Basic Mathematics Question Paper

Basic Mathematics

Course:Bachelor Of Science

Institution: Kenyatta University question papers

Exam Year:2009

KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2009/2010
FIRST SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE
SMA 102: BASIC MATHEMATICS

DATE: TUESDAY, 22ND DECEMBER 2009

TIME: 11.00 A.M. - 1.00 P.M.

INSTRUCTIONS:

QUESTION ONE (30 MARKS)

(a)
Find the possible values of x if

22x -
(
42 2x ) + 320 = 0

(4 marks)

(b)
The remainder when a polynomial P(x) is divided by (x - 2) is 3 and the

remainder when divided by (x + 1) is 6. Find the remainder when P(x) is
divided
by
(x - 2 ) (x + 1).

(5 marks)

(c) Prove
that

sin?
1+ cos?

+
= 2cos ?
ec

(5 marks)
1+ cos?
sin?

(d) Expand
3
1
( - 2x) in ascending powers of x, as far as the term in x3. (3
marks)

(e) If
3
cos A =
and
12
tan B =
where A and B are both reflex angles, find the
5
5
value
of
cos(A + B) .

(4 marks)

Page 1 of 4

(f)
A travelling squad for a basket ball team consists of two centres, five

forwards and four guards. The coach is to select a starting team of one centre,

two forwards and two guards. Find the number of ways he can do this.
(2 marks)

(g) Given
that
z = 3 + i
4 and w = 12 + 5i, write down the argument and
modulus
of
of
zw .

(3 marks)

(h)
Solve the equation cos 2x = sin x for o
o
0 = x = 360 .
(4
marks)

QUESTION TWO (20 MARKS)

(i)
State
the
remainder
theorem.
(2
marks)

(ii)
Use remainder theorem to show that (x - 2) and (x + 4) are factors of
24
f (x) = 2 4
x + 5 3
x -17 2
x -14x +

hence
factorize
f(x)
completely.
(8
marks)

(iii)
Find all the permutations of all the letters in the word M ISSISSIPPI
(3 marks)

(iv)
A school club has 13 members of whom 7 are boys and 6 are girls.

In how many different ways can a committee of 5 be selected if

(a)
The committee comprise of 2 boys and 3 girls.

(1 mark)

(b)
The committee has at least one girl

(3 marks)

(c)
The committee must include 1 boy who is the Chairman and one
girl
who
is
the
secretary
of
the
club.
(3
marks)

Page 2 of 4

QUESTION THREE (20 MARKS)

(a)
Solve the equation

o
o
4sin? + 3cos? - 2 = 0 for 0 = ? = 360

(6 marks)

(b)
Find the maximum and minimum values of 2sin? - 5cos? and the

corresponding values of ? , where 0o = ? = 360o.
(6
marks)

(c) If
sin(x + )
A = cos(x - B), find tan x in terms of A and B.

(4 marks)

(d)
Prove the Identity
cos 2A

= cos A - sin A

(4 marks)
cos A + sin A

QUESTION FOUR (20 MARKS)

(a)
Express the following in terms of (a + ß ) and aß
(i) 3
3
a + ß

(3 marks)
(ii)
2
(a - ß )

(3 marks)

(b) Given
that
3 2
x - 5x +1 = 0 has roots a and ß , determine the value of
2
2
?a
ß ?

+
+ (a - ß )2
?
?

(5 marks)
? ß
a ???

log 63
(c) If
log3 = x and log 7 = y express
in terms of x and y .
(3
marks)
log147

(d) Solve
for
x if (i) x+
1
3 1 = 243

(3 marks)

(ii) 2
3
1
- 3 x
4 x
3
2
8
x 8
÷ 27 =

(3
marks)
3

Page 3 of 4

QUESTION FIVE (20 MARKS)

(a)
Find the modulus and principal argument of the complex number
1- 2i

(5 marks)
2
1- 1
( - i)

(b)
(i)
Convert the complex number
2
? 2 + i ?

?
?
into
a
polar
form.
(3
marks)
? 3 - i ?

? 2
2
+ i ?
(ii)
Given
that
z = ?
? , determine the values of K if K 2 = z (6
marks)
? 3 - i ?

(c)
(i)
State and prove the Demoivres theorem.

(2 marks)

(ii)
Use the theorem in (i) above to show that
1

xm +
= 2cos ?
m if x = cos? + i sin?.
xm

(4 marks)

*****************
Page 4 of 4

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