PO LEUNG KUK CENTENARY LI SHIU
CHUNG MEMORIAL COLLEGE
F6 MOCK EXAMINATION
2022-2023
MATHEMATICS EXTENDED PART
MODULE 2
Name: ______________________
Class: ________ Class No.: _____
Time allowed: 2 hours and 30 minutes
Instruction
1. The full mark of this paper is 100 marks.
2. This paper consists of TWO sections,
Section A and Section B. Attempt ALL
questions.
3. Write your answers in the spaces provided
in this Question-Answer book.
4. Graph papers and supplementary answer
sheets will be provided on request. Write
down your name, class and class number on
each sheet and fasten them with strings.
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 necessary
drawn to scale.
2023 DSE MATHS M2-1
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FORMULAE FOR REFERENCE
sin( A  B)  sin A cos B  cos A sin B
cos( A  B)  cos A cos B sin A sin B
tan( A  B) 
tan A  tan B
1 tan A tan B
2sin A cos B  sin( A  B)  sin( A  B)
sin A  sin B  2sin
A B
A B
cos
2
2
sin A  sin B  2 cos
A B
A B
sin
2
2
cos A  cos B  2 cos
A B
A B
cos
2
2
cos A  cos B  2sin
A B
A B
sin
2
2
2sin A cos B  sin( A  B)  sin( A  B)
2sin A sin B  cos( A  B)  cos( A  B)
Section A (50 marks)
1. Let f ( x)  xe
x
2023 DSE MATHS M2-2
, find f (2  h)  f (2) . Hence, find f '(2) .
(4 marks)
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n
2. (a) Prove by Mathematical Induction that
 (3r  1)(r  2)  n(n  2)(n  3) is true for all positive
r 1
integer n.
100
(b) Hence, find  (3r  1)( r  2) .
(7 marks)
r 50
2023 DSE MATHS M2-3
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2023 DSE MATHS M2-4
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3. The constant term of the expansion of (2  9 x 2 ) n (3 
2 2
) is -23328, where n is a positive integer.
x3
Find
(a) n,
(b) The coefficient of x 2 .
(6 marks)
2023 DSE MATHS M2-5
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4. (a) Find
dy
of x 2 sin y  cos x 4  x
dx
(b) Hence, find the equation(s) of tangent for x  0 .
(6 marks)
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5. (a) Prove that ( x  1) is a factor of the cubic polynomial x 3  x 2  x  1 .
x3  x 2  x  1
dx .
(b) Hence, evaluate 
x2 1
(6 marks)
2023 DSE MATHS M2-7
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 sin 
6. Consider a 2  2 matrices M  
 cos 
matrices as I.
cos  
 , where 0     , define the 2  2 identity
 sin  
(a) Find M 2 .
(b) Hence find M 1 without finding M .
(c) If M 99  2M , find the value(s) of  .
(6 marks)
2023 DSE MATHS M2-8
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2023 DSE MATHS M2-9
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7. (a) Evaluate
e
2x
sin x cos xdx .
ln 2
(b) Hence, find the value of
e
2x
sin(ln x 2 )dx .
0
(7 marks)
2023 DSE MATHS M2-10
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8. Consider a curve  : y  xe x , it is given that the slope of tangent L of  is 2e , then
(a) Find the equation of L .
(b) Find the area bounded by L ,  and the y-axis.
(c) Let P be a point lines on  which x  n , where n  0 and P ' be the x -intercept of x  n .
Define the inclination of  at point P as  . If the Area of OPP ' with increases at a rate
of 3 square units/s, find the rate of change of  when x  3 .
(8 marks)
2023 DSE MATHS M2-11
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2023 DSE MATHS M2-12
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2023 DSE MATHS M2-13
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Section B (50 marks)
1 1 1
9. Let M and A be the 3  3 matrices, where A   1 1 0  . Define the 3  3 matrices identity
 1 0 1


matrices as I.
(a)
(i)
Prove that A3  A2  A  I  0
(ii)
Show that A is non-singular, hence using (a) (i), find A1 .
(3 marks)
(b)
(i)
Express (M  I )(M n  M n1  M n2 
(ii)
(iii)
Hence, show that A4  I
Using (a) and (b) (ii), prove that ( A1 )3  ( A1 )2  A1  I  0
 M  I).
(4 marks)
(c) Using (a) and (b) or otherwise, find X
Given that 0  p  4
4n  p
, where n and p is a positive integer.
(5 marks)
2023 DSE MATHS M2-14
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2023 DSE MATHS M2-15
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2023 DSE MATHS M2-16
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2023 DSE MATHS M2-17
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10. Consider the following system of linear equations:
(  2) x  6 y  2 z    1

(E)  7 x
 8 y  3 z  2 , where  ,  
 2 x
 3y 
z 
5

.
(a) Assume (E) has a unique solution.
(i)
Find the range of value(s) of  .
(ii)
Hence, solve (E).
(5 marks)
(b) Assume   6 and (E) is consistent.
(i)
Find the value(s) of  .
(ii)
Solve (E).
(4 marks)
(c) Consider the following system of linear equations:
(  2) x  6 y  2 z    1
 7x
 8 y  3 z  2

, where  ,   .
(F ) 

2
x

3
y

z

5

2

 y 2  z 2  10
 x
If the value(s) of  is equal to that found on (b) (i), someone claims that there are at least 2
solutions for (F), is he correct?
(4 marks)
2023 DSE MATHS M2-18
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2023 DSE MATHS M2-19
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2023 DSE MATHS M2-20
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2023 DSE MATHS M2-21
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x
11. (a) Let t  tan , where 0  x  
2
(i)
Evaluate
(ii)
Find
x
2
dx
x2
dt
.
dx

2
(iii)
dx
3  sin x  cos x
0
Hence, using (i) and (ii), find the value of 
(6 marks)
(b) Let f ( x ) be a continuous function such that f ( x)  f (  x) for all real number x,
where  is a constant.

(i)
Prove that


 xf ( x)dx  2  f ( x)dx
0
0

2
(ii)
Hence, find
x
 3  sin x  cos xdx
0
(4 marks)

(c) Using the above result, find
x
dx

 3  sin x  cos x
2
(4 marks)
2023 DSE MATHS M2-22
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2023 DSE MATHS M2-23
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2023 DSE MATHS M2-24
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2023 DSE MATHS M2-25
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12. Let the position vector of A, B and C be 4i  2k , 10i  2 j  8k and 2i  7 j  8k respectively. Let
M be the mid-point of AB.
(a) Consider ABC
(i)
Prove that CM  AB
(ii)
Let H be the orthocenter of
ABC , find OH
(6 marks)
(b) Consider the tetrahedron OABC, denote the plane ABC as 
(i)
Find the volume of OABC.
(ii)
Hence or otherwise, determine whether the projection of O
on  is the orthocenter of ABC .
(5 marks)
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END OF PAPER
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