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Pui Ying College(00-01)
F.7 Mock Examination
Pure Mathematics I
Time allowed : 3 hours
Name :
Class : 7 B
No.
Note :
1.
This paper consists of Section A and Section B.
2.
Answer ALL questions in Section A and any FOUR questions in Section B.
3.
You are provided with one AL(E) answer book and four AL(D) answer books.
Section A: Write your answers in the AL(E) answer book.
Section B: Use a separate AL(D) answer book for each question and put the question number
on the front cover of each answer book.
4.
The four AL(D) books should be tied together with the green tag provided. The AL(E) answer
book and the four AL(D) answer books must be handed in separately at the end of the
examination.
Formula for reference
00-01/F.7 Mock Exam/PMI/LCK/P.1 of 5
SECTION A (40 Marks)
Answer ALL questions in this section.
Write your answers in the AL(E) answer book.
1.
0
0
1


Let A    1  2  1 , I =
2
3
2 

1 0 0


 0 1 0 .
0 0 1


Prove by mathematical induction that for any integer n ≥ 3,
An – An-2 = A2 – I.
Hence find A100.
2.
(6 marks)
Prove that if n is a positive integer and x > 0, then
xn 
1
1
 x n -1  n 1 .
n
x
x
Find the necessary and sufficient condition for the equality to hold.
3.
Expand x(1+x)n and hence evaluate
n
4.
(4 marks)
C 0 n C1 n C 2
C


 ...  n n .
2
3
4
n2
Let z be a complex number satisfying
(5 marks)
z 1
 2.
z 1
(a) Show that the locus of z on an Argand diagram is a circle.
Find its centre and radius.
5.
6.
7.

z 1
3 
(b) Let S= z  C :
 2 and 0  argz   .
z 1
4

Draw and shade the region which represents S on an Argand diagram.
(6 marks)
Let P(x) = 2x5 + x3 + 3x2 + 1 and Q(x) = x3 + x + 1.
(a) Show that P(x) and Q(x) are relatively prime.
(b) Find two polynomials S(x) and T(x) such that P(x)S(x) + Q(x)T(x) = 1.
(6 marks)
The point (x, y) is transformed to the point (x’, y’) by means of the transformation
x'
1 2  x   1 
(Γ) :    
    
 y'   0 1  y   2 
(a) Describe the geometric interpretation of the transformation (Γ ) .
(b) Find the image of the line y = 2x under (Γ ) .
(6 marks)
Given that x3 + 8x2 + 5x – 50 = 0 has a double root. Find all the roots.
(7 marks)
00-01/F.7 Mock Exam/PMI/LCK/P.2 of 5
SECTION B (60 Marks)
Answer any FOUR questions in this section.
Use a separate AL(D) answer book for each question attempted.
8.
(a) Consider the following system of homogeneous equations
 u  v w 0

(I) : a 2 u  b 2 v  c 2 w  0
a 3 u  b 3 v  c 3 w  0

(i)
Show that this system (I) has only trivial solution if and only if a, b, c are all
distinct and ab + bc + ca ≠0.
(ii)
If a, b, c are all distinct, but ab + bc + ca = 0, find the solution of (I).
(6 marks)
(b) Consider the following system of linear equations
 x  y z3

( II) : a 2 x  b 2 y  c 2 z  a 2  b 2  c 2
a 3 x  b 3 y  c 3 z  a 3  b 3  c 3

(i)
Show that the system (II) has unique solution if and only if a, b, c are all distinct and
ab + bc + ca ≠0.
In such a case, find the solution of the system.
(ii) If a, b, c are all distinct, but ab + bc + ca = 0, find the complete solution of the
system (II).
(9 marks)
9.
Given three unit vectors a = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3) satisfying
a  b  0, b  c  0 and a  c  0 .
(a) Prove that the vectors (2a + 3b +c) and (2a – 3b +5c) are perpendicular.
(b) Show that a, b and c are linearly independent.
 a1

