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S.7 Mock Exam. 10– 11 / Pure Maths. 1 / p.1
2011-AL
Pure Mathematics
Paper 1
LIU PO SHAN MEMORIAL COLLEGE
(2010-2011)
MOCK EXAMINATION
SECONDARY SEVEN
PURE MATHEMATICS PAPER 1
Date : 22 - 2 – 2011
Time Allowed : 3 hours
1.
This paper consists of Section A and Section B.
2.
Answer ALL questions in Section A.
3.
Answer any FOUR questions in Section B.
4.
Unless otherwise specified, all working must be clearly shown.
FORMULAS FOR REFERENCE
sin  A  B = sin A cos B  cos A sin B
sin A + sin B = 2 sin
A+B
A-B
cos
2
2
sin A - sin B = 2 cos
A+B
A-B
sin
2
2
cos  A  B = cos A cos B  sin A sin B
tan  A  B =
tan A  tan B
1  tan A tan B
cos A + cos B = 2 cos
A+B
A-B
cos
2
2
cos A - cos B = -2 sin
A+B
A-B
sin
2
2
2 sin A cos B = sin (A + B) + sin (A - B)
2 cos A cos B = cos (A + B) + cos (A - B)
2 sin A sin B = cos (A - B) - cos (A + B)
S.7 Mock Exam. 10– 11 / Pure Maths. 1 / p.2
Section A (40 %)
Answer ALL questions in this section.
1. (a) (i)
1
x (x  2)
Resolve
into partial fractions.
n
(ii) Hence, or otherwise, show that
1
=
r
(r

2)
r =1

3
1
1
.
4
2(n  2)
2(n  1)
(b) Let a n  be a sequence of real numbers defined by
a1 =
1
4
and
a n +1 - a n =
Using (a), or otherwise, find
lim
n
1
n (n  2)
for n  1, 2, 3, , n.
an .
(7 marks)
2. (a) Resolve
(b) (i)
1
x (x  1) (x  3)
into partial fractions.
Prove that

7
1
=
.
36
k =1 k (k  1) (k  3)

1
(ii) Hence, or otherwise, evaluate 
.
(k

2)
(k

3)(
k

5)
k =1

(7 marks)
3. Denote the coefficient of x r in the expansion of ( 1  x ) n by C nr where n is a positive integer.
(a) Show that for any positive integer n  2,
C1n  2 2 C n x  33 C n x 2  ...  n 2 C n x n - 1  n ( 1  x ) n - 1 + n (n - 1) x ( 1  x ) n-2 .
2
3
n
(b) (i)
Show that
1 n 2 1 n 3
1
( 1  x ) n 1  1
n
n
n

1
C x   C x   C x  ... 
 Cn x
=
.
0
2 1
3 2
n 1
n 1
 
r 2 C 30
r
(ii) Hence, find 
31 - r
r =1
30
2
.
(7 marks)
4. Let f (x) = 2x + a3x + a2x + a1x + ao where ao, a1, a2, a3  R.
4
3
S.7 Mock Exam. 10– 11 / Pure Maths. 1 / p.3
2
When f (x) is divided by x + 2, the remainder is – 28. When f (x) is divided by x2 – 1, the reminder is
13x + 1. Let r (x) be the remainder when f (x) is divided by (x + 2)(x2 – 1), where r (x) is a quadratic
polynomial with real coefficients.
(a) Find r (x).
(b) If f (x) is divisible by x - 2, find f (x).
(6 marks)
5. Let S = {z C : 3| z  2i | = | z + 8  2i |}.
(a) Prove that S is represented by a circle on the Argand diagram.
Find the centre and radius of the circle.
(b) On the Argand diagram, P is the point representing the complex number 7 + 17i and Q is a point
representing any point z  S. Find the longest distance between P and Q.
(7 marks)
6. (a) Write down the matrix A representing the rotation in the Cartesian plane anticlockwise about the
origin by 60 ﾟ.
  3
(b) Let x and y be two 2  1 matrices relating y = A-1x + b where b =   and A as found in part (a).
 1 
(i)
Describe the geometric meaning of the transformation from x to y.
(ii) Find a 2  1 matrix c such that y = A-1(x + c).
(6 marks)
S.7 Mock Exam. 10– 11 / Pure Maths. 1 / p.4
Section B (60 %)
Answer any FOUR questions in this section.
7.
Consider the system of linear equations
y
z
 a
 x

