Yan Chai Hospital Lim Por Yen Secondary School
Mock Examination 2000/2001
Subject : Pure Mathematics
Paper :
I
Time Allowed : 3 hours
S. 7
IInnssttrruuccttiioonnss::
1.
2.
3.
4.
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 answer book and four answer sheets.
Section A: Write your answers in the answer sheets.
Section B: Write your answers in the answer book.
The answer book and answer sheets must be handed in separately at the end of the
examination.
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  tan B
1  tan A tan B
A B
A B
sin A  sin B  2 sin
cos
2
2
A B
A B
sin A  sin B  2 cos
sin
2
2
A B
A B
cos A  cos B  2 cos
cos
2
2
A B
A B
cos A  cos B  2 sin
sin
2
2
2 sin A cos B  sin( A  B )  sin( A  B )
tan( A  B ) 
2 cos A cos B  cos( A  B )  cos( A  B )
2 sin A sin B  cos( A  B )  cos( A  B )
p.1
SECTION A : (40 %)
Answer ALL questions in this section.
Write your answers in the answer sheets
1 2 3
1. (a) Expand
2 h 4
3 7 5
.
(b) Hence find the constants h and k such that the system of equations
 x  2 y  3z  k

(*)  2 x  hy  4 z  h
3x  7 y  5z  10

has infinitely many solutions.
(5%)
2. (a) Resolve
(b) Resolve
1
into partial fractions.
( x)( x  1)
1
into partial fractions.
x ( x  1) 2
2
1  2k 2

2
2
n  
k 1 k ( k  1)
n
(c) Using (a) and (b), or otherwise, evaluate lim
(6 %)
3. Given three vectors u, v and w in  3 , where u = (2, 1, 0) , v = (1, 0, 2 ) and
w = (0, 1, 1) .
(a) Show that u, v and w are linearly independent.
(b) Show that every vectors (x, y, z) in  3 can be expressed as a linear combination
of u, v and w .
(5%)
x
, show that
1 x
(i) if 0  p  q , then f ( p)  f (q) ;
(ii) if 0  p and 0  q , then f ( p  q)  f ( p)  f (q) .
4. (a) Let f ( x ) 
(b) Let a, b be 2 real numbers, using the results in (a), show that
|a|
|b|
| ab|


1 | a | 1 | b | 1 | a  b |
(6%)
5. By means of differentiation and integration, show that
n
(a)
 (k  1).C
k 0
n
k
 (n  2).2 n 1 .
(1) k n
1
.C k 

(n  1)( n  2)
k 0 k  2
n
(b)
where C kn is the binomial coefficient.
(5%)
p.2
6.
a 1
(a) Let A = 

0 b
where a , b  R and a  b .
 n a n  bn 
a

Prove that An = 
a  b  for all n  N .

bn 
0
100
1 2
(b) Hence , evaluate 
.

0 3 
7.
(6%)
Let OA  i + 2 j + k , OB  3 i + j + 2 k and OC  5 i + j + 3 k .
(a) Find ABx AC .
(b) Find the area of  ABC . Hence, find the distance from C to AB .
(c) Find the equation of the plane passing through A, B and C .
(7 %)
SECTION B (60 %)
Answer any FOUR questions in this section . Each question carries 15% .
Write your answer in your answer book.
8.
(a) Let U1 , U2 ,  , Un and
V1 , V2 ,  , Vn  C .
n
By considering the function f() =
U
r 1
show that
n

