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Unformatted text preview: STEP III, 2001 2 Section A: Pure Mathematics 1 Given that y = ln ( x + ( x 2 + 1) ) , show that d y d x = 1 ( x 2 + 1) . Prove by induction that, for n > 0 , ( x 2 + 1 ) y ( n +2) + (2 n + 1) xy ( n +1) + n 2 y ( n ) = 0 , where y ( n ) = d n y d x n and y (0) = y . Using this result in the case x = 0 , or otherwise, show that the Maclaurin series for y begins x- x 3 6 + 3 x 5 40 and find the next non-zero term. 2 Show that cosh- 1 x = ln ( x + ( x 2- 1)) . Show that the area of the region defined by the inequalities y 2 > x 2- 8 and x 2 > 25 y 2- 16 is (72 / 5) ln 2. 3 Consider the equation x 2- bx + c = 0 , where b and c are real numbers. (i) Show that the roots of the equation are real and positive if and only if b > 0 and b 2 > 4 c > 0, and sketch the region of the b- c plane in which these conditions hold. (ii) Sketch the region of the b- c plane in which the roots of the equation are real and less than 1 in magnitude. STEP III, 2001 3 4 In this question, the function sin- 1 is defined to have domain- 1 6 x 6 1 and range- 2 6 x 6 2 and the function tan- 1 is defined to have the real numbers as its domain and range- 2 < x < 2 . (i) Let g( x ) = 2 x 1 + x 2 ,- < x < . Sketch the graph of g( x ) and state the range of g. (ii) Let f ( x ) = sin- 1 2 x 1 + x 2 ,- < x < . Show that f( x ) = 2 tan- 1 x for- 1 6 x 6 1 and f( x ) = - 2 tan- 1 x for x > 1 . Sketch the graph of f( x ). 5 Show that the equation x 3 + px + q = 0 has exactly one real solution if p > 0 . A parabola C is given parametrically by x = at 2 , y = 2 at ( a > 0) . Find an equation which must be satisfied by t at points on C at which the normal passes through the point ( h, k ) . Hence show that, if h 6 2 a , exactly one normal to C will pass through ( h, k ) ....
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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