DSE
MATH EP
M2
WA YING COLLEGE
2022 − 2023 S6 Mock Examination
Name
MATHEMATICS Extended Part
6
Class
Module 2 (Algebra and Calculus)
Class Number
Question – Answer Book
11th January 2023
Question No.
1
hours)
2
(2
1
This paper must be answered in English.
2
INSTRUCTIONS
1.
Write your name, class and class number in the spaces
3
provided on this cover.
2.
Answer ALL questions in this paper.
Write your answers in
the spaces provided in this Question−Answer Book.
write in the margins.
Do not
Answers written in the margins will not
be marked.
3.
Write your name, class and class
number on each sheet and fasten them with string INSIDE
this book.
4.
5
6
Graph paper and supplementary answer sheet will be
supplied on request.
4
The Question−Answer book and the answer book will be
7
8
9
collected separately at the end of the examination.
5.
Unless otherwise specified, all working must be clearly
shown.
6.
11
Unless otherwise specified, numerical answers must be
exact.
7.
In this paper, vectors may be represented by bold−type
letters such as u, but candidates are expected to use

appropriate symbols such as u in their working.
8.
The diagrams in this paper are not necessary drawn to scale.
DSE–MATH – EP(M2) − 1
10
12
Total
Marks
FORMULAS 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
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 A + sin B = 2 sin
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 = −2 sin
A −B
A +B
sin
2
2
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
Section A
1.
(50 marks)
Let f(x) = x 2e x .
Find f '(1) from first principles.
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(4 marks)
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DSE–MATH – EP(M2) − 2
2.
n
Define (1 − 5x)n =
∑ µk xk , where n is an integer greater than 1.
It is given that µ 2 =−15µ1 .
k =0
(a) Find n.
2
3

