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Pui Ying College
F.7 Mock Examination(2003-04)
Pure Mathematics 1
Time allowed : 3 hours
Name :
Class : 7 B
No.
1.
2.
3.
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 one AL(E) answer book and four AL(D) answer books.
Section A : Write your answers in 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.
Unless otherwise specified, all working must be clearly shown.
5.
6.
SECTION A (40 Marks)
Answer ALL questions in this section.
Write your answers in the AL(C1) answer book.
1
1. Resolve
into partial fractions.
x ( x  1)
n
Hence show that
r 1
2.
1
 r2  2

and determine whether 
1
r 1 r
2
exists.
(6 marks)
Sketch the locus of a point which moves so that |z – 1| = |z – 3i| in the Argand
Diagram.
Hence find z when |z| has its least value on the above locus.
(6 marks)
2003-04/F7 MOCK EXAM/PM1/ LCK/p.1 of 5
3.
For any positive integer n, let Cnk be the coefficien t of x k in the expansion
of (1 + x)n. Evaluate
(a)
C1n  C n2  C3n  ...  C nn ,
(b)
C 0n
(c)
C1n C n2
C nn


 ... 
,
2
3
n 1
1 n 2 n
n
C1  C 2  ... 
C nn .
2
3
n 1
(7 marks)
4.
 
 


Given two lines (L1) : x  a  c and (L2) : x  tb  c, where λ, t  R ,



a  (4, 2, 3), b  (5, 4, 3) and c  (1, - 1, 2).
(a) Show that (L1) and (L2) are coplanar.
(b) Find the equation of the plane containing (L1) an (L2).
(6 marks)
5.
Solve the equation 375x 3  475x 2  190x  24  0 if the roots are in geometric
sequence.
(7 marks)
6.
Given 0  a1  a 2  ...  a n  ...  a and lim a n  a.
n 
Define x n  n
Prove that
an
n
n
a1n  a n2  ...  a nn
for n = 1, 2, 3,… .
n
 x n  a and hence evaluate lim x n .
n 
(8 marks)
SECTION B (60 Marks)
Answer any FOUR questions in this section.
Use a separate AL(D) answer book for each question attempted.
7.
(a) Show that (a + b) is a root of the equation
x 3  3abx  (a 3  b3 )  0.
(b) (i)
(3 marks)
Express the equation x 3  6x  6  0 in the above form.
Hence find a real root of this equation.
(ii) By transforming the equation x 3  3x 2  3x  11  0 in the form
y3  py  q  0 , find one of its roots.
(12 marks)
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8.
Let p and q be two real numbers.
(a) (i) Show that the system (E) of linear equations
 xy z 6

(E) 3x  y  11z  6
2x  y  pz  q

in x, y, z has a unique solution if and only if p  4.
(ii) Let p = 4 in the system (E).
Find q such that the system (E) has no solution.
Find q such that the system (E) has infinitely many solutions and solve
(E).
(9 marks)
(b) Show that for all values of p and q, the system ( E  ) of linear equations
 x y z 6

( E ) 3x  py  11z  6
2x  y  pz  q

in x, y, z has a unique solution.
(6 marks)
9.
The Fibonacci sequence is defined inductively as follows:
a1  a 2  1 and a n  a n1  a n2 for n  3, 4, 5,...
a 
1 1 
1 0
 and I  
.
Let U k   k 1 , A  
1 0 
0 1
 ak 
(a) Show that U n  A n1U1 , for all natural numbers n.
(3 marks)
(b) Find the eigenvalues λ1 < λ2 such that det(A – λiI) = 0.
(3 marks)
1
   2  1 0  1  2 


 .
(c) Show that A   1
 1 1  0  2  1 1 
(4 marks)
n
(d) Find a formula for A and hence deduce that
n
n

1 5  
1  1  5 

 
 
an 
 2   for all natural numbers n.
5  2 

 

(5 marks)
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 

     
10. Given a set of orthogonal non-zero vectors u, v and w i.e. u  v  v  w  u  w  0.
(a) (i)
 

Prove that u, v and w are linearly independent.
(ii) Determine whether linearly independence implies orthogonality.
(4 marks)
 u1

(b) Show that  u 2
u
 3
v1
v2
v3
w1  x 
 u1   v1   w1 
 
    

w 2  y   x  u 2   y v 2   z w 2  .
u  v  w 
w 3  z 
 3  3  3
Hence deduce that
u1
u2
u3
v1
v2
v3
w1
w 2  0,
w3
 u1 
 v1 
 w1 

     
 
where u   u 2 , v   v 2 , and w   w 2 .
u 
v 
w 
 3
 3
 3

(c) Using (b), deduce that any a in R 3 can be uniquely expressed as



xu  yv  zw for some x , y, z  R.
(4 marks)
 

(d) Furthermore, u, v and w are unit vectors.
         

Show that for any a  R 3 , a  (a  u )u  (a  v) v  (a  w ) w.
11. (a) Define a function f : R+→ R by f ( x ) 
(3 marks)
ax
mm x
where a is a positive constant and m  N, m  2.
 a 
Show that the least value of f(x) is 

 m 1
m1
m
(4 marks)
,
.
(4 marks)
(b) Let a 1 , a 2 , a 3, ... be positive real numbers.
(i)
Using (a), or otherwise, prove that for any k  N,
1
k
a 1  a 2  ...  a k 1
 a  a 2  ...  a k  k 1
 a kk11  1
 .
k 1
k


1
a1  a 2  ...  a k .
k
(ii) Using (b)(i) and induction, or otherwise, prove that for any n  2,
with equality holds iff a k 1 
a1  a 2  ...  a n n
 a1a 2 ...a n
n
with equality holds iff a1  a 2  ...  a n .
(7 marks)
(c) Prove that for any n  N,
2n
1
4n
 r  2n  1 .
r 1
(4 marks)
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12. Let n be a positive integer and α be a real number.
(a) Express the two roots of the equation x 2  2x cos n  1  0 in the form
a + bi, where a and b are real numbers.
(3 marks)
(b) Write down n distinct nth roots of cosθ + isinθ.
Hence show that the n distinct nth roots of cosθ – isinθ are
   2k 
   2k 
cos
  i sin 
, where k  0,1,2,..., n - 1.
 n 
 n 
(4 marks)
(c) Using (a) and (b), or otherwise, show that
z
2n
n 1

2k  

 2z cos n  1   z 2  2z cos  
 1 .
n  

k 0 
n
(4 marks)
(d) Using (c), or otherwise, show that
n 1

 sin 2   
k 0
k 
1
2
  2n 2 sin (n),
n  2
where θ is any real number.
(4 marks)
END OF PAPER
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