Shatin Pui Ying College s1
Shatin Pui Ying College
Final Examination (2004-05)
Pure Mathematics
Time allowed : 3 hours
Name : Date :
Class : F.6B No. : Marks :
There are 13 questions in this paper. Attempt ALL questions.
1. Show that ex £ ex for all x > 0. (3 marks)
2. Evaluate (3 marks)
3. Using Mean Value Theorem, show that for any b > 0,
. (3 marks)
4. Let f(x) = (1 – x2)n. Using Leibniz’s Rule,
show that
(4 marks)
5. Let f(x) = .
Discuss the continuity and differentiability of f(x) at x = 0. (4 marks)
6. Let
Using Binomial Theorem, show that
Hence evaluate (5 marks)
7. (a) Find
(b) Let R be the region bounded by the curves y = ln x, x = 1, x = e and the x-axis.
Find the volume of the solid of revolution of R about the x-axis.
(6 marks)
8. A sequence is defined by
(a) Prove that and hence deduce that xn+1 < xn for n ³ 2.
(b) Using, or otherwise, show that {xn} is convergent and find its limit.
(6 marks)
9. Let f(x) and g(x) be continuous functions on [a, b].
Define a function F(x) on [a, b] by
.
Prove that
Hence deduce that
(6 marks)
10. Let
(a) (i) Find
(ii) Doesexist?
(4 marks)
(b) Find the values of x for each of the following cases :
(i)
(ii)
(iii)
(iv)
(4 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion f(x).
(3 marks)
(d) Find all asymptote(s) of the graph of f(x).
(2 marks)
(e) Sketch the graph of f(x).
(2 marks)
11. F(x) is a function defined on R satisfying
(1) F(x + y) = F(x)F(y)
(2) 1 + x £ F(x) £ 1 + xF(x)
(a) Show that
(i) F(0) = 1,
(ii) F(x) ³ 1 for any x ³ 0,
(iii) F(x) ³ 0
Deduce that F(x) is increasing.
(8 marks)
11. (b) Show that F(x) is continuous at x = 0.
(3 marks)
(c) Show that F(x) is differentiable at x = 0 and find
(4 marks)
12. Let f0(x) be a continuous function and for each positive integer n and for x ³ 0,
fn(x) =
(a) For any positive integers m, n and x ³ 0. Using integration by part, show that
(5 marks)
(b) Using (a), show that for each positive integer n and x ³ 0,
(5 marks)
(c) Suppose that |f0(x)| £ M for all 0 £ u £ 1, where M is a positive constant.
If 0 £ x £ 1, show that |fn(x)| £ and hence deduce that
(5 marks)
13. Given two ellipses
(E) :
(F) :
where 0 < b < a.
(a) Show that the line mx + ny = 1 is tangent to (E) if and only if a2m2 + b2n2 = 1.
(4 marks)
(b) Find the equations of the common tangents to (E) and (F).
(5 marks)
(c) R(h, k) is a point outside (E). The two tangents drawn from R to (E) touch (E) at two points S and T. Find the equation of the straight line through S and T.
(2 marks)
(d) It is furthermore given that the line through S and T in (c) is tangent to (F). Find and name the locus of R.
(4 marks)
END OF PAPER
2004-05/Final Exam/F.6PMaths/LCK/p.1 of 3