2002_Paper II

# 2002_Paper II - STEP II, 2002 Section A: 1 2 Pure...

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STEP II, 2002 2 Section A: Pure Mathematics 1 Show that Z π/ 4 π/ 6 1 1 - cos 2 θ d θ = 3 2 - 1 2 . By using the substitution x = sin 2 θ , or otherwise, show that Z 1 3 / 2 1 1 - (1 - x 2 ) d x = 3 - 1 - π 6 . Hence evaluate the integral Z 2 / 3 1 1 y ( y - ( y 2 - 1 2 )) d y . 2 Show that setting z - z - 1 = w in the quartic equation z 4 + 5 z 3 + 4 z 2 - 5 z + 1 = 0 results in the quadratic equation w 2 + 5 w + 6 = 0. Hence solve the above quartic equation. Solve similarly the equation 2 z 8 - 3 z 7 - 12 z 6 + 12 z 5 + 22 z 4 - 12 z 3 - 12 z 2 + 3 z + 2 = 0 . 3 The n th Fermat number, F n , is deﬁned by F n = 2 2 n + 1 , n = 0 , 1 , 2 , . . . , where 2 2 n means 2 raised to the power 2 n . Calculate F 0 , F 1 , F 2 and F 3 . Show that, for k = 1, k = 2 and k = 3 , F 0 F 1 . . . F k - 1 = F k - 2 . ( * ) Prove, by induction, or otherwise, that ( * ) holds for all k > 1. Deduce that no two Fermat numbers have a common factor greater than 1. Hence show that there are inﬁnitely many prime numbers.

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STEP II, 2002 3 4 Give a sketch to show that, if f( x ) > 0 for p < x < q , then R q p f( x ) d x > 0 . (i) By considering f( x ) = ax 2 - bx + c show that, if a > 0 and b 2 < 4 ac , then 3 b < 2 a +6 c . (ii) By considering f( x ) = a sin 2 x - b sin x + c show that, if a > 0 and b 2 < 4 ac , then 4 b < ( a + 2 c ) π . (iii) Show that, if a > 0, b 2 < 4 ac and q > p > 0 , then b ln( q/p ) < a ± 1 p - 1 q ² + c ( q - p ) . 5 The numbers x n , where n = 0, 1, 2, . . . , satisfy x n +1 = kx n (1 - x n ) . (i) Prove that, if 0 < k < 4 and 0 < x 0 < 1, then 0 < x n < 1 for all n . (ii) Given that x 0 = x 1 = x 2 = ··· = a , with a 6 = 0 and a 6 = 1, ﬁnd k in terms of a . (iii) Given instead that x 0 = x 2 = x 4 = ··· = a , with a 6 = 0 and a 6 = 1, show that ab 3 - b 2 + (1 - a ) = 0, where b = k (1 - a ) . Given, in addition, that x 1 6 = a , ﬁnd the possible values of k in terms of a .
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## This note was uploaded on 04/01/2012 for the course MATH 1016 taught by Professor Rotar during the Spring '12 term at Central Lancashire.

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2002_Paper II - STEP II, 2002 Section A: 1 2 Pure...

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