# STEP I - Section A 1 Pure Mathematics A proper factor of an...

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Section A: Pure Mathematics 1 A proper factor of an integer N is a positive integer, not 1 or N , that divides N . (i) Show that 3 2 × 5 3 has exactly 10 proper factors. Determine how many other integers of the form 3 m × 5 n (where m and n are integers) have exactly 10 proper factors. (ii) Let N be the smallest positive integer that has exactly 426 proper factors. Determine N , giving your answer in terms of its prime factors. 2 A curve has the equation y 3 = x 3 + a 3 + b 3 , where a and b are positive constants. Show that the tangent to the curve at the point ( - a, b ) is b 2 y - a 2 x = a 3 + b 3 . In the case a = 1 and b = 2, show that the x -coordinates of the points where the tangent meets the curve satisfy 7 x 3 - 3 x 2 - 27 x - 17 = 0 . Hence ﬁnd positive integers p , q , r and s such that p 3 = q 3 + r 3 + s 3 . 3 (i) By considering the equation x 2 + x - a = 0 , show that the equation x = ( a - x ) 1 2 has one real solution when a > 0 and no real solutions when a < 0 . Find the number of distinct real solutions of the equation x = ( (1 + a ) x - a ) 1 3 in the cases that arise according to the value of a . (ii) Find the number of distinct real solutions of the equation x = ( b + x ) 1 2 in the cases that arise according to the value of b .

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4 The sides of a triangle have lengths p - q , p and p + q , where p > q > 0 . The largest and smallest angles of the triangle are α and β , respectively. Show by means of the cosine rule that 4(1 - cos α )(1 - cos β ) = cos α + cos β . In the case α = 2 β , show that cos β = 3 4 and hence ﬁnd the ratio of the lengths of the sides of the triangle.
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STEP I - Section A 1 Pure Mathematics A proper factor of an...

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