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2002_Paper III

# 2002_Paper III - STEP III 2002 Section A 1 2 Pure...

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STEP III, 2002 2 Section A: Pure Mathematics 1 Find the area of the region between the curve y = ln x x and the x -axis, for 1 6 x 6 a . What happens to this area as a tends to inﬁnity? Find the volume of the solid obtained when the region between the curve y = ln x x and the x -axis, for 1 6 x 6 a , is rotated through 2 π radians about the x -axis. What happens to this volume as a tends to inﬁnity? 2 Prove that arctan a + arctan b = arctan ± a + b 1 - ab ² when 0 < a < 1 and 0 < b < 1 . Prove by induction that, for n > 1 , n X r =1 arctan ± 1 r 2 + r + 1 ² = arctan ± n n + 2 ² and hence ﬁnd X r =1 arctan ± 1 r 2 + r + 1 ² . Hence prove that X r =1 arctan ± 1 r 2 - r + 1 ² = π 2 . 3 Let f ( x ) = a x - x - b , where x > b > 0 and a > 1 . Sketch the graph of f ( x ) . Hence show that the equation f ( x ) = c , where c > 0, has no solution when c 2 < b ( a 2 - 1 ) . Find conditions on c 2 in terms of a and b for the equation to have exactly one or exactly two solutions. Solve the equations (i) 3 x - x - 2 = 4 and (ii) 3 x - x - 3 = 5 .

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STEP III, 2002 3 4 Show that if x and y are positive and x 3 + x 2 = y 3 - y 2 then x < y . Show further that if 0 < x 6 y - 1, then x 3 + x 2 < y 3 - y 2 . Prove that there does not exist a pair of positive integers such that the diﬀerence of their cubes is equal to the sum of their squares. Find all the pairs of integers such that the diﬀerence of their cubes is equal to the sum of their squares. 5 Give a condition that must be satisﬁed by p , q and r for it to be possible to write the quadratic polynomial px 2 + qx + r in the form p ( x + h ) 2 , for some h . Obtain an equation, which you need not simplify, that must be satisﬁed by t if it is possible to write ( x 2 + 1 2 bx + t ) 2 - ( x 4 + bx 3 + cx 2 + dx + e ) in the form k ( x + h ) 2 , for some k and h .
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2002_Paper III - STEP III 2002 Section A 1 2 Pure...

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