Download 2004 Pure Maths Paper 2

2004 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
1. Evaluate
(a)
1
x
lim (tan 3x  cos 4 x) ,
x 0
(b) lim (cos 2004  x  cos x ) .
x 
(7 marks)
2.
 x 2  ax  b when

Let f ( x)   sin x
when



x  1,
x  1.
If f is differentiable at 1, find a and b.
(6 marks)
3.
(a) Evaluate
 sec
3
 d .
(b) Consider the curve C : x 2  2 y , 0  x  1 . Find the length of C.
(7 marks)
4.
Using the substitution u 
1
, prove that
x
3n
Hence, or otherwise, evaluate lim
n 

k 1

2
1
2
ln x
dx  0 .
1 x2
 1 k 
ln  2 (  ) 
 2 2n 
.
1 k 2

2 n 1  (  ) 
2 2n 

(6 marks)
5.

For each non-negative integer n, define I n   e  x x n dx .
(a)
0
(i)
Evaluate I 0 .
(ii)
Prove that if I n is convergent, then I n 1  (n  1) I n .
(iii)
Express I n in terms of n.
(b) For each non-negative integer n, define J n  

0

x2 

e 1 
 n 1
2
x
dx.
Find a non-negative integer m such that J m  J n for all n.
(8 marks)
6.
Consider the two planes  1 : x  y  z  1 and  2 : x  y  z  2 .
(a) Find a parametric equation of the line of intersection of  1 and  2 .
(b) Find the equation(s) of the plane(s) containing all the points which are
equidistant from  1 and  2 .
(6 marks)
Section B
7.
Let f ( x) 
x x3
(x  3 2) .
x 2
(a) (i) Find f ( x) and f ( x) for x > 0.
(ii) Write down f ( x) and f ( x) for x < 0.
(iii) Prove that f (0) exists.
3
(iv) Does f (0) exist? Explain your answer.
(5 marks)
(b) Determine the range of 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.
(3 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of f(x). (2 marks)
(d) Find the asymptote(s) of the graph of f(x).
(3 marks)
(e) Sketch the graph of f(x).
(2 marks)
8.
(a)
For any non-negative integers m and n, define
x
I m, n ( x)   cos m  cos n d for all x R.
0
Prove that I m 1, n 1 ( x) 
(b) Evaluate
(c) Evaluate


2
0


2
0
cos m 1 x sin( n  1) x
m 1

I m , n ( x) .
mn2
mn2
(6 marks)
cos 4  cos 3 d .
(4 marks)
sin 4  cos 3 d .
(5 marks)
9.
x2 y2

 1 , where a and b are two positive constants
a2 b2

with a > b. Let P be the point a cos  , b sin   , where 0    .
2
Consider the ellipse E :
(a) Prove that P lies on E.
(1 mark)
(b) Let L be the tangent to E at P. L cuts the x-axis and the y-axis at P1 and
P2 respectively. Find
(i)
(ii)
the equation of L,
the coordinates of P1 and P2 .
(4 marks)
(c) Consider the two circles C1 : x 2  y 2  a 2 and C 2 : x 2  y 2  b 2 . Also
consider the two points P1 and P2 described in (b). For k = 1, 2, let Lk
be the tangent to C k from Pk , with the point of contact Qk lying in the
first quadrant.
(i)
Prove that L1 is parallel to L2 .
(ii)
Find the coordinates of Q1 and Q2 .
(iii) Let l be the straight line passing through Q1 and Q2 . Is l a
common normal to C1 and C2 ? Explain your answer. (10 marks)
10. (a) (i) Evaluate
 1

(ii) Prove that
0
1
2
dx
2x  x 2
.
2 2
dx   .
1  2x  x 2
2 2  4x  2 2x 2
dx   .
0
1 x4
Let k be a non-negative integer. Prove that
1
(iii) Using (a)(ii), deduce that
(b) (i)
k
 x 4 k  4   (1) n x 4 n 
n 0
2
(6 marks)
1
 x 4 k  4 for all real numbers x.
1 x4
(ii) Using (b)(i) and (a)(iii), or otherwise, prove that

1 n 2
2
1 



  .
 4n  1 4n  2 4n  3 
 ( 4 )
n 0
Candidates may use the fact, without proof, that for any
given polynomial p(x), lim

k  0
1
2
x k p( x)dx  0 .
(9 marks)
11. For any real number x, let [ x ] denote the greatest integer not greater than x. Let
f : R→R be defined by
1
 2
f ( x)  
 x  [ x]  1

2
when x is an integer ,
when x is not an integer .
(a) (i) Prove that f is a periodic function with period 1.
(ii) Sketch the graph of f(x), where  2  x  3 .
(iii) Write down all the real number(s) x at which f is discontinuous.
(6 marks)
x
(b) Define F( x)   f (t )dt for all real numbers x.
0
(i)
(ii)
x2  x
.
2
Is F a periodic function? Explain your answer.
If 0  x  1 , prove that F( x) 
(iii) Evaluate
12. (a)


0
F( x)dx .
(9 marks)
Let f : R→R be a twice differentiable function. Assume that a and b are
two distinct real numbers.
(i) Find a constant k (independent of x) such that the function
h( x)  f ( x)  f (b)  f ( x)( x  b)  k( x  b) 2 satisfies h(a) = 0.
Also find h(b).
(ii)
(b)
Let I be the open interval with end points a and b. Using Mean Value
Theorem and (a)(i), prove that there exists a real number c  I such
f (c)
(b  a) 2 .
that f (b)  f (a)  f (a)(b  a) 
(7 marks)
2
Let g : R→R be a twice differentiable function. Assume that there exists a
real number   0, 1 such that g ( x)  g (  )  1 for all x  0, 1 .
(i) Using (a)(ii), prove that there exists a real number   0, 1 such that
g ( )
(1   ) 2 .
2
If g ( x)  2 for all x  0, 1 , prove that g(0) + g(1)  1 .
g(1)  1 
(ii)
(8 marks)