Further Pure Mathematics PAPER 1

International GCSE in Further Pure Mathematics Formulae sheet Mensuration Surface area of sphere = 4πr2 Curved surface area of cone = πr × slant height Volume of sphere = 4 3 πr3 Series Arithmetic series Sum to ...

International GCSE in Further Pure Mathematics Formulae sheet Mensuration Surface area of sphere = 4πr2 Curved surface area of cone = πr × slant height Volume of sphere = 4 3 πr3 Series Arithmetic series Sum to n terms, Sn = + n[ ] a n − d 2 2 1 ( ) Geometric series Sum to n terms, S a r r n n = −− ( ) ( ) 1 1 Sum to infinity, S a r r ∞ = 1 − < 1 Binomial series ( ) ( ) ! ( ) ( ) ! 1 1 1 , 2 1 1 + = x n + + x n n − x2 + + n n − − n r + + ∈ 1 r x x n n r    for <  Calculus Quotient rule (differentiation) d d f g f g g x [g xx x x x x x ( ) ( ) ( ) ( ) f( ) ( ) ( )]   = −' ' 2 Trigonometry Cosine rule In triangle ABC: a2 = b2 + c2 – 2bccos A sin tan cos θ θ θ = sin(A + B) = sin A cos B + cos A sin B sin(A – B) = sin A cos B – cos A sin B cos(A + B) = cos A cos B – sin A sin B cos(A – B) = cos A cos B + sin A sin B tan( ) tan tan tan tan A B A B A B + = + 1 − tan( ) tan tan tan tan A B A B A B − = − 1 + Logarithms log log a log b b x xa = http://britishstudentroom.wordpress.com/*P66024A0336* Turn over 3 Answer all ELEVEN questions. Write your answers in the spaces provided. You must write down all the stages in your working. 1 The quadratic equation 3(k + 2)x2 + (k +5)x + k = 0 has real roots. Find the set of possible values of k. 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(Total for Question 1 is 6 marks) http://britishstudentroom.wordpress.com/4 *P66024A0436* 2 Angle α is acute such that cos α = 3 5 Angle β is obtuse such that sinβ = 1 2 (a) Find the exact value of (i) tanα (ii) tanβ (3) (b) Hence show that tan(α + β) = m n n m 3 3 − + where m and n are positive integers whose values are to be found.