2014 F5 DSE
MATH EP
M2
13 – 01 – 2015.
8.15 a.m. – 9.45 a.m. ( 1
1
2 hours)
This paper must be answered in English
INSTRUCTIONS
1.
Write your Name, Class and Class Number in the spaces provided on Page 1.
2.
This paper consists of Section A and Section B.
Answer ALL questions in this paper.
3.
Write your answers in the spaces provided in this
Question-Answer Book. Do not write in the margins. Answers written in the margins will not be marked.
4.
Graph paper and supplementary answer sheets will be supplied on request. Write your Name and mark the question number box on each sheet, and fasten them with string INSIDE this book.
5.
Unless otherwise specified, all working must be clearly shown.
6.
Unless otherwise specified, numerical answers must be exact.
7.
The diagrams in this paper are not necessarily drawn to scale.
Name
Class ( )
Marker’s
Use Only
Examine r’s
Use Only
Question No.
1
2
3
4
5
6
7
8
9
10
11
12
Total
Marker No. Examiner No.
Marks Marks
A&C1_M5_14
 
1


sin

A

B

 sin cos

A

B

A cos B
 cos A sin B
 cos A cos B  sin tan( A

B )
 tan
1 
A tan
 tan
A tan
B
B
A sin B
2 sin A cos B
 sin( A

B )
 sin( A

B )
2 cos A cos B
 cos

A

B

 cos

A

B

2 sin A sin B
 cos

A

B

 cos

A

B
 sin sin cos cos
A
A
A


 sin sin cos
B
B
B

2


2
2 sin cos cos
A

B cos
A
A
2

A
2

2
B sin
B cos
A

2

A
2

2
B
B
B
A
 cos B
 
2 sin
A

2
B sin
A

2
B
**********************************************************
1. Find d d x
 cos 2 x

from first principles. (4 marks)
Answers written in the margins will not be marked .
A&C1_M5_14

2
 Page total

2. Find
 sin
5 x cos
4 x d x .
3. (a) Let f ( x )
 x
3 sin
4 x . Show that f ( x ) is an odd function.
(b) Evaluate


2

2
 x
3 sin
1
 x
2
4 x d x .
(4 marks)
(4 marks)
Answers written in the margins will not be marked .
A&C1_M5_14

3
 Go on to the next page Page total

d y
4. The slope of a curve at any point ( x , y ) is given by d x
 
( , ), find the equation of the curve.
2 4
= sin
2 x
2
. If the curve passes through
(5 marks)
5. Evaluate
(a)
(b) x lim

0 x lim

0 sin sin
3 x
4 x
2 x


2 x sin
, x
.
3 x
(6 marks)
Answers written in the margins will not be marked .
A&C1_M5_14

4
 Page total

Answers written in the margins will not be marked .
A&C1_M5_14

5
 Go on to the next page Page total

6. Find
(a) d y
for the following functions. d x y
 ln
 sec( 1
 e
3 x
)

(b) xy
 cos( x
 y )
7. Evaluate
(a)
 x x

1 d x ,
(b)
 x
2 cos x d x .
Answers written in the margins will not be marked .
A&C1_M5_14

6

(6 marks)
(6 marks)
Page total

Answers written in the margins will not be marked .
A&C1_M5_14

7
 Go on to the next page Page total

8. (a) Find
 x ln x d x , where x

0 .
(b) Figure 1 shows the shaded region bounded by the curves y
 x , y
 x ln x and the straight line x = 1.
Find the area of the shaded region.
(6 marks)
O y
Answers written in the margins will not be marked .
A&C1_M5_14

8
 y
 x ln x y
 x x

1
Figure 1 x
Page total

9. Consider a continuous function f( x )

4 x
2 x
2

1

1
. The following table is given. x x
 
1
3

1
3

1
3
 x

0 0 0
 x

1
3 f ' ( x ) f '' ( x )
< 0
< 0
< 0
0
< 0
> 0
0
0
> 0
> 0
(a) Find all the maximum and/or minimum point(s) and point(s) of inflexion.
(b) Find the asymptote(s) of the graph of y
 f ( x ) .
(c) Sketch the graph of y
 f ( x ) on page 10.
1
3
> 0
0 x

> 0
< 0
1
3
(6 marks)
Answers written in the margins will not be marked .
A&C1_M5_14

9
 Go on to the next page Page total

Answers written in the margins will not be marked .
A&C1_M5_14

10
 Page total

10. (a) Solve sin 2
  tan

, where 0
  

2
(b) In Figure 2, the shaded region is bounded by the curves y
 sin 2 x and y
 tan x .
.
Find the volume of the solid of revolution if the shaded region is revolved about the x -axis.
(7 marks)
Answers written in the margins will not be marked .
A&C1_M5_14

11
 y
O y
 tan x
Figure 2 y
 sin 2 x
Go on to the next page x
Page total

11. In Figure 3, P is a moving point on the circle x
2  y
2 
1 .
The tangent to the circle at P cuts the x -axis at the point Q .
Let


0
  

2
be the angle subtended by OP and the positive x -axis.
(a) Find the coordinates of P in terms of

.
(b) Show that the equation of tangent to the circle at P is x cos
  y sin
 
1 .
(c) Suppose

increases at the rate of along the positive x -axis. When
 
1
4
 rad/s and Q is moving
, find the moving
6 speed of Q .
Figure 3
(10 marks)
Answers written in the margins will not be marked .
A&C1_M5_14

12
 Page total

Answers written in the margins will not be marked .
A&C1_M5_14

13
 Go on to the next page Page total

12. (a) Evaluate
(b) Show that


1
1

0 a
1

1 x
2 d x . f ( x ) d x

1
2

0 a
[ f ( x )
 f ( a
 x )] d x .
(c) (i) By using the substitution of y
 cos x , or otherwise, evaluate

0
 sin
3 x
1
 cos
2 x d x .
(ii) Evaluate

0
 x sin
3
1
 cos
2 x x d x .
(11 marks)
Answers written in the margins will not be marked .
A&C1_M5_14

14
 Page total

Answers written in the margins will not be marked .
A&C1_M5_14

15
 Go on to the next page Page total