KIANGSU-CHEKIANG COLLEGE (SHATIN)
FINAL EXAMINATION 2006-2007
F.6 PURE MATHEMATICS
PAPER 1
Monday, June 25, 2007
8.30 am - 10.45 am (2 hours 15 minutes)
This paper must be answered in English
1. This paper consists of Section A and Section B.
2. Answer ALL questions in Section A and any THREE questions in Section B.
3. Write your answers in the answer book provided.
4. Unless otherwise specified, all working must be clearly shown.
© S.M Fan
All Rights Reserved 2007
FORMULAS FOR REFERENCE
cos (A ± B) = cos A cos B sin A sin B
tan (A ± B) =
sin A + sin B =
sin A - sin B =
cos A + cos B =
cos A - cos B = -
2 sin A cos B = sin (A + B) + sin (A - B)
2 cos A cos B = cos (A+B) + cos (A-B)
2 sin A sin B = cos (A - B) - cos (A + B)
Section A (35 marks)
Answer ALL questions in this section.
Write your answers in the answer book.
HKALE 1986I Q5a
1. Let m and n be two positive integers with m > n.
(a) Show that .
(b) By considering the expression or otherwise, find
where m - n £ k £ m.
(5 marks)
HKALE 2007 I Q3
2. (a) (i) Resolve into partial fractions.
(ii) Using differentiation, or otherwise, resolve into partial fractions.
(b) Evaluate .
(6 marks)
HKALE 2004 I Q3
3. (a) Let R be the matrix representing the rotation in the Cartesian plane anticlockwise about the origin by 60o.
(i) Write down R and R6.
(ii) Let A = . Verify that A-1RA is a matrix in which all elements are integers.
(b) Using the results of (a), or otherwise, find a 2 ´ 2 matrix M, in which all the elements are integers, such that M3 = I but M ¹ I, where I = .
(7 marks)
HKALE 1997 I Q7
4. (a) Let A be a 3 ´ 3 non-singular matrix. Show that
det (A-1 - xI) = -det (A - x -1I).
(b) Let A = .
(i) Show that 4 is a root of det(A - xI) = 0 and hence find the other roots in surd form.
(ii) Solve det(A-1 - xI) = 0
(6 marks)
5. Let f : R¥{1} ® R¥{1} be a function defined by .
(a) Prove that f is a surjective function.
(b) If furthermore, f is an injective function, find
(i) f -1(x);
(ii) .
(6 marks)
HKALE 1990 I Q6
6. Solve the inequality | x - 1 | - | x + 2 | > 2.
(5 marks)
SECTION B (45 marks)
Answer any THREE questions in this section. Each question carries 15 marks.
Write your answers in the AL(A) answer book.
97f6e31 Q10
7. (a) Solve the following system of equations: (I) : .
(2 marks)
(b) Find all possible values of p and q such that the following system of equations is solvable :
(II) : . (7 marks)
(c) Find the solutions, if possible, of the system of equations
(III) : . (6 marks)
Intell Q&A F.6 QA1
8. (a) Prove that for any positive integer n,
(i) ;
(ii) ;
(iii) Hence, or otherwise, deduce that for n ³ 3.
(8 marks)
(b) Using (a) and Mathematical induction, prove that for any positive integer
n ³ 6,
(7 marks)
HKALE 2007 Q10
9. Let a1 and a2 be real numbers and p, q be positive constants such that . For each n = 1, 2, 3, ... , define .
(a) Prove that an+4 - an+2 = (an+2 - an). (3 marks)
(b) Suppose that a2 ³ a1 .
(i) Prove that
(1) a2n+1 ³ a2n-1 ,
(2) a2n+2 £ a2n ,
(3) a2n ³ a2n-1 .
(ii) Prove that and both exist.
(9 marks)
(c) Suppose that a2 < a1. Do and exist? Explain your answer.
(3 marks)
Note: You are not supposed to find an explicitly in doing this question.
HKAL 1982 I Q7
10. Let A = and let v denote a 2´1 matrix.
(a) Find the two real values l1 and l2 of l with l1 < l2 such that the matrix equation
(1) Av = lv
has non-zero solutions. (4 marks)
(b) Let v1 and v2 be non-zero solutions of (1) corresponding to l1 and l2, respectively. Show that if v1 = and v2 = , then the matrix
V = is non-singular.
(4 marks)
(c) (i) Using (a) and (b), show that AV = V .
(ii) Hence or otherwise, evaluate , n Î N.
(7 marks)
THE END
2007-F.6-P MATH 1-6