Quod Erat Demonstrandum



Pure Math Paper I Pre-mock Examination (2010-01-25)

SECTION A (40 marks)

Answer ALL questions in this section.

1. (a) Resolve [pic] into partial fractions.

(b) Express [pic]in the form of [pic], where A, B, C and D

are constants.

(c) Evaluate [pic].

(7 marks)

2. (a) Prove that [pic].

(b) Suppose |x| < 1. Find the sum of the following expressions:

(i) [pic];

(ii) [pic].

[Hint to (ii): you may consider [pic]for |x| < 1.]

(7 marks)

3. (a) Let a, b and c be three numbers such that a + b + c = 0. Prove that [pic].

(b) Using (a), solve the equation

[pic]

(6 marks)

4. Let T be the rotation in the Cartesian plane anticlockwise about the origin by an angle θ, where

0 < θ < 2(. It is given that P1, P2, P3, … are the points in the Cartesian plane, where P1 = ((5,12),

P2 = ((12,(5) and T transforms Pk to Pk + 1 for each positive integer k.

(a) Find θ.

(b) Let A be the matrix representing T . Find A2007.

(c) Write down the coordinates of Pn for all positive integers n.

(7 marks)

5. (a) Let n ( [pic]; ( , ( ( [pic].

By considering [pic], or otherwise, show that

[pic].

(b) Evaluate [pic].

(7 marks)

6. Let S = [pic], where a1, a2, …, an are positive real numbers.

(a) Let G = [pic]. Using A.M. ( G.M., prove that

(i) [pic],

(ii) [pic].

(b) Using (a), or otherwise, prove that [pic].

(6 marks)

SECTION B (60 marks)

Answer any FOUR questions in this section. Each question carries 15 marks.

7. (a) Consider the system of linear equations in x, y, z

(E): [pic], where a, b ( [pic].

(i) Find the range of values of a for which (E) has a unique solution. Solve (E) when (E) has a

unique solution.

(ii) Suppose that a = (2. Find the value(s) of b for which (E) is consistent, and solve (E) for such

value(s) of b.

(8 marks)

(b) Is the system of linear equations

[pic]

consistent? Explain your answer.

(3 marks)

(c) Solve the system of linear equations

[pic]

(4 marks)

8. Let M = [pic].

(a) Suppose that [pic], where r > 0 and 0 < θ < 2( . Find r and θ.

(2 marks)

(b) A real matrix of the form [pic] is called a 2×2 diagonal matrix. Find all positive integers

integers n such that Mn is a 2×2 diagonal matrix, and evaluate Mn for such values of n.

(3 marks)

(c) Let A be a 2×2 real matrix such that AM = MA . Prove that

[pic].

(2 marks)

(d) Denote the 2×2 identity matrix by I. Is 2I ( M a singular matrix? Explain your answer.

(2 marks)

(e) Evaluate [pic].

(6 marks)

9. (a) By differentiating f(x) = xlnx ( x , prove that xlnx ( x + 1 ( 0 for all x > 0.

(4 marks)

(b) Let a be a positive real number. Define g(x) = [pic] for all x > 0. Prove that g is increasing.

(3 marks)

(c) Let p and q be real numbers such that p > q > 0.

(i) Suppose that a1, a2, …, an are positive numbers satisfying [pic].

Using (b), prove that [pic].

(ii) Suppose that b1, b2, …, bn are positive numbers.

Using (c)(i), prove that [pic].

Hence prove that [pic].

(8 marks)

10. Let p(x) be a polynomial of degree k with real coefficients satisfying the following conditions:

(1) p(x) = p(x ( 1) + x100 for all x ( [pic];

(2) p(1) = 1.

(a) Find k and the coefficient of xk in p(x).

(3 marks)

(b) Find p(0) and p((1).

(2 marks)

(c) Prove that p(x) + p((x (1) = p(x ( 1) + p((x) for all x ( [pic].

Hence, prove that p(n) + p((n (1) = 0 for all n ( [pic].

(4 marks)

(d) Prove that p(x) + p((x (1) = 0 for all x ( [pic].

(3 marks)

(e) Prove that p(x) is divisible by x(x + 1)(2x + 1).

(3 marks)

11. Consider the sequence {an}, defined as a1 = 1 and [pic].

(a) (i) Show that {an} is decreasing.

(ii) Show that [pic].

(iii) Using the result in (a)(ii) and Squeezing Principle, show that [pic].

(4 marks)

(b) Given that [pic] and [pic]. Show that {un} is convergent.

(3 marks)

(c) Given that [pic] and [pic].

Let [pic].

(i) Show that [pic] exists. (ii) Show that [pic] exists.

