Download Section A ( 40 marks ) 1. Consider the system of linear equationsin x

Section A ( 40 marks )
1.
Consider the system of linear equationsin
a  1x 
E  :  x

 x


y

a  1y 
y

x,y
and
z.
z
 1
z
 b
a  1z  b 2
where a , bR .
(a) Find the range of values of
a
and
b
for which (E) has a unique solution.
(b) For each of the following cases, find the value(s) of b
for which
(E)
is consistent:
(1) a = 2 ,
(2) a = 1 .
2.
(a) Prove that
( 6 marks )
C rn 1

r!
nr
, where
n,r
are positive integers and
(b) If a1 , a2 , an are positive real numbers and
n r .
s  a1  a2    an , using “A.M.  G.M.”and
(a) , or otherwise, prove that
s2 s3
sn
1  a1 1  a2 1  an   1  s      .
2! 3!
n!
3.
P
( 6 marks )
is a point in an Argand diagram representing the complex number
z
such that
z  2  i   1 .
Let
C
be the locus of
(a) Sketch C
P.
on an Argand diagram
(b) S is a point on
C
such that the modulus of the complex number represented by S
greatest. Find the complex number represented by
(c) T
is a point on
represented by T
C
is the
S.
such that the principal value of the argument of the complex number
is the least. Find the complex number represented by
T.
( 7 marks )
1
4.
A sequence a1 , a 2 , a n  of real numbers is defined by
a0  0 , a1  1
and
an  an1  an2
Show that for all non-negative integers
x 2  x 1  0
Also prove that
5.
Let
is
with
lim
n 

1
 n  n
5

, where
,
are the roots of
a n 1
 .
an
( 7 marks )
f ( x ) be a polynomial with real coefficients. It is given that the leading coefficient of f ( x )
2 and deg f ( x ) = 4 . When f ( x ) is divided by x + 2 , the remainder is
( x ) is divided by
(a) Find
(b) If
Let
an 
> 0 , < 0 .
x 2  1 , the remainder is
f ( x ) is divided by
6.
n,
n=2,3….
for all
x  2x 2  1
28 . When
13x  1 . Let r ( x ) be the remainder when f
.
r(x).
f ( x ) is divisible by x– 2 , find f ( x ) .
a 0 0


A   0  a 0  , where
0 0 a


It is given that
a
( 7 marks )
is a real number.
3  3 non-singular matrix such that
B is a


det BAB 1  8 .
(a) Find the value(s) of a .
(b) If there exist real numbers
1  p1  q1  r 
.
p , q
and
r
such that
 p 0 0


BAB 1   0 q 0  , find
0 0 r


( 7 marks )
2
Section B ( 60 marks )
Answer any FOUR questions in this section. Each question carries 15 marks.
7.
Let
n be a positive integer.
(a) (i)
z 2n  1  0 .
Find all the roots of
(ii) By factorizing z 2 n  1 into a product of quadratic factors with real coefficients, or
otherwise, prove that
zn 
n 1
2k  1  for all
1
1


 z   2 cos


n
z
2n 
z
k 0 
z0.
( 7 marks )
(b) Using (a), or otherwise, prove that,
(i)
n 1
2k  1  for any R

cosn  2 n 1   cos   cos

2n 
k 0 
(ii) sin
8.
(a) Let
A
2n  1  2 .

3
sin
sin
4n
4n
4n
2n
and
B
( 8 marks )
be two square matrices of the same order. If AB = BA = 0 , show that for
any positive integer
n,
A  B n  A n   1n B n .
(b) Let
(i)
a b 
 , where a , b , c , dR , c≠ 0 and
A = 
c d 
 kc kd 
 for some kR .
Show that A = 
c d 
(ii) Show that for all positive integers
A n  a  d 
n 1
(c) Let
(i)
( 3 marks )
detA = 0 .
n≥ 2 ,
A .
( 5 marks )
5 6 
 .
P= 
 4  5
Find two
2  2
singular matrices
C
and
D
such that
C–D = P
and
 r 0
 , where r , sR .
C + D = 
0 s 
(ii) Using the above results, or otherwise, find
P1997 .
( 7 marks )
3
9.
(a) Consider the system of linear equations
x 
2 x 
E  : 
3  p y
py
 5z  2
 10 z  4
where pR .
Find the value(s) of p
so that
(E) is consistent.
Solve (E) in each case.
( 5 marks )
(b) Consider the system of linear equations
x 
F  : 2 x 
3x 

3  p y
py
8y
 5z  2
 10 z  4
 15 z  q
where p , qR .
Find the value(s) of p
and
q
so that
(F)
Solve (F) in each case.
is consistent.
( 6 marks )
(a) Consider the system of linear equations
x

* : 2 x
3 x
x





3  p  y
py
8y
y
Find all solutions of (*) .
 5 z
 10 z
 15 z

z




2
4
6
8
( 4 marks )
4
P0  x  , P1 x  , P2 x  , … are defined by
10. The polynomials
P0 x   1 , P1 x  x ,
(a) Show that
Pr k 
Pr x  
xx  1 x  r  1
,
r!
is an integer for r 0
[Hint: For r 2 , consider the cases
binomial coefficients
where r = 2 , 3 , …
and for any integer
r k , 0  k< r
and
k.
k< 0
and use the fact that the
C qp are integers.]
( 4 marks )
n
(b) Let
Px    a r Pr x  , where
r 0
If
a0 , a1 ,, a m1
a0 , a1 ,, an
are constants.
am 0  m  n
are integers but
is not, show that
Pm
is not an
integer.
Deduce that if
(i)
P0 , P1 , … , Pn
a0 , a1 ,, an
are integers,
(ii) Pk  is an integer for any integer
(c) Find a polynomial
but not all of a , b
are integers, then
k.
Qx   ax 2  bx  c
( 8 marks )
such that
and c are integers.
Qk 
is an integer for any integer
k,
( 3 marks )
5
11. (a) Let
(i)
f x   x p  px
p> 1 and
f x 
Find the absolute minimum of
x p  1  px  1
(ii) Deduce that
(b) (i)
for all
Let
,,
and
x> 0 .
(0,).
on the interval
for allx> 0 .
( 4 marks )
 be positive numbers such that
1 1
  1 and     1 .
 
By taking
x  
and  respectively, prove that, for
p> 1 ,
 p 1  p   p 1 p  1 ,
where the equality holds if and only if
(ii) Deduce that, if
ab


 a 
(c) Suppose
p 1
a,b,c
and
ab
c 

 b 
p
a1 , a2 , an 
and
d
p 1
 =  = 1 .
are positive and
d p  c  d 
p
p> 1 , then
.
b1 , b2 ,bn 
( 7 marks )
are two sequences of positive numbers
andp>1 .
1
1
By considering
1
p
 n
p 
a    a j 
 j 1

and
1
 n pp
b    b j  , prove that
 j 1

1
p
 n pp  n pp  n
p 
  ai     bi     ai  bi   ,
 i 1

 i 1

 i 1

wherethe equality holds if and only if
a
a1 a2
a
.

 n 
b1 b2
bn b
( 4 marks )
END OF PAPER
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