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Unformatted text preview: Section A: Pure Mathematics 1 Let f( x ) = sin 2 x + 2 cos x + 1 for 0 6 x 6 2 . Sketch the curve y = f( x ), giving the coordinates of the stationary points. Now let g( x ) = a f( x ) + b c f( x ) + d ad = bc , d =- 3 c , d = c . Show that the stationary points of y = g( x ) occur at the same values of x as those of y = f( x ), and find the corresponding values of g( x ). Explain why, if d/c <- 3 or d/c > 1, | g( x ) | cannot be arbitrarily large. 2 Let I( a, b ) = Z 1 t a (1- t ) b d t ( a > , b > 0) . (i) Show that I( a, b ) = I( b, a ), (ii) Show that I( a, b ) = I( a + 1 , b ) + I( a, b + 1). (iii) Show that ( a +1)I( a, b ) = b I( a +1 , b- 1) when a and b are positive and hence calculate I( a, b ) when a and b are positive integers. 3 The value V N of a bond after N days is determined by the equation V N +1 = (1 + c ) V N- d ( c > , d > 0) , where c and d are given constants. By looking for solutions of the form V T = Ak T + B for some constants A, B and k , or otherwise, find V N in terms of V . What is the solution for c = 0? Show that this is the limit (for fixed N ) as c 0 of your solution for c > 0. 4 Show that the equation (in plane polar coordinates) r = cos , for- 2 6 6 2 , represents a circle. Sketch the curve r = cos 2 for 0 6 6 2 , and describe the curves r = cos 2 n , where n is an integer. Show that the area enclosed by such a curve is independent of n . Sketch also the curve r = cos 3 for 0 6 6 2 . 1 5 The exponential of a square matrix A is defined to be exp( A ) = X r =0 1 r ! A r , where A = I and I is the identity matrix. Let M =- 1 1 . Show that M 2 =- I and hence express exp( M ) as a single 2 2 matrix, where is a real number. Explain the geometrical significance of exp( M )....
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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.
- Spring '12