(iii) Given an example of bn such that [pic] exists but [pic] does not exist.

(iv) Using the following theorem: “If [pic], then [pic] exists.”,

show that [pic] exists.

(8 marks)

END OF PAPER I

Pure Math Paper II Pre-mock Examination (2010-01-26)

SECTION A (40 marks)

Answer ALL questions in this section.

1. Evaluate (a) [pic]. (b) [pic].

(6 marks)

2. The function Fn : [0, 1] [pic] R is defined as [pic].

(a) Sketch the graph of [pic].

(b) If x is fixed and [pic], show that [pic].

(c) Let An be the area enclosed by [pic] and the axes.

A student claims, “As [pic], [pic]”.

Do you agree? Explain your answer.

(6 marks)

3. Let f : [pic] ( [pic] be defined as f(x) = | x − 1 | − | x + 1 |.

(a) Sketch the graph of y = f(x).

(b) Is f a surjective function? Explain your answer.

(c) Let g : [pic] ( [pic] be defined as g(x) = f(x − 1) − f(x + 1) + 1.

(i) Prove that g(x) is an even function.

(ii) Sketch the graph of g(x).

(7 marks)

4. (a) (i) Find [pic].

(ii) Using the result of (a)(i), or otherwise, find [pic].

(b) Evaluate [pic].

(7 marks)

5. (a) Find (i) [pic], (ii) [pic].

(b) The figure below shows the curve

[pic], where [pic].

[pic]

The region bounded by ( and the x-axis is revolved about the x-axis. Find the volume of the

solid of revolution generated.

(7 marks)

6. Let P be a point in the first quadrant at which the ellipse E : [pic] and the straight line

L1 : x = 3 intersect.

(a) Find the coordinates of P.

(b) Let L2 be the straight line passing through Q((3,0) and perpendicular to the tangent to E at P.

(i) Find the equation of L2.

(ii) Let R be the point of intersection of L1 and L2 . Is △PQR isosceles? Explain your answer.

(7 marks)

SECTION B (60 marks)

Answer any FOUR questions in this section. Each question carries 15 marks.

7. Let f(x) = [pic] for x ( 2.

(a) (i) Is f(x) differentiable at x = 0 ? Explain your answer.

(ii) Find f’(x) and f’’(x) for x ( 0.

(4 marks)

(b) Solve each of the following inequalities:

(i) f’(x) > 0,

(ii) f’’(x) > 0.

(2 marks)

(c) Find the relative extreme point(s) and point(s) of inflexion of the graph of y = f(x).

(4 marks)

(d) Find the asymptote(s) of the graph of y = f(x).

(3 marks)

(e) Sketch the graph of y = f(x).

(2 marks)

8. (a) For each non-negative integer n, define [pic].

(i) Verify that [pic].

Hence evaluate I0 .

(ii) Prove that [pic].

Hence prove that [pic].

(iii) Prove that [pic].

Hence, or otherwise, prove that [pic].

(12 marks)

(b) Using (a), evaluate [pic].

(3 marks)

9. (a) Let f : [pic] ( [pic] be a continuous function.

(i) Using integration by substitution, prove that [pic].

(ii) Using (a)(i), prove that if f(x) is an odd function, then [pic].

(3 marks)

(b) Let g : [pic] ( [pic] be a function with derivatives of any order. Suppose that g(−x) + g(x) = 1 for

all x ( [pic].

(i) Prove that [pic].

(ii) Prove that [pic] is an even function and [pic]is an odd function for any positive

integer n. Hence, prove that [pic] for any positive integer n.

(7 marks)

(c) Let g : [pic] ( [pic] and G : [pic] ( [pic] be defined by g(x) = [pic] and G(x) = x3g(x). Using (b),

or otherwise, evaluate [pic].

(5 marks)

10 Consider the function, [pic] such that [pic].

Consider the sequence [pic], [pic].

(a) (i) Prove that [pic].

(ii) By writing [pic], prove that

[pic].

(5 marks)

(b) Prove that [pic].

(2 marks)

(c) Given that [pic] is well defined for any real x.

(i) Using the result in (a), show that [pic].

(ii) Using the above results and considering [pic],

show that [pic].

(8 marks)

11. The equation of the hyperbola H is [pic], where a, b > 0.

Let P be the point [pic], where [pic].

(a) Using differentiation, find the equation of the tangent at P to H.

(2 marks)

(b) Suppose that the tangent at P to H meets the asymptotes of H at the points

Q and R.

(i) Find the coordinates of Q and R .

(ii) Let C be the circum-centre of triangle OQR where O is the origin.

Find the coordinates of C.

(iii) Find the equation of the locus of C .

(13 marks)

END OF PAPER II

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