C3 Algebraic Fractions Questions June 2006 1. (a) Simplify 3x 2 x 2 . x2 1 (3) 3x 2 x 2 1 (b) Hence, or otherwise, express – as a single fraction in its simplest form. 2 x( x 1) x 1 (3) Jan 2008 1. Given that dx e 2 x 4 3x 2 x 1 (ax2 + bx + c) + 2 , 2 ( x 1) ( x 1) find the values of the constants a, b, c, d and e. (4) June 2009 7. The function f is defined by f(x) = 1 – (a) Show that f (x) = x 8 2 + , x ℝ, x ≠ −4, x ≠ 2. ( x 2)( x 4) ( x 4) x3 . x2 (5) Jan 2007 f(x) = 1 – 2. (a) Show that f(x) = 3 3 + , x –2. x 2 ( x 2) 2 x2 x 1 , x –2. ( x 2) 2 (4) (b) Show that x2 + x + 1 > 0 for all values of x. (3) (c) Show that f(x) > 0 for all values of x, x –2. (1) June 2007 2. f(x) = (a) Show that f (x) = 2x 3 9 2x – , 2 x2 2 x 3x 2 x> 1 2 . 4x 6 . 2x 1 (7) (b) Hence, or otherwise, find f (x) in its simplest form. (3) Jan 2010 Jan 2011 June 2011 C3 Algebraic Fractions Questions ANSWERS (45 marks) June 2006 Jan 2008 June 2009 Jan 2007 June 2007 Jan 2010 Jan 2011 June 2011

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