This preview shows page 1. Sign up to view the full content.
Unformatted text preview: s) (b) If v = λa + µb , where λ, µ ∈ R and λ , µ ≠ 0 , and v is also a
linear combination of vectors c and d , show that
λ : µ = (−b ⋅ c × d) : (a ⋅ c × d) .
Hence evaluate λ : µ .
(5 marks) (c) Suppose the four points A , B , C and D are coplanar.
(i) Find r . (ii) Let E be the intersection of AB and CD . Using (b) or
otherwise, find BE : EA and the position vector of E .
(7 marks) 2001ALP MATH 1−9 −8− Go on to the next page 13. (a) Let P( x) = x 4 + ax 3 + bx 2 + cx + d where a, b, c, d ∈ R .
(i) Show that if α is a complex root of P(x) = 0 , then α is also a
root of P(x) = 0 . (ii) For any α ∈ C , show that ( x − α )( x − α ) is a quadratic
polynomial in x with real coefficients.
Hence show that P(x) can be factorized as a product of two
quadratic polynomials with real coefficients.
(6 marks) (b) Let f(x) = x 4 + 8 x 3 + 23 x 2 + 26 x + 7 and g(x) = f(x + k) where k ∈ R
and the coefficient of x 3 in g(x) is zero.
(i) Find k and the coefficients of g(x) . (ii) Suppose g(x) = ( x 2 + px + q)( x 2 + rx + s) where
p, q, r, s ∈ R . By comparing coefficients or otherwise, show
that p 6 − 2 p 4 + 5 p 2 − 4 = 0 .
Hence find p , q , r and s . (iii) Find all roots of f(x) = 0 .
(9 marks) 2001ALP MATH 1−10 −9− (a) If a , b are two real numbers such that a ≤ 1 ≤ b , show that
a + b ≥ ab + 1 and the equality holds if and only if a = 1 or b = 1 .
(3 marks) (b) 14. Show by induction that if x1 , x 2 , ! , x n are n (n ≥ 2) positive real
numbers such that x1 x 2 " x n = 1 , then x1 + x 2 + " + x n ≥ n and
the equality holds if and only if x1 = x 2 = " = x n = 1 . (6 marks)
(c) Let a1 , a 2 , ! , a n be n (n ≥ 2) positive real numbers. Using (b)
a1 + a 2 + " + a n n
≥ a1 a 2 " a n and the
n
equality holds if and only if a1 = a 2 = " = a n .
(3 marks)
or otherwise, show that (d) For u ≥ 0 and n = 2, 3, 4, ! , using the identity (1 + u ) n − 1 = u[(1 + u ) n −1 + (1 + u ) n −...
View
Full
Document
 Fall '02
 lo
 Math

Click to edit the document details