2004 Pure Maths Paper 2

Formulas For Reference

Section A

  1. Evaluate

(a),

(b).

(7 marks)

  1. Let

If f is differentiable at 1, find a and b.

(6 marks)

  1. (a) Evaluate .

(b)Consider the curve , . Find the length of C.

(7 marks)

  1. Using the substitution , prove that .

Hence, or otherwise, evaluate .

(6 marks)

  1. (a) For each non-negative integer n, define .

(i)Evaluate .

(ii)Prove that if is convergent, then .

(iii)Express in terms of n.

(b)For each non-negative integer n, define dx.

Find a non-negative integer m such that for all n.

(8 marks)

  1. Consider the two planes and .

(a)Find a parametric equation of the line of intersection of and .

(b)Find the equation(s) of the plane(s) containing all the points which are equidistant from and .

(6 marks)

Section B

  1. Let .

(a)(i) Find and for x > 0.

(ii) Write down and for x < 0.

(iii) Prove that exists.

(iv) Does exist? Explain your answer.

(5 marks)

(b)Determine the range of values of x for each of the following cases:

(i)> 0,

(ii)< 0,

(iii)> 0,

(iv)< 0.

(3 marks)

(c)Find the relative extreme point(s) and point(s) of inflexion of f(x). (2 marks)

(d)Find the asymptote(s) of the graph of f(x). (3 marks)

(e)Sketch the graph of f(x). (2 marks)

  1. (a) For any non-negative integers m and n, define

for all R.

Prove that . (6 marks)

(b)Evaluate . (4 marks)

(c)Evaluate .

  1. Consider the ellipse , where a and b are two positive constants with ab. Let P be the point , where .

(a)Prove that P lies on E. (1 mark)

(b)Let L be the tangent to E at P. L cuts the x-axis and the y-axis at and respectively. Find

(i)the equation of L,

(ii)the coordinates of and . (4 marks)

(c)Consider the two circles and . Also consider the two points and described in (b). For k = 1, 2, let be the tangent to from , with the point of contact lying in the first quadrant.

(i)Prove that is parallel to .

(ii)Find the coordinates of and .

(iii)Let l be the straight line passing through and . Is l a common normal to and ? Explain your answer. (10 marks)

  1. (a) (i) Evaluate .

(ii) Prove that .

(iii) Using (a)(ii), deduce that . (6 marks)

(b) (i) Let k be a non-negative integer. Prove that for all real numbers x.

(ii) Using (b)(i) and (a)(iii), or otherwise, prove that .

(9 marks)

  1. For any real number x, let [ x ] denote the greatest integer not greater than x. Let f : R→R be defined by

(a) (i) Prove that f is a periodic function with period 1.

(ii) Sketch the graph of f(x), where .

(iii) Write down all the real number(s) x at which f is discontinuous.

(6 marks)

(b) Define for all real numbers x.

(i) If , prove that .

(ii) Is F a periodic function? Explain your answer.

(iii) Evaluate . (9 marks)

  1. (a) Let f : R→R be a twice differentiable function. Assume that a and b are

two distinct real numbers.

(i) Find a constant k (independent of x) such that the function satisfies h(a) = 0.

Also find h(b).

(ii) Let I be the open interval with end points a and b. Using Mean Value Theorem and (a)(i), prove that there exists a real number such that . (7 marks)

(b) Let g : R→R be a twice differentiable function. Assume that there exists a real number such that for all .

(i) Using (a)(ii), prove that there exists a real number such that .

(ii) If for all , prove that g(0) + g(1). (8 marks)