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# Sleeping Patterns

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FINAL SEMESTER EXAMINATION
(MAY 2012 SESSION)
MATHEMATICS I
FMT0013
LECTURER :……………………………………………………………………………
MATRIC NO

: ………………………………………….

VENUE

PROGRAMME : ..……………………………………

: ………………………………………….

OCTOBER 2012

DURATION : 3 HOURS

INSTRUCTIONS:

1. Candidates must read all questions carefully.
2. The examination script consists of the followings:
Part
A

Types of
Questions

B

Instruction

Used

THIS EXAMINATION SCRIPT CONTAINS 7 PAGES INCLUDING THE FRONT COVER

DO NOT OPEN THE EXAMINATION SCRIPT UNTIL FURTHER INSTRUCTION

PART A

CONFIDENTIAL

FMT0013
(70 marks)
Time: 2 hours

INSTRUCTION
This part consists of FIVE (5) questions. Answer ALL questions in the answer booklet provided. 1.

(a)

Simplify the following expressions without using the calculator:
i.

2

5

2

5

(2 marks) ii. 3 3

3n 2

36 3

10 3

n

3n 1

3

(3 marks)
(b)

Solve the following inequalities:
i.

2x

5

3

6

(2 marks) ii. 2x x 1
1

(4 marks)
(c)

It is given that the quadratic equation x 2

5x

0 with two roots α and β.

2

5

Form a quadratic equation whose roots are

5

and

.
(4 marks)

2.

(a)

Given that 2 x
i.

5

3
(
is a factor of P x ) 2 x

13 x

2

mx

20 .

Find the value of m .
(2 marks)

ii.

(
Find a , b and c such that P x ) 2 x

5 ax

2

bx

c .

(3 marks) iii. 3
(
Factorise P x ) 2 x

13 x

2

mx

20 completely.

(2 marks) iv. (
Solve the equation P x ) 0 .

(2 marks)
(b)

(
Given the improper algebraic fractions P x )

FMT0013 MATHEMATICS I

x x 2

3

1
2x

.
3

1/6

CONFIDENTIAL

i.

FMT0013

By using long division, simplify the given P x ).
(
(2 marks)

ii.

Then, express the P x )in (i) as sum of partial fraction.
(
(4 marks)

3.

(a)

Find the number of terms in each of the following arithmetic sequence:
3.2, 4.3, 5.4, ..., 31.8

(2 marks)
(b)

The second and fifth terms of a geometric sequence are -12 and 324 respectively. Find the seventh term.
(3 marks)

(c)

Expand 1

2x

10

in ascending powers of x , up to and including the 4th term.
10

Hence, evaluate 1 . 0002

, correct to four decimal places.
(5 marks)

1

4.

(a)

Given the matrix Q

2

1
3

1
0 . Find the value of Q .

1
1

1

(2 marks)

(b)

2

Given
R

1

3

x

0

i.

x
4

2

and R

25

.Find

5

1

3

the value of x .
(3 marks)

ii.

(4 marks)

iii.

inverse of matrix R .
(2 marks)

(c)

By using matrix Q in question 4(a) and matrix R in question 4(b), find matrix
1
P so that QR

P

4
2

0

1
2 .

1
1

1

(4 marks)

5.

(a)

Given two points A (5,2) and B(-4,-5). Find
i.

FMT0013 MATHEMATICS I

2/6

CONFIDENTIAL

FMT0013
(2 marks)

ii.

the equation of straight line that passing through points A and B in general form
(2 marks)

iii.

the equation of straight line that passing through points A that is parallel to the line 2 x

3y

5

.

2

(2 marks)
(b)

Functions f and g are defined by :
(x ) f 2 x g x) 4
(

x

2

Evaluate
i.

h

gf

(2 marks) ii. fg

1
3

(3 marks)
(c)

State the domain and range for each of the following functions:
i.

(x ) 3 x f 2

1

(2 marks) ii. (x ) 2 f 3 x (2 marks)

PART B
(30 marks)
Time: 1 hour

FMT0013 MATHEMATICS I

3/6

CONFIDENTIAL

FMT0013

INSTRUCTION
This part consists of THREE (3) questions. Answer any TWO (2) questions in the answer booklet provided.

1.

(a)

A bicycle rider is riding down a hill. He travels 4 feet for the first second. In each following second, he travels 5 feet further than in the previous second.
i.

Express the distance traveled in sequence form.
(1 mark)

ii.

Find the distance at 8 seconds.
(2 marks)

iii.

(b)

If the rider reaches the bottom of the hill in 11 seconds, find the total distance traveled.
(3 marks)

A certain culture initially contains 1000 bacteria and increases by 20% every hour as shown in the sequence below
1000, 1200, 1440,1688,…
If N t )is the number of bacteria present after t hours, find
(
i.

The general formula for N t )
( .
(2 marks)

ii.

How many bacteria are in the culture at the end of 10 hours?
(3 marks)

(c)

Write down the first three terms in the binomial expansion of

show that

3

0 .8

0 . 928

3

1

2 x . Hence,

.
(4 marks)

2.

A fast food company sells three specials. The number of hamburgers, french fry orders, and sodas in each special is summarized below. The day's receipt indicates that there were 14 hamburgers, 16 french fries, and 15 sodas sold.

FMT0013 MATHEMATICS I

4/6

CONFIDENTIAL

FMT0013
Special 1
1
1
1

Hamburger
French fries
Sodas

Special 2
3
4
3

Special 3
3
3
4

Let x, y and z represents the number of hamburger, French fries and sodas respectively. Then,
(a)

Find a system of linear equations to represent the above information.
(3 marks)

(b)

Express the system of linear equations in (a) as a matrix equation, AX
B.
(3 marks)

(c)

Find the determinant of the coefficient matrix, A .
(2 marks)

(d)

Find the inverse of A .
(3 marks)
Solve for x, y and z using Cramer’s Rule

(e)

(4 marks)
Given that f x

3.

ln x

3 ,x

3.

Sketch the graph of f x .

(a)

(2 marks)
State the domain and range of f x .

(b)

(2 marks)
Find the inverse function, f

(c)

1

x .

(3 marks)
State the domain and range of f

(d)

1

x

.
(2 marks)

Sketch the graph of f

(e)

1

x

on the same axes of f x .
(2 marks)

Show that f  f

(f)

1

f

1

 f.

(4 marks)

END OF QUESTION PAPER
FORMULAE:
Tn

a

n

1d

FMT0013 MATHEMATICS I

5/6

CONFIDENTIAL

Tn

ar n Sn

FMT0013

n 1

2a

n

1d

2 a r

Sn

n

1

r

r

1

or S n

1

n

a1

r

a

S
1

a+b

n

r n n

= k =0

1

x

n

1

a

n -k

b

k

k

n n

nx

1

x

2

n n

2! sin A

B

cos A

2

x

3!

3

n n

1 n

2  n

r

1

x

r

r!

cos A sin B

cos A cos B  sin A sin B

B

tan A

sin A cos B

1 n

B

tan A

tan B

1  tan A tan B

a s in A + b c o s A = R c o s A - α o r R s in A + α
2

2

s in A + c o s A = 1
2

2

ta n A + 1 = s e c A
2

2

1+ cot A = csc A

sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A
= 1 – 2 sin2 A
= 2 cos2 A – 1 tan 2A = 2 tan A_
1 – tan2 A

FMT0013 MATHEMATICS I

6/6

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