Shatin Pui Ying College
F.7 Mock Examination (2006-07)
Pure Mathematics 1
Time allowed : 3 hours
Name :
Class : 7 B
1.
2.
3.
4.
5.
6.
No.
This paper consists of Section A and Section B.
Answer ALL questions in Section A and any FOUR questions in Section B.
You are provided with two answer books. Write your answers for Section A and B in these two
answer books separately.
The two answer books must be handed in separately at the end of the examination.
Unless otherwise specified, all working must be clearly shown.
SECTION A (40 marks)
Answer ALL questions in this section.
1.
The complex numbers z and w are represented by points P and Q in an Argand diagram.
If z(1 – w) = w and P describes the line 2x + 1 = 0, prove that Q describes a circle and find its
centre and radius.
(6 marks)
2.
n 1
n
i 0
k 1
 (1  x) i   C kn (1) k 1 x k 1 .
(a) Show that
n 1
(Hint :
 (1  x)
i
is a geometric series.)
i 0
(b) Using integration, or otherwise, show that
1
(1) n 1 n
1 1
1
C1n  C 2n  ... 
C n  1    ...  .
2
n
2 3
n
(6 marks)
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3.
(a) Let α,β,γ be the roots of the equation x 3  ax 2  bx  c  0 and
S n  α n  β n  γ n for n  1, 2, 3.
Show that
(i)
S 2  S1  2b  0 ,
2
(ii) S3  aS 2  bS1  3c  0.
(b) Using (a), or otherwise, solve the simultaneous equations in x, y, z
x  y  z  1
 2
2
2
 x  y  z  9.
x3  y 3  z 3  1

(7 marks)
4.
(a) Resolve
2x  3
into partial fractions.
( x  1) x( x  1)
(b) Prove that for any positive integer n  2,
2k  3 3 1
5
 

.
3
 k 4 2n 2(n  1)
k 2
n
k
2k  3
.
3
n 
k
k 2
n
Hence evaluate lim
k
(7 marks)
5.
Let {xn } be a sequence of real numbers such that
x1  6 and xn1  6  xn for any n  N .
(a) Show that for 0  xn  3 for any n N.
(b) Show that { xn } is a monotonic increasing sequence.
Hence deduce that { xn } is convergent and find its limit.
(7 marks)
6.
(a) Let 0 <  < 1. Show that x   (1   )  x for all x > 0.
(b) Given that x1 , x2 , x3 ,..., xn  0.
Using (a), or otherwise, show that
1 n 3 
1 n
  xi   3  xi .
n  i 1
n i 1

(7 marks)
SECTION B (60 marks)
Answer any FOUR questions in this section.
7. Consider the system of linear equations in x, y and z

xyz  a

(E) : 
2 x  y  2 z  b ,
 x  (2  3) y  2 z  c

where a, b and c  R.
(a) Show that (E) has a unique solution if and only if   R \ {2} and a, b, c  R.
Solve (E) for   1 and a, b, c  R.
(6 marks)
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(b) Let   2.
(i) Find the conditions for a, b and c so that (E) has infinitely many solutions.
(ii) Solve (E) when a  1, b  2 and c  3.
(6 marks)
(c) Given the following system of linear equations in x, y and z
x  y  z  5  3


( F ) :  2 x  2 y  2 z  2  2 , where   R.

x  y  4z    1

Using the results obtained in (b), solve (F).
(3 marks)
8.
 cos   sin  
 .
(a) Let A = 
 sin  cos  
Prove by mathematical induction that for any natural number n,
 cos n  sin n 
 .
An = 
 sin n cos n 
(3 marks)
 a  b 

 : a, b  R and n be a positive integer.
(b) Let M  
 b a 

(i)
For any X , Y  M , show that
(I) XY  M ,
(II) XY =YX,
(III)
 0 0
, then X 1 exists and X 1  M .
if X  
 0 0
(ii) For any X  M , show that there exists r ≥ 0 and   R such that
 cos   sin  
 .
X  r 
 sin  cos  
Hence find all X  M such that
1 0
.
X n  
0 1
(iii) If Y , B  M and Y n  B n , show that there exists X  M such that
1 0
 and Y  BX .
X n  
0 1
Hence find all Y  M such that
n
 1 2
 .
Y  
  2 1
n
(12 marks)
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9.
Let  be a constant and 0   

4
.
(a) Let z1 and z2 be the two roots of the equation
z2 = cos 2  i sin 2
If Re(z2) < 0, express z1 and z2 in terms of  .
(3 marks)
(b) Let 1 and 2 be the roots of the equation
  12  cos 2  i sin 2
...(*)
with Re( 1 ) < Re( 2 ).
(i)
Using (a), express the roots 1 and 2 of (*) in terms of z1 obtained in (a).
(ii)

Show that 2 is purely imaginary and
1
 2

 1
20

 is purely real for all values of  .

Hence, or otherwise,


(1)
Show that arg( 1 ) 
(2)
Find the value(s) of  if 120   220 is real.
2

2
.
(12 marks)
10. (a) Let a, b and c be real numbers and c  0. Consider the equation
x3  3ax 2  3bx  c  0 ….(*)
Let  ,  and  be the roots of the equation (*).
(i)
(ii)
(iii)
If  ,  and  are positive, using A.M.  G..M. to show that a 3  c.
Show that  ,  and  are in A.S. if and only if 2a 3  3ab  c  0.
1 1
1
Find a cubic equation whose roots are
, and .
 

(9 marks)
(b) Let  ,  and  be the roots of the equation x 3 
It is given that
3
2
x 2  3x  c  0, where c > 0.
1 1
1
, and are in A.S..
 

Using (a), or otherwise, find c.
Hence solve the equation x 3 
3
2
x 2  3x  c  0.
(6 marks)
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11. (a) Let f(x) =
ax
bm x
for any x  0, where a, b are positive constants and m  N \ {1,2} .
1
m a 
Show that the least value of f(x) is


b  m 1
1
m
.
(4 marks)
(b) Let a1 , a2 ,..., an ,... be positive real numbers.
Using (a), or otherwise, prove that for k  N ,
k
a1  a 2  ...  a k 1
(k  1) k 1 a1 .a 2 ...a k 1
 a  a 2  ...  a k  k 1
 1

 k k a1 .a 2 ...a k 
with equality holds if and only if a k 1 
1
a1  a 2  ...  a k .
k
(Hint : Let b  (k  1)k 1 a1a2 ...ak . )
Hence, by induction, or otherwise, prove that for any n  N,
1
a1  a2  ...  an
 a1a2 ...an n .
n
with equality holds if and only if a1  a 2  ...  a n .
(7 marks)
(c) Prove that for any n  N,
2n
1
4n
 r  2n  1 .
r 1
(4 marks)
END OF PAPER
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