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Unformatted text preview: Section A: Pure Mathematics 1 Show that, if n is an integer such that ( n 3) 3 + n 3 = ( n + 3) 3 , ( * ) then n is even and n 2 is a factor of 54. Deduce that there is no integer n which satisfies the equation ( * ). Show that, if n is an integer such that ( n 6) 3 + n 3 = ( n + 6) 3 , ( ** ) then n is even. Deduce that there is no integer n which satisfies the equation ( ** ). 2 Use the first four terms of the binomial expansion of (1 1 / 50) 1 / 2 , writing 1 / 50 = 2 / 100 to simplify the calculation, to derive the approximation 2 1 . 414214. Calculate similarly an approximation to the cube root of 2 to six decimal places by considering (1 + N/ 125) a , where a and N are suitable numbers. [You need not justify the accuracy of your approximations.] 3 Show that the sum S N of the first N terms of the series 1 1 . 2 . 3 + 3 2 . 3 . 4 + 5 3 . 4 . 5 + + 2 n 1 n ( n + 1)( n + 2) + is 1 2 3 2 + 1 N + 1 5 N + 2 . What is the limit of S N as N ? The numbers a n are such that a n a n 1 = ( n 1)(2 n 1) ( n + 2)(2 n 3) . Find an expression for a n /a 1 and hence, or otherwise, evaluate n =1 a n when a 1 = 2 9 . 1 4 The integral I n is defined by I n = Z ( / 2 x ) sin( nx + x/ 2) cosec ( x/ 2) d x, where n is a positive integer. Evaluate I n I n 1 , and hence evaluate I n leaving your answer in the form of a sum. 5 Define the modulus of a complex number z and give the geometric interpretation of  z 1 z 2  for two complex numbers z 1 and z 2 . On the basis of this interpretation establish the inequality  z 1 + z 2  6  z 1  +  z 2  ....
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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
 rotar
 Math

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