STEP III - Section A: Pure Mathematics 1 The points S , T ,...

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Unformatted text preview: Section A: Pure Mathematics 1 The points S , T , U and V have coordinates ( s, ms ), ( t, mt ), ( u, nu ) and ( v, nv ), respectively. The lines SV and UT meet the line y = 0 at the points with coordinates ( p, 0) and ( q, 0), respectively. Show that p = ( m- n ) sv ms- nv , and write down a similar expression for q . Given that S and T lie on the circle x 2 + ( y- c ) 2 = r 2 , find a quadratic equation satisfied by s and by t , and hence determine st and s + t in terms of m , c and r . Given that S , T , U and V lie on the above circle, show that p + q = 0. 2 (i) Let y = ∞ X n =0 a n x n , where the coefficients a n are independent of x and are such that this series and all others in this question converge. Show that y = ∞ X n =1 na n x n- 1 , and write down a similar expression for y . Write out explicitly each of the three series as far as the term containing a 3 . (ii) It is given that y satisfies the differential equation xy- y + 4 x 3 y = 0 . By substituting the series of part (i) into the differential equation and comparing coef- ficients, show that a 1 = 0. Show that, for n > 4, a n =- 4 n ( n- 2) a n- 4 , and that, if a = 1 and a 2 = 0, then y = cos( x 2 ) . Find the corresponding result when a = 0 and a 2 = 1. 3 The function f( t ) is defined, for t = 0, by f( t ) = t e t- 1 . (i) By expanding e t , show that lim t → f( t ) = 1 . Find f ( t ) and evaluate lim t → f ( t ) . (ii) Show that f( t ) + 1 2 t is an even function. [ Note: A function g( t ) is said to be even if g( t ) ≡ g(- t ).] (iii) Show with the aid of a sketch that e t (1- t ) 6 1 and deduce that f ( t ) = 0 for t = 0. Sketch the graph of f( t )....
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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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STEP III - Section A: Pure Mathematics 1 The points S , T ,...

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