2001 Pure Maths Paper 2
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  B ) 
tan A  tan B
1  tan A tan 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  cos B  2 cos
A B
A B
cos
2
2
A B
A B
sin
2
2
2 sin A cos B  sin( A  B)  sin( A  B)
cos A  cos B  2 sin
2 cos A cos B  cos( A  B)  cos( A  B)
2 sin A sin B  cos( A  B)  cos( A  B)
Section A (40 marks)
1.
(a)
Find
e
lim
x0

2
 e x
.
1  cos 2 x
x
(b) Prove that lim x 2 cos
x 0
2.
3.
(6 marks)
Evaluate
(a)
x3
 1  x 2 dx,
(b)
x
2
tan 1 x dx.
Let a1  1 and an1 
n N.
Hence show that
4.
1
 0.
x
 an 
(5 marks)
4  an
2
2
for n N. Show that 1  an  2 for all
is convergent and find its limit.
(6 marks)
Let P be the parabola y 2  4ax where a is a non-zero constant, and

2
 
2

A at1 , 2at1 , B at 2 , 2at 2 be two distinct points on P.
(a)
(b)
Find the equation of chord AB.
If A and B move in such a way that chord AB always passes through (a, 0),
find the equation of the locus of the mid-point of AB.
(5 marks)
5.
Let f be a real-valued function continuous on [0, 1] and differentiable in (0, 1).
Suppose f satisfies
A.
f(0) = 0,
B.
f(1) =
C.
1
,
2
0 < f  (t) < 1 for t  0, 1 .
Define F(x) = 2 f (t ) d t  f  x  for x  0, 1 .
x
2
0
6.
7.
(a)
Show that F (x) > 0 for x  0, 1 .
(b)
Show that
1
 f (t ) dt >
0
1
.
8
 x 3  12 x

Let f : R → R be defined by f(x)   x
x
 2  x
e
e
(6 marks)
when
x  2,
when
x  2.
(a)
Show that lim f ( x)  lim f ( x)  0.
(b)
Is f differentiable at x = 2? Explain your answer.
x 2
x 2
Figure 1 shows the curve
 x  cos 3 t ,
:
0  t  2 .
 y  sin 3 t ,
(a)
(b)
Find the length of  .
Find the area enclosed by  . (7 marks)
Section B (60 marks)
Answer any FOUR questions in this section.
2
8.
Let f(x) = x 3 6  x 3 .
(a)
1
(i) Find f ( x) for x  0, 6 .
(ii) Show that f (0) and f (6) do not exist.
(iii) Show that f ( x) 
(b)
(c)
(d)
(e)
9.
8
4
3
x x  6
5
3
for x  0, 6 .
Determine the values of x for each of the following cases:
(i) f ( x) > 0
(ii) f ( x) < 0
(iii) f ( x ) > 0
(iv) f ( x ) < 0
(4 marks)
(3 marks)
Find all relative extreme points and points of inflexion of f(x). (3 marks)
Find all asymptotes of the graph of f(x).
(2 marks)
Sketch the graph of f(x).
(3 marks)
In Figure 2, C is the circle x 2   y  1  4 2 and F
2
is the point (0, 1). For any point P on C, let L p be
the perpendicular bisector of the line segment FP. It
appears that as P moves on C, all the L p ’s are
tangents to an ellipse inside C.
(a)
Suppose for every P on C, the line L p is tangent to the same ellipse.
Write down the equations of the two horizontal tangents and the two
vertical tangents to the ellipse.
Hence guess the equation of the ellipse.
(Note that C is symmetric about the y-axis and F lies on this line of
symmetry.)
(5 marks)
(b)
Let E be the ellipse found in (a).
(i) For any point P(p, q) on C, let M be the point (m, n) where
3p
4q  1
and n 
. Show that
m
7q
7q
M lies on E,
the tangent at M to E is the perpendicular bisector of FP.
(I)
(II)
(ii)
For any point M on E, show that there is a point P on C such that the
perpendicular bisector of FP is the tangent to E at M.
(10 marks)
10. (a) Prove by induction that lim xln x   0 for any non-negative integer n.
n
x0
(3 marks)
(b)
Let n be a positive integer.
 ln x 
n
n 1
dx = xln x   n  ln x 
n
(i)
Show that
(ii)
Show that the improper integral
dx.
1
 ln x dx is convergent
0
1
(i.e. lim  ln x dx exists) and find its value.
h0
h
Hence deduce that the improper integral
1
 (ln x)
0
and find its value.
(c)
n
dx is convergent
(8 marks)
Let n be a positive integer and  be a positive real number. For 0 < h < 1,
show that

