the following expression
i)(1 − √5)(1 + √5)
ii) √90 × √600000
[4mks]
i) How many ways can first and second place be awarded to 10 people?
[2mks]
j) The 6th term of an arithmetic progression is 17 and the 13th term is 38. Determine the
19th term.
[3mks]
QUESTION TWO (20 MARKS)
a) Solve the cubic equation
𝑥 3 − 2𝑥 2 − 5𝑥 + 6 = 0 by using the factor
theorem.
[10mks]
2
2
b) Show that 𝑥 − 6𝑥 = (𝑥 − 3) − 9. Hence use completing the squares to solve
𝑥 2 − 6𝑥 = 5.
[5mks]
c) Use elimination method to solve the following pair of simultaneous equations
[5mks]
1
10
𝑥+𝑦 =
3
3
1
11
2𝑥 − 𝑦 =
4
4
QUESTION THREE (20 MARKS)
a) Evaluate (1.002)9 using binomial theorem correct to
i) 3 decimal places
ii) 7 significant figures
b) Determine the remainder when
𝑥 3 − 2𝑥 2 − 5𝑥 + 6 is divided by
i)
𝑥−1
ii) 𝑥 + 2
[4mks]
[2mks]
[2mks]
[2mks]
iii) Hence factorize the cubic expression
[4mks]
c) Ten different letters of alphabet are given. Words with five letters are formed from
these given letters. Then the number of words that have at least one letter repeated is?
[6mks]
QUESTION FOUR (20 MARKS)
a) The radius of a cylinder is reduce by 4% and its height is increased by 2%. Determine the
approximate percentage change in
i)
Its volume
[6mks]
ii)
Its curved surface area (neglect the product of small numbers)
[4mks]
b) In a geometric progression, the sixth term is 8 times the third term and the sum of the seventh
and eighth term is 192. Determine
i)
The common ration
[3mks]
ii)
The first term
[3mks]
iii)
The sum of the fifth to eleventh terms, inclusive
[4mks]
QUESTION FIVE (20 MARKS)
1
1
a) The first, twelfth and last term of an arithmetic progression are 4, 31 2, and 376 2
respectively. Determine
i)
The number of terms in the series
ii)
The sum of all the terms
iii)
The 80th term
[4mks]
[4mks]
[4mks]
b) Solve for 𝑥 using grouping terms ; 𝑥 3 + 3𝑥 2 − 4𝑥 − 12 = 0.
[4mks]
c) In how many ways a committee consisting of 3 men and 2 women, can be chosen from 7
men and 5 women?
[4mks]