Re U r Vr  
 r 1

r
 Vr
where   R ,
2
n
 n
2 
2
  U r   Vr 
 r 1
 r 1

(4 %)
 n

(b) Let Arg  U r Vr    , by considering Wk  U k (cos  i sin  ) and
 r 1

n
using (a) , show that
U V
r 1
r
r
n
 n
2 
2
   U r   Vr 
 r 1
 r 1

n
 n
2 
2
  Wr   Vr 
 r 1
 r 1

n

[ Hint : Re Wr Vr  
 r 1

]
(6 %)
(c) Let V1 , V2 ,  , Vn be the roots of the equation
Zn + Zn-1 +  + Z + 1 = 0 .
(i) Find Vr
for r = 1 , 2 , 3 , . . . n .
n
(ii) Hence, show that
n
Ur

n
Ur


r 1 Vr
r 1
p.3
2
.
(5%)
9.
10.
Let p be a real number greater than 1 . { Xn } is a sequence such that
(1) X1 > p , and
p2  Xn
(2) Xn+1 
for n = 1 , 2 , . . . . . . . . .
1 Xn
(a) Express X2n+1 in terms of X2n-1 .
(3 %)
(b) Prove that X2n-1 > p for all n  N .
Hence, show that X2n+1 < X2n-1 for all n  N .
(6 %)
(c) Let Yn = X2n-1 for all n  N .
Show that { Yn } converges and find its limit.
(6 %)
(a) By considering f(x) = ex-1 – x , show that
ex-1  x
for all x  R
(3%)
(b) Let a1 , a2 , . . . . . . , an and
 n a i  
n

 i 1 bi  

e
Show that
b1 , b2 , . . . . . . , bn be positive numbers.
n

i 1
n
ai
bi
ai
 n , then

i 1 bi
Hence, show that if
.
n
n
 a  b
i
i 1
i
.
(4 %)
i 1
(b) Using the result in (b) , show that for any positive numbers
1
 n n 1 n
 ai   n  ai
i 1
 i 1 
a1 , a2 , . . . . . . , an ;
n
Hence, show that
1
a
i 1

i

1 n
1
  0, where m =  ai .
n i 1
m
(8 %)
 5 3
 and let x denote a 2 x 1 matrix.
11. Let A  
 2 0
(a) Find the two real values 1 and  2 of  with 1  2 such that the matrix
equation Ax =  x …. (*) has non-zero solution.
(3%)
(b) Let x1 and x2 be non-zero solution of (*) corresponding to 1 and  2 , respectively.
x 
x 
Show that if x1 =  11  and x2 =  21  , then the matrix
 x12 
 x 22 
x 21 
x
 is non-singular.
P   11
 x12 x 22 
(6%)
0

 ,
(c) Using (a) and (b), show that AP  P 1
 0 2 
 1 n
0  1
n
 P , where n is a positive integer.
and hence A  P
n
0

2 

 5 3

Evaluate 
 2 0
20
.
(6%)
p.4
12. (a) Suppose u and v are two non-zero complex numbers such that
u+v+1=0.
Show that | u | = | v | = 1 if and only if
1 1
 1 0 .
u v
Hence , by considering (u  v  1) 2 and the previous result, show that
| u | = | v | = 1 if and only if u 2  v 2  1  0 .
(9%)
(b) Let A, B and C be three distinct points on the complex plane representing
the complex numbers z1 , z 2 and z 3 respectively.
Using the second result of (a), show that ABC is an equilateral triangle if
2
2
2
and only if z1  z 2  z 3  z1 z 2  z 2 z 3  z 3 z1 .
z  z3
z  z2
(Hint: Putting u  2
and v  3
)
z1  z 3
z1  z 3
(6%)
13. (a) (i) Solve x 11  1  0 .
(ii) Hence, or otherwise, show that
5
2k
x 11  1  ( x  1). ( x 2  2 x cos
 1)
11
k 1
5
(iii) By putting x = -1 in (ii), show
 4 cos
k 1
2
k
1 .
11
(5%)
(b) Using (a) or otherwise, show that
5
(1  z ) 11  (1  z ) 11  2 z ( z 2  tan 2
k 1
k
) .
11
(4%)
(c) Hence, deduce that
5
k
sec 2
 1024
(i)

11
k 1
5
(ii)
 sec
k 1
2
k
 60
11
(6%)
***END OF PAPER I ***
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