(b) Find the coefficient of x in the expansion of (1 − 5x)  x −  .
x

n
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(5 marks)
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DSE–MATH – EP(M2) − 3
3.
(a) Using mathematical induction, prove that
2n
1
2n
∑ (5k − 4)(5k + 1) = 10n + 1 for all positive
k =1
integers n.
1
.
k =9 (5k − 4)(5k + 1)
28
(b) Using (a), evaluate ∑
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(7 marks)
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DSE–MATH – EP(M2) − 4
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DSE–MATH – EP(M2) − 5
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4.
(a) Using integration by parts, find ∫ x 2 3 x dx .
5
2
(b) Consider the curve C: y = x 2 (3 x ) , where x ≥ 0 . Let R be the region bounded by C, the
straight line x = 1 and the x−axis. Find the volume of the solid of revolution generated by
revolving R about the x−axis.
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(7 marks)
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DSE–MATH – EP(M2) − 6
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DSE–MATH – EP(M2) − 7
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5.
(a) Let x be the angle which is not a multiple of 30°.
(i)
=
cos3x 4 cos3 x − 3 cos x ,
(ii)
4 cos x cos(x + 60o )cos(x − 60o ) =
cos3x .
Prove that
(b) Using (a)(ii), prove that 4 sin 21o sin81o sin39o = sin 63o .
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(6 marks)
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DSE–MATH – EP(M2) − 8
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DSE–MATH – EP(M2) − 9
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6.
Let f(x) be a continuous function defined on R.
Denote the curve y = f(x) by Γ.
It is given
that Γ passes through the point (1, − 4) and f '(x)
= 3x − 12x for all x ∈R .
2
(a) Find the equation of Γ.
(b) Let L be a tangent to Γ such that L passes through the point (0, 9) and the slope of L is
negative. Denote the point of contact of Γ and L by P. Find
(i)
the coordinates of P,
(ii) the equation of the tangent to Γ at P.
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(8 marks)
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DSE–MATH – EP(M2) − 10
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DSE–MATH – EP(M2) − 11
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7.
ABC is a triangle with AB = x cm , BC = y cm , AC = 13 cm and ∠ABC =
120o .
(a) Show that x 2 + y 2 + xy − 169 =
0.
(b) If x increases at a rate of
1
cm / s , find the rate of change of the area of ∆ABC when x = 7 .
2
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(6 marks)
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DSE–MATH – EP(M2) − 12
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DSE–MATH – EP(M2) − 13
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8.
 1 0
 −3 −1
Define P = 
.
 and Q = 
0 0
 12 4 
where a, b and c are real numbers.
 1 a
Let M = 
 such that M = 1 and PM = MQ ,
b c 
(a) Find a, b and c.
1
 4
(b) Define R = 
.
 −12 −3 
(i)
Evaluate M−1RM .
(ii) Prove that (αP + βR)23 = α 23P + β23R for any real numbers α and β.
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(7 marks)
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DSE–MATH – EP(M2) − 14
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DSE–MATH – EP(M2) − 15
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Section B
9.
(50 marks)
Let f(x) =
(x − 5)3
for all real numbers x ≠ −1.
(x + 1)2
Denote the graph of y = f(x) by H.
(a) Find the asymptote(s) of H.
(3 marks)
(b) Find f ''(x) .
(2 marks)
(c) Somebody claims that there are two turning points of H.
answer.
Do you agree?
Explain your
(2 marks)
(d) Find the point(s) of inflexion of H.
(2 marks)
(e) Sketch H.
(1 mark)
(f)
(3 marks)
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Find the area of the region bounded by H, the x−axis and the y−axis.
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DSE–MATH – EP(M2) − 16
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DSE–MATH – EP(M2) − 17
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DSE–MATH – EP(M2) − 18
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DSE–MATH – EP(M2) − 19
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10. (a) Evaluate
3
1
∫ 9 + t dt .
(b) Let 0 ≤ x ≤
(2 marks)
2
0
π
.
4
Show that cos 2x =
1 − tan2 x
.
1 + tan2 x
π
3
1
dx .
5 + 4 cos 2x
0
(c) Evaluate ∫
(1 mark)
(3 marks)
−f(x) for all x ∈R .
(d) Let f(x) be a continuous function defined on R such that f( − x) =
a
a
−a
0
x
∫ f(x)ln(1 + e )dx =
∫ xf(x)dx for any a ∈R .
Prove that
(e) Evaluate
π
3
sin 2x
∫ (5 + 4 cos 2x) ln(1 + e )dx .
x
(4 marks)
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−π
3
2
(3 marks)
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DSE–MATH – EP(M2) − 20
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DSE–MATH – EP(M2) − 21
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11. (a) Consider the system of linear equations in real variables x, y, z
(a − 1)y
b
=
 x +

(E) : 2x +
3ay + (2 − a)z =−6 , where a, b ∈R
3x + (4a − 1)y −
z =
4b

(i)
Assume that (E) has a unique solution.
(1) Prove that a ≠ 3 and a ≠ −2 .
(2) Solve (E).
(ii) Assume that a = 3 and (E) is consistent.
(1) Prove that b = −2 .
(2) Solve (E).
(9 marks)
(b) Is there a real solution of the system of linear equations
satisfying x 2 − 3y 2 + 3y − z < −9 ?
Explain your answer.
(3 marks)
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DSE–MATH – EP(M2) − 22
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=
−2
 x + 2y

−6
2x + 9y − z =
3x + 11y − z =
−8

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DSE–MATH – EP(M2) − 23
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DSE–MATH – EP(M2) − 24
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DSE–MATH – EP(M2) − 25
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


−32i + 32 j + 6k .
12. Let OA =−2i − j + 3k , OB = 4i + 5 j + 6k and OC =
contains O, A and B by Π. Let D be the projection of C on Π.
 
(a) (i) Find OA × OB .
  
= OA + OB , find the area of the quadrilateral OASB.
(ii) If OS

(ii) Find DC .
Denote the plane which
(4 marks)
(5 marks)
(c) Find the angle between ∆ABC and Π.
(3 marks)
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(b) Let E be a point lying on AB such that CE is perpendicular to AB.

(i) Find CE .


(ii) Is AB perpendicular to DE ? Explain your answer.
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DSE–MATH – EP(M2) − 26
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DSE–MATH – EP(M2) − 27
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END OF PAPER
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DSE–MATH – EP(M2) − 28
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