Pui Ying College
F.7 Mock Examination (2004-05)
Pure Mathematics 2
Time allowed : 3 hours
Name :
Class : 7 B
No.
1. This paper consists of Section A and Section B.
2. Answer ALL questions in Section A and any FOUR questions in Section B.
3. You are provided with two answer books. Write your answers for Section A and
B in these two answer books separately.
4. The two answer books must be handed in separately at the end of the
examination.
5. Unless otherwise specified, all working must be clearly shown.
6.
SECTION A (40 Marks)
Answer ALL questions in this section.
1. Evaluate
(a)
x  cos x
,
x  x  1
lim
(b) lim
x 0
3x  2 x
1
.
x3
(6 marks)
2.
Let f : R→R be a differentiable function and f ( x )  1 for all x  R.
α is called a fixed point of f if and only if f(α) = α. Using Mean Value or
otherwise, prove that f has at most one fixed point.
(6 marks)
3.

1
1
1
1
Evaluate lim 


 ... 

2
n 
n 2  2n
n 2  3n
n2  n2
 n n

.


(Hint : Express the limit as a definite integral.)
(7 marks)
2004-05/F7 MOCK EXAM/PM2/ LCK/p.1 of 4
4.
Let f : R→R be a twice differentiable function satisfying the following
conditions :
(1) f ( x )  f ( x ) for all x  R,
(2) f(0) = 1 and
(3) f (0)  0.
Prove that f(x) = cos x for all x  R.
(Hint : Differentiate h(x) = [f(x) – cos x]2 + [f (x)  sin x]2 . )
(7 marks)
5.
(a) Evaluate

x 3 1  9 x 4 dx.
(b) Find the area of surface of revolution generated by revolving the arc of the
curve y3 = x between the points (0, 0) and (1, 1) about the y-axis.
(7 marks)
6.
x  2 y  8
x  3 y 1 z 1


.
Given two lines L1 : 
and L 2 :
1
3
2
3x  2z  14
(a) Prove that L1 and L2 intersect and find their common point.
(b) Find the equation of the plane π which contains L1 and L2.
(7 marks)
SECTION B (60 Marks)
Answer any FOUR questions in this section.
7.
5
3
Let f ( x )  x  5x
2
3
for x R.
(a) (i) Is f differentiable at x = 0? Explain your answer.
(ii) Find f ( x ) and f ( x ) for x  0.
(4 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.
(4marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of f(x).
(3 marks)
(d) Find the asymptote(s) of the graph of f(x).
(1 mark)
(e) Sketch the graph of f(x).
(3 marks)
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8.
Let f : R→R be a function which is not identically zero and satisfies
f(x + y) = f(x)f(y) for any x, y  R.
(a) Show that
(i) f ( x )  0 for any x  R,
(ii) f(x) > 0 for any x  R,
(ii) f(0) = 1.
(6 marks)
(b) Suppose that f is continuous.
(i)
1 x
x
Prove that for any x  R,
1
f (u )du  f ( x )  f ( v)dv.
0
(ii) By differentiating the equation in (i) with respect to x,
show that there exists a real number α such that f ( x )  f ( x ).
Hence show that f(x) = eαx for all x  R.
(Hint : Differentiate e-α xf(x).)
(9 marks)
9.
n
dx
n
(1  x )(1  x 2 )
Given a positive integer n and a real number a, let In(a) =  1
a
.
(a) (i) Find In(0).
(ii) Using partial fractions, find In(1) and In(-1).
(7marks)
(b) (i)
Using the substitution x =
1
, show that
u
xa
dx.
a
2
n (1  x )(1  x )
n
I n (a )   1
(ii) Deduce that In(a) is independent of a.
(iii) Find the improper integral


0
xa
dx.
(1  x a )(1  x 2 )
(8marks)
10. (a) Let x > 0 and define a sequence {an} by
a1  x and a n 
a 2n 1
for n  2.
2a n 1  2
(i)
Show that a n  0 and a n  a n1 for all positive integers n.
(ii)
Show that lim a n  0.
n 
(8 marks)
(c) Let f : [0, ∞]→R be a continuous function satisfying
 x2 
 for all x  0.
f (x)  f 
 2x  2 


Using (a), show that f(x) = f(0) for all x  0.
(7 marks)
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11
(a) Let f(x) be a strictly decreasing and continuous function defined for x  1.
Show that f(t + 1) <
Deduce that
n 1
1

t 1
t
f (x)dx < f(t) for any real number t  1.
n
f ( x )dx   f (k )   f ( x )dx  f (1) for any n  1.
n
1
k 1
(4 marks)
n
1
 ln( n )  1 for any n  1.
k
k 1
(b) Prove that 0  
n
Hence discuss the convergence of
1
 k.
k 1
(4 marks)
(c) (i)
By putting f(x) =
1
in (a), or otherwise, show that
xn
2n
1
1
 2n  1 
 2n 
ln 
for any n  1.
    ln 

 n  1  k n 1 k
 n 1 n 1
2n
(ii) Prove that
1
is convergent and find its limit.
k  n 1 k

(7 marks)
x  t 2
12. The parametric equations of the parabola y = 4x are 
.
y  2t
2
(a) Write down the equation of tangent at the point t.
(2 marks)
(b) Prove that for all values of t and c,
x2 + y2 – (3t2 + 4 – c)x + (t3 – ct)y + ct2 = 0
is the equation of a circle which touches the parabola at t.
(4 marks)
(c) (i) Find the equation of the circle which passes through the focus F(1, 0)
of the parabola and touching the parabola at t.
(ii) Let t1 and t2 be two points on the parabola such that t1Ft2 is a focal
chord of the parabola.
(1) Prove that t1t2 = – 1.
(2) Two circles are drawn through the focus F to touch the parabola at
t1 and t2 respectively. Prove that these two circles are orthogonal.
(Hint : You may use the fact that two circles
Ci : x2 + y2 + aix + biy + ci = 0
are orthogonal iff a1a2 + b1b2 = 2(c1 + c2).)
(9 marks)
END OF PAPER
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