Yearly Examination, 2010-2011

QUEEN’S COLLEGE

Yearly Examination, 2010-2011

Pure Mathematics

Secondary 6E, S Date : 21/ 6 / 2011 Time : 8:30 - 11:30 am

Instructions :

(1) Answer ALL questions in Section A and Section B.

(2) All workings must be clearly shown.

(3) Unless otherwise specified, numerical answers must be exact.

(4) This paper consists of 4 pages with 200 marks.

FORMULA FOR REFERENCE

Section A : Answer ALL questions in this section. (80 marks)

1. (a) Show that (a + b) is a root of the equation .

(b) Hence find a real root of the equation . Write your answer in surd form.

(10 marks)

2. (a) Let be two square matrices and ,

find and .

(b) Use (a) to evaluate . (12 marks)

3. (a) Find the value of k such that the system

has non-trivial solution.

(b) Determine the ratio x : y : z if the value of k obtained in (a) is an integer.

(12 marks)

4. (a) Write in partial fractions.

(b) Hence evaluate .

(12 marks)

5. Factorize .

(10 marks)

6. (a) If is a purely imaginary number, find the locus of the point representing z and sketch the locus in an Argand diagram.

(b) Find the complex number z in polar form on the locus in (a) such that is the largest. Write your answer in polar form.

(12 marks)

7. (a) Let .

By evaluating , or otherwise, find

(b) Evaluate the infinite sum :

(12 marks)

Section B : Answer ALL questions in this section. (120 marks)

8. Let and .

(a) (i) Evaluate . (4 marks)

(ii) Hence evaluate . (4 marks)

(b) Using (a) , or otherwise, prove that for all integers n

with the convention that . (8 marks)

(d) Use (b) to evaluate . (7 marks)

9. For k > 1, define , where n = 0, 1, 2, ….

(a) Show that , where n ³ 2. (11 marks)

(b) Let , find cos q in terms of t. Then find I0 and hence I1. (11 marks)

10. (a) If , show that . (4 marks)

Given …. (*)

where are constants , and k is a real constant, k > 0 and k ¹ 1.

(b) Rewrite the given equation (*) in the form ,

where A, B are functions of a, b and k. (10 marks)

(Hint: rewrite equation in (b) in complex center-radius form) (10 marks)

(d) Find the center and radius of the circle (6 marks)

11. Let

(a) If (E1) has a unique solution, find the value(s) of k and find your solution in

terms of k . (10 marks)

(b) If (E1) has infinitely many solutions, find the value(s) of k and express your solution in terms of a parameter t , where t Î R . (5 marks)

Let

(d) If (E2) has a unique solution, find all possible value(s) of k and the corresponding solution.

(10 marks)

< End of Paper >

QUEEN’S COLLEGE

Yearly Examination, 2010-2011

Pure Mathematics

Solution

1. (a) Let f(x) = 1M

f(a + b) = 1M

= 1A

By Factor Theorem, (a + b) is a root of f(x) = 0 . 1M

(b) Let 1M

Then 1M + 1A

Hence one of the solution is . 1A

The equation can be expressed as

By (a) , we obtain a real root . 1A

2. (a) 2A

(b) From (a) , 2M

Hence 2M

3. (a) 2M

(E) has non-trivial solution iff 1M

\ 1A

(b) If k is an integer, then k = 2. 1A

is an dependent set of equations. 2M

Using the first two equations, we get:

4. (a) 1M + 2A

(b) 1M

1M + 1A

Put x = dx = 2M

\ 2A

\ 1A

5. 1M

= 2M

6. (a) Let z = x + yi 1M

Since is purely imaginary number,

x(x – 2) + y(y – 1) = 0 Ù (x – 2)(y – 1) – xy ¹ 0 Ù ( x ¹ 2 Ù y ¹ 0) 1M

x2 + y2 – 2x – y = 0 Ù x + y ¹ 2 Ù ( x ¹ 2 Ù y ¹ 0)

x2 + y2 – 2x – y = 0 Ù (x, y) ¹ (2, 0) Ù (x, y) ¹ (0, 1) (for circle) 1A

The locus is a circle excluding the points z = 2 and z = i. 1A

Centre of the circle = G =

Radius =

Diagram : 2A

(b) Let O(0, 0) .

Let OG produce to meet the circle again in A.

Thus OA is the diameter of the circle. The required z is at point A. 2M

, `Arg z =

\ 2A

7. (a) Let

Then 1A

Hence,

= 2M

, taking v0 = 0. 1A

(b)

1M + 1A

8. (a) (i) 2A

\ …. (1)

(ii) Multiply (1) by , we get

…. (2) 1A

(b) Let P(n) be the proposition , Use M.I. 1M

For P(0) ,

\ P(0) is true . 1A

Assume P(k) is true for some integer k,

i.e. …. (3) 1M

For P(k +1), , by (3)

, by (1) 1M

\ P(k +1) is true .

For P(k – 1), , by (3)

, by (2) 1M

\ P(k – 1) is true .

By the Principle of Mathematical Induction, P(n) is true "nÎZ . 1A

\ 1A

\ is singular . 1A

, by (b)

\ is singular . 1A

(d) 2A

9. (a) For n ³ 2,

2M + 1A

(b) 1A

Let . 2A

1M + 1A

10. (a) Let z = a + bi, 2M + 2A

(b) 2M

1M+1A

(d) a = i b = –i , k = 3 2A

Centre = and radius = 2A + 2A

11. (a) Applying row operations to the augmented matrix, we obtain:

When k ¹ –1, 3 , the system has a unique solution. We have: 2A

(b) When k = –1 , the system of equation has infinity many solutions. The system becomes: 2A

The solution is given by 2A

The last equation shows that the original system is inconsistent, and has no solution. 1M

(d) The first 3 equations of

is the same as (E1) . 2M

(i) When k ¹ –1, 3 (E1) has unique solution, 1A

substitute the result of (a) in the last equation of (E2): x – y + 8z =0 , 1M

\ k =1 1A

For this value of k , x = 6 , y = –2, z = –1. 1A

(ii) When k = –1 , (E1) has infinitely many solution, 1A

substitute the result of (b) in the last equation of (E2): x –y + 8z =0 1M

(– 2t – 2) – (t – 1) + 8t = 0 1A

and only this value of t satisfies all equations in (E2) and therefore (E2) has unique solution.