Fergus Sir 2021 m2 Free Mock (E)

2021-MOCK
MATH EP Please stick the barcode label here.
M2
FERGUS SIR MATHEMATICS
HONG KONG ONLINE DSE MOCK 2021 Candidate Number
MATHEMATICS Extended Part

Marker’s Use Only
Module 2 (Algebra and Calculus)
Question-Answer Book Question Marks
No.
Time Allowed: 21/2 hours
1 /4
This paper must be answered in English
2 /5
INSTRUCTIONS
3 /5
(1) After the announcement of the start of the 4 /6
examination, you should first write your
Candidate Number in the space provided on 5 /6
Page 1 and stick barcode labels in the spaces
provided on Pages 1, 3, 5, 7, 9 and 11.
6 /7
(2) This paper consists of TWO sections, A and B.

7 /8
(3) Attempt ALL the questions in this paper. Write your 8 /9

answers in the spaces provided in this Question-
Answer Book. Do not write in the margins. 9 /11
Answers written in the margin will not be
10 /12
marked.
(4) Supplementary answer sheets will be supplied 11 /13

on request. Write your Candidate Number,
12 /14
mark the question number box and stick a
barcode label on each sheet, and fasten them Total /100
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) No extra time will be given to candidates for

sticking on the barcode labels or filling in the
question number boxes after the ‘Time is up’
announcement.
All Rights Reserved
© Fergus Sir Mathematics. 2021
2021-DSE-MOCK-MATH-EP(M2)-1 1
************************************************************************************
SECTION A (50 marks)

d
1. Find csc 4x from first principle. (4 marks)
dx
Answer written in the margins will not be marked.
Please stick the barcode label here.
2. Let n be a positive integer and α be a constant. In the expansion of (1 + a x)2(1 + 2x)n , the coefficients
of x and x 2 are 8 and 9 respectively.
(a) Find n and α .
(b) Find the coefficient of x 3 .
(5 marks)
2021-DSE-MOCK-MATH-EP(M2)-3 3 Go on to the next page

3. Consider the curve C : x (2 y ) + x 2 y = 11 .
dy
(a) Find .
dx
(b) Find the equation of normal to C at the point (1,3) .
(5 marks)
∫
4. (a) Find x x 2 + 9d x .
(b) At any point (x , y) on the curve Γ , the slope of tangent to Γ is 6x x 2 + 9 . Also, Γ passes
through (4,7) . Find the equation of Γ .
(6 marks)

π −1 + 5 2π 4π 2π 4π
5. (a) Using the fact that sin = , evaluate cos + cos and cos cos .
10 4 5 5 5 5
(b) Prove that sin 3θ + sin 5θ + sin 7θ + sin 9θ + sin 11θ = sin 7θ (4 cos2 2θ + 2 cos 2θ − 1) .
π
(c) Solve the equation sin 3x + sin 5x + sin 7x + sin 9x + sin 11x = 0 , where 0 < x < .
2
(6 marks)

3n
(−1)k k 2(2k + 1) = (−1)n(28n 3 + 17n 2 + n) for all
∑
6. (a) Using mathematical induction, prove that
k=n
positive integers n .
900
k 2
(−1)k (
∑
(b) Using (a), evaluate ) (2k + 1) .
k=100
20
(7 marks)

7. In Figure 1(a) , sector OA B is a thin paper card. OA = r cm and ∠ AOB = θ radian . It is given that the
area of OA B is 50π cm2 . By joining OA and OB together, OA B can be folded to form a right circular
cone as shown in Figure 1(b) .
A B
O
Figure 1(a) Figure 1(b)
(a) Express the volume of the cone in terms of r .
(b) Find the maximum volume of the cone.
(c) Suppose ∠ AOB is increasing at a constant rate of π radian per second . Find the rate of change of
OA when the volume of the cone formed by joining OA and OB together is maximum.

(8 marks)

1 1 1
8. (a) Factorize a 2 b 2 c 2 .
bc ac ab
(b) Consider the system of linear equation in real variables x , y , z
x +y +z =7
(E ) : 4x + λ 2 y + 9z = 31 , where λ , μ ∈ R .
3λ x − 6y − 2λ z = μ
(i) Assume (E ) has a unique solution.
(1) Find the range of values of λ .
(2) Express y in terms of λ and μ .
(ii) Assume λ = − 1 and (E ) is consistent. Find μ .
(9 marks)

