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

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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 0 = 1 and a 2 = 0, then y = cos( x 2 ) . Find the corresponding result when a 0 = 0 and a 2 = 1.
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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 0 f( t ) = 1 . Find f ( t ) and evaluate lim t 0 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 ). 4 For any given (suitable) function f, the Laplace transform of f is the function F defined by F( s ) = Z 0 e - st f( t )d t ( s > 0) . (i) Show that the Laplace transform of e - bt f( t ), where b > 0, is F( s + b ).
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