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)

n  3n  2  2. Evaluate lim  . (3 marks) n 3n 1 

3. Using Mean Value Theorem, show that for any b > 0, b  tan 1 b  b . (3 marks) 1 b 2

4. Let f(x) = (1 – x2)n. Using Leibniz’s Rule,

2 n (n) r n r nr show that f (x)  (1)   n!(1 x) (1 x) . r0  r  (4 marks) ln(1 x)  if x  (-1,0)  (0,) 5. Let f(x) =  x . 1 if x  0. Discuss the continuity and differentiability of f(x) at x = 0. (4 marks)

6. Let n n  n  1 x n , where n is a natural number and x n  0.

2 2 2 Using Binomial Theorem, show that x n   . n 1 n (n 1)

Hence evaluate lim n n  n . (5 marks) n

7. (a) Find  lnx2 dx. (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)

2004-05/Final Exam/F.6PMaths/LCK/p.1 of 3 3 2x n  27 8. A sequence is defined by x1 1, x n1  2 . 3x n

2 x n-1  3 2x n1  3 (a) Prove that x n  3  2 and hence deduce that xn+1 < xn for n  2. 3x n1

(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

x x x 2 F(x)   f 2 (t)dt g 2 (t)dt   f(t)g(t)dt .   a   a    a 

x Prove that F (x)  f (t)g(x)  f (x)g(t)2 dt  a

b b b 2 Hence deduce that  f 2 (x)dx g 2 (x)dx   f(x)g(x)dx .   a   a    a  (6 marks)

2 10. Let f (x)  xe x , where x  0. (a) (i) Find f  (x) and f  (x) for any x  0. limf (x) (ii) Does x0 exist?

(4 marks) (b) Find 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) (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) x, y  R, (2) 1 + x  F(x)  1 + xF(x) x  R

(a) Show that (i) F(0) = 1, (ii) F(x)  1 for any x  0, (iii) F(x)  0x  R, Deduce that F(x) is increasing. (8 marks)

2004-05/Final Exam/F.6PMaths/LCK/p.2 of 3 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 F(0). (4 marks)

12. Let f0(x) be a continuous function and for each positive integer n and for x  0,

x fn(x) = f n1 (u)du.  0 (a) For any positive integers m, n and x  0. Using integration by part, show that

x x m1 1 m (x  u) f n (u)du  (x  u) f n1 (u)du.  0 m  0 (5 marks) (b) Using (a), show that for each positive integer n and x  0,

x 1 n1 f n (x)  (x  u) f 0 (u)du. (n 1)! 0 (5 marks)

(c) Suppose that |f0(x)|  M for all 0  u  1, where M is a positive constant. M If 0  x  1, show that |fn(x)|  and hence deduce that lim f n (x)  0. n! n (5 marks)

13. Given two ellipses x 2 y 2 (E) :   1, a 2 b 2 x 2 y 2 (F) :   1, b 2 a 2 where 0 < b < a.

2 2 2 2 (a) Show that the line mx + ny = 1 is tangent to (E) if and only if a m + b n = 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

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