HKALE Pure Maths 2001 Paper01

I find r ii let e be the intersection of ab and cd

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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) 2001-AL-P 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) 2001-AL-P 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 −...
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