(c) Let M   a 2
a
 3
b1
b2
b3
(2 marks)
(4 marks)
c1 

c2  .
c 3 
Evaluate MTM and hence deduce that M is non-singular.
(3 marks)
(d) Let u = (u1, u2, u3) be a vector in R3. By considering the system of equations
a 1 u 1  a 2 u 2  a 3 u 3  0

b1 u 1  b 2 u 2  b 3 u 3  0 ,
c u  c u  c u  0
2 2
3 3
 1 1
show that if u  a  0, u  b  0 and u  c  0, then u  0.
(e) Using (d), deduce that for any v  R 3 ,
v = ( v  a )a  ( v  b)b  ( v  c)c.
[Hint : Put u = v – ( ( v  a )a  ( v  b)b  ( v  c)c )]
(3 marks)
(3 marks)
00-01/F.7 Mock Exam/PMI/LCK/P.3 of 5
2x  y  3z  0
10. Given the line (L) : 
and the plane  : 2x  y  z  2  0.
x  2 y  3z  1  0
(a) Find the intersection point of the line (L) and the plane   .
(2 marks)
(b) (i) Find h, k if P(h – 2 , h – k, k – 1) lies on (L).
(ii) Find the equation of the line (L1) passing through P and perpendicular to   .
(iii) Find the intersection point of (L1) and   . Hence find the mirror image of P with
respect to   .
(7 marks)
(c) Using (b) or otherwise, find the equation of the following lines:
(i) the orthogonal projection of (L) onto   .
(ii) the mirror image of (L) with respect to   .
11. (a) Let a, b ≥ 0. Show that (a + b)n ≥ an + nan-1b for any positive integer n.
(b) Let a1 ≤ a2 ≤ …≤ an be positive numbers and n a natural number .
Using (a) or otherwise, show that
(6 marks)
(2 marks)
 a 1  a 2  ...  a n 

 ≥ Kn + (an – K)Kn-1,
n


n
a 1  a 2  ...  a n 1
.
n 1
where K =
(5 marks)
(c) Show by mathematical induction that
 a 1  a 2  ...  a n 

 ≥ a1a2….an .
n


n
(4 marks)
(d) Prove that for any n (  2) distinct positive numbers a1, a2, a3,…, an,
1
1
1
n2
+
+ ... +
>
a1 a 2
a n a 1 + a 2 + ... + a n
.
(4 marks)
12. Let a1, a2,…, an be n ( 2) distinct real numbers, f(x) = (x – a1)(x – a2)…(x – an) and f ( x ) the
derivative of f(x).
(a) Express f (a i ) (i = 1, 2, 3, …, n) in terms of a1, a2, …, an .
(3 marks)
(b) Let g(x) be a real polynomial of degree less than n.
(i)
Show that there exist unique real numbers A1, A2,…, An such that
n
gx    A i x  a 1 ...x  a i 1 x  a i 1 ...x  a n . .......... (*)
i 1
(Hint : Resolve
g( x )
into partial fractions. )
f (x)
(ii) Using (i), or otherwise, show that if g(x) is of degree less than n – 1, then
n
g(a i )
 0.

i 1 f (a i )
(iii) By taking ai = i (i = 1, 2, 3,…, n) and a suitable g(x) in (b)(ii), show that, for any
non-negative integer m ≤ n – 2,
n
 (1)
i 1
n i
im
 0.
i  1!n  i !
(12 marks)
00-01/F.7 Mock Exam/PMI/LCK/P.4 of 5
13. (a) Solve the equation x2n + 1 = 0, where n  N.
Hence deduce that the roots of the equation
(1 + z)2n + (1 – z)2n = 0
are  i tan
(2k  1) π
, where k = 0, 1, 2,…, (n – 1).
4n
(8 marks)
(b) Prove that
n 1
(2k  1) π


(1 + z)2n + (1 – z)2n = 2∏  tan 2
 z2  .
4n

k 0 
n 1
Hence or otherwise, deduce that
n 1
and find
 sin
k 0
2k  1.

k 0
1 2 n

2k  1
cos
2 2
4n
(7 marks)
4n
END OF PAPER
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