HKALE Pure Maths 2001 Paper01

Such that x1 1 x 2 and the equality holds iff x1 x 2

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: + s = −1 ps − pq = −2 Putting r = − p , ⇒ q + s = p 2 − 1 2 q − s = p ⇒ 2 2 2 q = p − 1 + p 2 s = p 2 − 1 − 2 p 2001 Paper 1 Section B Since qs = −1 , therefore 2 2 2 2 p − 1 + p − 1 − = −4 p p (p 3 )( ) − p + 2 p 3 − p − 2 = −4 p 2 6 4 2 p − 2p +5p −4 = 0 (p ) − 1 ( p 4 − p 2 + 4) = 0 p = ±1 Hence or 2 p = 1, q = 1 , r = − 1 , s = − 1 p = −1, q = −1 , r = 1 , s = 1 (iii) From (b)(ii), g(x) = ( x 2 − x − 1)( x 2 + x + 1) . − 1 ± 3i 1± 5 or . 2 2 Since f(x) = g(x + 2) , ∴ g(x) = 0 if x = ∴ f(x) = 0 if x = −3± 5 − 5 ± 3i or . 2 2 2001 Paper 1 Section B 14. (a) If a ≤ 1 ≤ b , then a + b − ab − 1 = (a − 1)(1 − b) .........(1) ≥0 ∴ a + b ≥ ab + 1 From (1), the equality holds iff a − 1 = 0 or 1 − b = 0 iff a = 1 or b = 1 . (b) If x1 , x 2 are positive real numbers such that x1 x 2 = 1 , without loss of generality, we may assume that x1 ≤ 1 ≤ x 2 . From (a), x1 + x 2 ≥ x1 x 2 + 1 = 2 and ∴ iff x1 = 1 or x 2 = 1 iff the equality holds x1 = x 2 = 1 (since x1 x 2 = 1 ) The statement holds for n = 2. Assume the statement holds for n = k where k ∈ N and k ≥ 2 . Let x1...
View Full Document

Ask a homework question - tutors are online