 x ln x
1
h
1
n
dx =
1

n 1
 ln x 
1
Hence show that the improper integral
find its value.
n
h
dx.

 x ln x
1
0
1
n
dx is convergent and
(4 marks)
11. (a)
Let f and g be real-valued functions continuous on [a, b] and differentiable
in (a, b).
(i)
(ii)
By considering the function
h(x) = f(x) g(b)  g(a)  g( x)[f (b)  f (a)] on [a, b], or otherwise,
show that there is c  a, b such that
f (c)[g(b)  g(a)]  g(c)[f (b)  f (a)] .
Suppose g( x)  0 for all x  a, b . Show that g( x)  g(a)  0
for any x  a, b .
f ( x )
If, in addition,
is increasing on (a, b), show that
g ( x )
P(x) =
f ( x)  f (a )
is also increasing on (a, b).
g ( x)  g (a )
 e x cos x  1
if

(b) Let Q(x) =  sin x  cos x  1
1
if

(9 marks)
 
x   0, ,
4

x  0.
 
Show that Q is continuous at x = 0 and increasing on 0,  .
 4
x
 
Hence or otherwise, deduce that for x  0,  ,  Q(t ) dt  x .
 4 0
(6 marks)
12. (a) Evaluate
(i)
(ii)
(a)
x
2
1
dx ,
 x 1
x2  1
dx .
x2  x  1 x2  x  1



(6 marks)
For n = 1, 2, 3, … and 0  x  1 , define g n ( x)  r 1
n
x 6 r 5
and
6r  5
(i)
x 6 r 1
. For any fixed x  0, 1 ,
6r  1
show that the sequences g n ( x) and h n ( x) are increasing ;
(ii)
deduce that lim g n ( x) and lim h n ( x) exist.
h n ( x)  r 1
n
(b)
n 
n 
(3 marks)
6 r 5
n  x
x 6 r 1 

For n = 1, 2, 3, … and 0  x  1 , define f n ( x)  r 1 

 6r  5 6r  1 
and let f(x) = lim f n ( x) .
n
(i)
For any fixed x  0, 1 , evaluate f n ( x) and show that
lim f n ( x) exists.
n
(ii)
Assuming that f ( x)  lim f n ( x) for any fixed x  0, 1 and f is
n
continuous on
x  0, 1 , show that f ( x) 
Hence find f(x).
1  x2
for x  0, 1 .
1  x2  x4
(6 marks)
13. Let f : (0,  )  (0,  ) be a continuous function satisfying f[f(x)] = x and
f ( x)
f(1+x) =
for all x.
1  f ( x)
(a)
Show that for any n N and x R, f n  x  
(b)
Define xn  1 
f ( x)
.
1  nf ( x)
(3 marks)
n 1
f (1)
for any n N. Show that f ( xn ) 
.
n2
1  nf (1)
Hence, by considering lim xn , show that f(1) = 1.
n
1
1
and f ( )  n .
n
n
For any q  N and q  2 , let S(q) be the statement
Deduce that f (n) 
(c)
(5 marks)
 p q
“ f   
for all p  N with 0 < p < q.”
q p
(i) Show that S(2) is true.
(ii) Assume that S(h) is true for 2  h  q and h N. Use (a) to show
 q  1
p
 
that f 
for 0 < p < q + 1.
 p  q 1
Hence deduce that S(q + 1) is true.
(iii)
Use (a) to show that for any positive rational number x, f ( x) 
1
.
x
(7 marks)