Download Paper 1_07-08

Shatin Pui Ying College
F.7 Mock Examination (2007-08)
Pure Mathematics 1
Time allowed : 3 hours
Name :
Class : 7 B
1.
2.
3.
4.
5.
No.
This paper consists of Section A and Section B.
Answer ALL questions in Section A, using the AL(E) answer book.
Answer any FOUR questions in Section B, using the AL(A) answer book.
Unless otherwise specified, all working must be clearly shown.
SECTION A (40 marks)
Answer ALL questions in this section.
Write your answers in the AL(E) answer book.
1.
(a) Show that (1  i ) n  C0n  C2n  C4n  C6n  ...  i C1n  C3n  C5n  C7n  ... , where i 2  1 ,
C rn is the coefficient of x r of the expansion of (1  x) n and n N.
(b) Using (a), or otherwise, show that
C
n
0
 C2n  C4n  C6n  ...  C1n  C3n  C5n  C7n  ...  2 n.
2
2
(6 marks)
2.
(a) Resolve
2x  1
into partial fractions.
x ( x  1) 2
2
(b) Using (a), or otherwise,
n
(i)
show that
r
r 1

(ii) evaluate
r
r 1
2r  1
 1,
(r  1) 2
2
2r  1
.
(r  1) 2
2
(7 marks)
2007-08/F7 MOCK EXAM/PM1/ LCK/p.1 of 4
3.
Let f ( x)  x 4  2 x 3  3x 2  x  2 and g ( x)  x 2  x  1.
(a) Show that f(x) and g(x) only have constant common factors.
4.
(b) Find a polynomial p(x) of the lowest degree such that f(x) + p(x) is divisible by g(x).
(6 marks)
A 2  2 matrix M is the matrix representation of a transformation T which transforms (1, 1)
and (0, 1) to (–2, 2) and (0, 2) respectively.
a b
 a b  k 0 
(a) Find k > 0 such that M can be decomposed as 
 1.

, where
c d
 c d  0 k 
Hence describe the geometric meaning of T.
(b) If T transforms the point (u, v) to (4, 0), find the values of u and v.
(7 marks)
n
5.
 6n
6 2
  
(a) Prove that for any positive integer n, 
0 6
0
2n6 n 1 
.
6 n 
 5 1
 1 0
(b) Let A  
, B  
 and C  BAB 1 .
  4 1
 2 1
Find C.
Hence find A10 .
(7 marks)
6.
Show that for all positive real numbers x,
xn
1
by using

2
2n
1  x  x  ...  x
2n  1
(a) A.M.  G.M.,
(b) Cauchy-Schwarz’s inequality.
(7 marks)
SECTION B (60 marks)
Answer any FOUR questions in this section. Each question carries 15 marks.
Write your answers in the AL(A) answer book.
7.
 x  (2k  1) y  kz  

Consider the system of linear equations ( E ) : 3x  ky  z  
.
5 x  (2  3k ) y  z  

(a) Find the value(s) of k such that (E) has a unique solution for all real values of , , .
(3 marks)
(b) (i) When k = 0, show that (E) is consistent iff  = 2 + . Hence solve (E).
(ii) When k = 1, show that (E) is consistent iff  = 2 – . Hence solve (E).
(8 marks)
x  3 y  5z  0

(c) Let (a, b, c) be a non-zero solution of ( F ) : (2k  1) x  ky  (2  3k ) z  0 .
kx  y  z  0

Show that if (E) is consistent then a + b + c = 0.
(4 marks)
2007-08/F7 MOCK EXAM/PM1/ LCK/p.2 of 4
8.
(a) Let p, q and r be real numbers such that p  q , r > 0 and r  1.
Find the roots of the equation ( x  p)3  r 3 ( x  q)3 in terms of p, q, r and  ,
where   cos
2
2
 i sin
.
3
3
(4 marks)
(b) Let f ( x)  ax  3bx  3cx  d , where a, b, c, d  R and ac  b  0.
If f(x) can be expressed in the form A( x  p) 3  B( x  q) 3 , where A, B, p, q  R,
3
show that p  q 
2
2
ad  bc
bd  c 2
and
pq

.
ac  b 2
ac  b 2
(5 marks)
(c) Using (a) and (b), or otherwise, find all the roots of the equation
x 3  9 x  12 = 0
in terms of   cos
9
Let x1 , x2 ,..., xn
2
2
 i sin
.
3
3
(6 marks)
be n real numbers such that not all xi ' s are equal and 0  x1  x2  ...  xn .
xn
 1.
x1 x 2 ... x n
(i) Prove that f is strictly increasing on [ x1 , xn ].
(ii) Prove that the equation f(x) = 0 has one and only one real root in ( x1 , x n ).
(a) For x  0 , let f ( x ) 
(4 marks)
x
.
x1 x 2 ... x n
Using (a)(i), show that the sequence {a n } is a monotonic increasing sequence.
(b) Let ai 
(i)
n
i
(ii) Explain why there exists a positive integer k such that
a1  a2  ...  ak  1  ak 1  ...  an .
Hence show that a1a2 ...an  n  a1 n a2 n ...an n .
xk
Deduce that
x1 x2 ... xn 
x1  x2 ...  xn
n
x1
x2
xn
 x1 1 x2 2 ... xn n .
x
x
x
(11 marks)
10. Let {an } be a sequence satisfying
(1) a1  a2  0,
(2) an2  pan1  qan for n = 1, 2, 3,…,
where p, q are positive constants and p + q = 1.
(a) Show that for any n  1, an 1  an  ( 1) n q n 1 ( a1  a2 ).
Hence deduce that the sequence {a1 , a3 , a5 ,...}is strictly decreasing and that the sequence
{a2 , a4 , a6 ,...} is strictly increasing.
(6 marks)
(b) For any positive integers m and n, show that a2 m  a2 n1.
(4 marks)
(c) Show that two sequences {a1 , a3 , a5 ,...}and {a2 , a4 , a6 ,...} are convergent and their limits
are the same.
(5 marks)
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11. (a) Using De Moivre’s Theorem, show that
sin 7  sin  64 cos6   80 cos 4   24 cos 2   1 .


(3 marks)
(b) It is given that  is not a multiple of . Prove that
sin 7
 8 y 3  4 y 2  4 y 1 ,
sin 
where y  cos 2 .
(4 marks)
(c) Using (b), or otherwise, prove that the roots of 8 y 3  4 y 2  4 y  1  0 are
cos
2
4
6
, cos
and cos
.
7
7
7
Hence evaluate
(i)
(ii)
cos
2
4
4
6
6
2
cos
 cos
cos
 cos cos
and
7
7
7
7
7
7
cos 2
2
4
4
6
6
2
cos 2
 cos 2
cos 2
 cos 2
cos 2
.
7
7
7
7
7
7
(8 marks)
END OF PAPER
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