SECTION B (50 marks)
2x 3 + a x 2 + 48x − 18
9. Let a be a constant. Define f (x) = for all real number x ≠ 3 . Denote the graph
(x − 3)2
of y = f (x) by G . It is given that the straight line y = 2x − 5 is an asymptote of G .
(a) Find a . (2 marks)
(b) Find f ′(x) and f ′′(x) . (2 marks)
(c) Find the maximum and/or minimum point(s) of G . (2 marks)
(d) Does G have a point of inflexion? Explain your answer. (2 marks)
(e) Let R be the region bounded by G , the straight line y = 1 , the straight line x = − 1 and the
y-axis. Find the volume of the solid of revolution generated by revolving R about the straight
line y = 1 . (3 marks)

2021-DSE-MOCK-MATH-EP(M2)-16
16

10. The position vectors of the points A , B , C and D are i − 4j , 4i − j + 3k , j − 4k and 8i + 3j − k
respectively. Denote the plane which contains A , B and C by Π . Let E be the projection of D on Π .
(a) Find
(i) A B ⋅ AC ,
(ii) A B × AC ,
(iii) the volume of tetrahedron A BCD ,
(iv) DE .
(6 marks)
(b) Let F be a point such that A , B and F are collinear. It is given that DF is perpendicular to A B .
(i) Find EF .
(ii) Are A , C , E and F concyclic? Explain your answer.
(6 marks)

20

π
11. It is given the fact that sin−1 x + cos−1 x = for all x ∈ [−1,1] .
2
∫
(a) (i) Find tan4 x d x .
∫
(ii) Using integration by part, find x tan3 x sec2 x d x .
(5 marks)
1
(1 − x)cos−1 x
∫0
2
(b) Using the substitution x = cos 2θ , find dx . (4 marks)
(1 + x)3
x sin−1 x sin−1 x
(c) Define f (x) = and g(x) = for all x ∈ (−1,1] . Let R be the region bounded
(1 + x)3 (1 + x)3
by the curve y = f (x) , the curve y = g(x) , the x-axis and the straight line 2x − 1 = 0 . Find the
area of R . (4 marks)

24
25
( c d) (0 1) (0 0)
a b 1 0 0 0
12. Let M = , where a , b, c, d ∈ R . Define I = and O = .
(a) Prove that
(i) M 2 − (a + d )M + det(M ) ⋅ I = O ,
(ii) det(M − x I ) = x 2 − (a + d )x + det(M ) , where x ∈ R .

(3 marks)
(b) Suppose a d < b c . Let λ and μ be the roots of the equation det(M − x I ) = 0 , where λ < μ .
1 1
Define A = (M − μ I ) and B = (M − λ I ) .
λ −μ μ−λ
(i) Find A + B , A B and B A .
(ii) Prove that A 2 = A and B 2 = B .
(iii) Prove that M n = λ n A + μ n B for all positive integer n .
(8 marks)

2021
(9 −6)
2 1
(c) Using the results from the previous parts, find . (3 marks)
27
28
29
END OF PAPER
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2021-MOCK

MATH EP Please stick the barcode label here.

FERGUS SIR MATHEMATICS

HONG KONG ONLINE DSE MOCK 2021 Candidate Number

MATHEMATICS Extended Part

(2) This paper consists of TWO sections, A and B.

(3) Attempt ALL the questions in this paper. Write your 8 /9

(4) Supplementary answer sheets will be supplied 11 /13

(5) Unless otherwise specified, all working must be

(6) Unless otherwise specified, numerical answers

(7) No extra time will be given to candidates for

All Rights Reserved

© Fergus Sir Mathematics. 2021

SECTION A (50 marks)

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

(a) Find n and α .

(b) Find the coefficient of x 3 .

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

2021-DSE-MOCK-MATH-EP(M2)-3 3 Go on to the next page

(b) Find the equation of normal to C at the point (1,3) .

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

2021-DSE-MOCK-MATH-EP(M2)-5 5 Go on to the next page

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

2021-DSE-MOCK-MATH-EP(M2)-7 7 Go on to the next page

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

2021-DSE-MOCK-MATH-EP(M2)-9 9 Go on to the next page

(a) Express the volume of the cone in terms of r .

(b) Find the maximum volume of the cone.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

2021-DSE-MOCK-MATH-EP(M2)-11 11 Go on to the next page

(b) Consider the system of linear equation in real variables x , y , z

(i) Assume (E ) has a unique solution.

(1) Find the range of values of λ .

(2) Express y in terms of λ and μ .

(ii) Assume λ = − 1 and (E ) is consistent. Find μ .

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

Answer written in the margins will not be marked.

2021-DSE-MOCK-MATH-EP(M2)-13 13 Go on to the next page

(a) Find a . (2 marks)

(b) Find f ′(x) and f ′′(x) . (2 marks)

(c) Find the maximum and/or minimum point(s) of G . (2 marks)

(d) Does G have a point of inflexion? Explain your answer. (2 marks)

Answer written in the margins will not be marked.