(S) :  2x  ( λ  2)y
 2z
 b
 x  (2 λ  1)y  ( λ  2) 2 z  c

where  R.
(a) Show that (S) has a unique solution if and only if  is non-zero.
Solve (S) for  = 1.
(7 marks)
(b) Let  = 0.
(i)
Find the conditions on a, b and c so that (S) is consistent.
(ii) Solve (S) when a = -1, b = -2 and c = 3.
(5 marks)
(c) Let  and p be two real constants.
 x  y  z  8  3μ
 2x  2y  2z  4  2 μ

.
(T) : 
x

y

4z

μ

 xyz  p
Find the range of values of p so that (T) is consistent.
(3 marks)
S.7 Mock Exam. 10– 11 / Pure Maths. 1 / p.5
8. (a)
By considering the function f (x) = ln x - x + 1
for x >0, or otherwise, show that for x > 0,
ln x  x - 1 .
(4 marks)
(b)
Let x , x , ... , x n and q , q , ... , q n be 2n positive real numbers such that
1 2
1 2
n
n
q
x

 i i  q i  1.
i=1
i=1
Using (a), or otherwise, show that
q
q
q
x1 1 x 2 2 x 3 3 ... x n
qn
 1.
(4 marks)
(c)
Prove that for any 2n positive real numbers a , a , ... , a n and p , p , ... , p n ,
1 2
1 2
1
 a p1 a p 2 a p 3 ... a p n  p1  p 2 ...  p n 
3
n
 1 2

a p  a p  a p
1 1
2 2
n n .
p  p  p
1
2
n
(3 marks)
(d)
For any positive integer n, show that
(i)
 n 1 ,
n!  

 2 
(ii)
1 2 3
n
1
2 3
n
n

2

 11  2 2    n 2

 1 2    n



n(n 1)
.
(4 marks)
 xn 
9. A sequence of real numbers
is defined as follows :
2x
x
n -1 n - 2
x =
n x
x
n -1
n-2
2ab
Show that, if 0 < a < b and h =
, then
ab
0  x1  x 2
(a) (i)
S.7 Mock Exam. 10– 11 / Pure Maths. 1 / p.6
and
for any integer n  3.
a < h < b.
(ii) Hence, arrange x1 , x 2 , x 3 in ascending order.
(iii) Show, by induction, that for any positive integer n,
0  x 2n-1  x 2n1  x 2n2  x 2n .

(b) Show that the sequences x
(5 marks)
2n-1
 and x 2n  are both convergent and they tend to the
same limit.
(5 marks)
(c) (i)
Show that
- 1  1
1 
1
1

for k = 3, 4, 5, … .

2 x
x
x
x
k-2 
 k -1
k
k -1
Hence, deduce that
(ii) Show that
1
1
1
1
.
 

x
2x
2x
x
n
n -1
1
2
3x x
1 2 .
x =
n
2x
x
n
1
2
lim
(5 marks)
10. (a) Show that a polynomial equation f (x) = 0 has a repeated root  if and only if f (  ) = 0 and
f ’(  ) = 0.
(4 marks)
(b) Using (a), or otherwise, show that if the equation x3 + bx + c = 0 where b  0 has a repeated root,
then
(i)
4b3 + 27c2 = 0,
(ii) the repeated root is 
3c
.
2b
(4 marks)
(c) Let f (x) = x3 + 6x2 + 9x + a where a > 0.
(i)
Find the value of h so that the coefficient of x2 in f (x – h) is 0.
(ii) If f (x) = 0 has a repeated root, find the value of a and hence solve f (x) = 0.
(7 marks)
11. Let  be a constant and 0   

4
S.7 Mock Exam. 10– 11 / Pure Maths. 1 / p.7
.
(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  c o s2  i s i n2
... (*)
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 real for all values of  .

Hence, or otherwise,
(1) show that arg (1 ) 

2


2
.
(2) find the value(s) of  if 120  220 is real.
(12 marks)
END OF PAPER