Shatin Pui Ying College

Shatin Pui Ying College First Examination (06-07) F.6 Pure Mathematics Time allowed : 3 hours

Name : Class : F.6B No. : Marks :

There are 13 questions in this paper. Attempt ALL questions.

1. Factorize (a + b + c)3 – a3 – b3 – c3. (4 marks)

2. Show that if n is a natural number, then n2 1 is not a natural number.

(4 marks) 3. Let n be a positive integer and f(x) = x2n+1 – 80x – 3. (a) Write down all the possible integral roots of f(x) = 0. (b) If  is an integer and f(x) is divisble by x – , find the values of n and . (c) Find a polynomial g(x) with integral coefficients such that f(x) + g(x) is divisible by x2 – 1. (5 marks) 4 4. Let f : [0, ) → R defined by f(x) = . x2 1 (a) Find f(x). Hence show that f is injective. (b) Find a subset of R to replace R such that f is bijective and find its inverse. (5 marks)

7 25 5. Given a sequence {an} defined by a1 = 4, a2 = , a3 = and 2 12 5 1 1 an+3 = an+2 + an+1 – an for nN. 6 3 6

n-1 n-2  1   1  Prove that an  2       for all nN.  2   3 

(5 marks) sin1 x d 2 y dy 6. Let y = . Show that 1 x2   3x  y  0. 1 x2 dx2 dx (5 marks)

a 7. If f(x) is an odd function, prove that f (x)dx  0. a

1 Hence evaluate (e x  ex ) tan 2 xdx . 1 (5 marks) 2006-07/1st Exam/F.6PMaths/LCK/p.1 of 3 8. Given a function f(x) defined by ln(1 x)  if x  0 x  f (x)  1 if x  0 ,  1 x  1 x  if 1 x  0  x (a) Prove that f is continuous at x = 0. (b) Determine whether f is differentiable at x = 0. (6 marks)

9. Evaluate the following limits : cos1 x (a) lim , x1 x 1 (4 marks)  1 1  (b) lim   , x0 x sin x  (4 marks) lim tan xtan 2x (c)   . x 4 (5 marks)

10. Find the following indefinite integrals: (a)  cos5 xsin10 xdx , (4 marks) 1 2x dx (b)  2 , 1 2x  x (4 marks) (c)  2x sin xdx . (4 marks)

1 2x  2 1 1 11. (a) Evaluate dx and dx . 0 x 2  2x  2 0 x 2  2x  2 (4 marks) 2x2  9x  3 (b) Resolve into partial fractions. x 1x2  2x  2 (4 marks) 2 1 2x  9x  3 (c) Using (b), or otherwise, evaluate dx .  0 x 1x 2  2x  2 (4 marks)

2006-07/1st Exam/F.6PMaths/LCK/p.2 of 3  4 n 12. For any positive integer n, define I n  tan xdx. 0

(a) Evaluate I0 and I1. (4 marks) 1 (b) Show that I   I for any n  2. n n 1 n2   Hence evaluate 4 tan 3 xdx and 4 tan 4 xdx. 0 0 (4 marks) n k 1 (1) n (c) Prove by induction that for any nN, I2n1    (1) ln 2 . k 1 2(n  k 1) (4 marks)

13 (a) Let f : [0, ) → R be a function defined by  f(x) = 2 sin x tdt . 0 x 1 Show that for any x ≥ 0, f(x + 2) = f (x) . x  2 (5 marks) (b) Let g : [0, ) → R be a function defined by g(x) = (x + 1)f(x) f(x+1) Show that g is a periodic function with period 1.

     Hence evaluate  2 sin 9 tdt  2 sin10 tdt .  0  0    

(7 marks)

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