F
F.6 Pure Mathematics Quiz (Binomial Theorem, Determinants, Matrix)
Time allowed: 70 minutes Total score: 40 Attempt ALL questions
1. (a) Let A = [pic]. Determine A(1.
(b) Hence, or otherwise, solve [pic].
(7 marks)
2. (a) Determine [pic]. Show that [pic].
(b) Determine [pic].
(8 marks)
3. Suppose A is a square matrix and I is the identity matrix with the same order as that of A.
Given that A3 = I, and det(A ( I) ( 0. Find det(I + A(I + A)).
(4 marks)
4. (a) Prove that [pic].
(b) Hence, or otherwise, factorize [pic].
(8 marks)
5. Given that, for n ( [pic], [pic] for a ( 1.
Evaluate [pic]. [You may express the answer in the form of nCr.]
(4 marks)
6. For n ( [pic], show that
(a) [pic]; (b) [pic].
(9 marks)
END OF PAPER
Marking Scheme of F.6 Pure Mathematics Quiz
(Binomial Theorem, Determinant, Matrix)
1.
(a) [pic] 1A for determinant + 1A
(b) [pic] [pic] 1M
[pic] 1M
[pic] 1M
[pic] i.e. x = [pic], y = ([pic] and z = 0 2A
2.
(a) [pic]= [pic] 1A
(b) [pic]
[pic] 1A for[pic]
( [pic]
(c) By (b), [pic]
[pic] 1M
[pic] 1M
[pic] 2M
= [pic] = [pic] 2A
3. A3 = I => I ( A3 = 0
(I ( A)(1 + A + A2) = 0 1M
det(I ( A)(1 + A + A2) = det0 1M
det(I ( A)det(1 + A + A2) = 0 1M
det(1 + A + A2) = 0 (∵ det(I ( A) ( 0) 1
4.
(a) [pic] 1M
= [pic] 1M
= [pic] 1M
= [pic] = [pic] 1
(b) [pic]
= [pic] (2xyzR1 + R3 ( R3) 1M
= [pic] 1A
= [pic] [by (a)] 1M
= [pic]
= [pic] 1A
5. [pic] 1M
= [pic] = [pic] 1A
By comparing coefficient of x10 of expansions in both sides, we have 1M
[pic] = [pic] 1A
6.
(a) [pic]
Differentiate w.r.t. x, [pic] 1M
[pic] 1M
Differentiate w.r.t. x, [pic] - - - (*) 1M
Put x = 0
[pic] 1M
[pic] 1
(b) By (*),[pic]
[pic]
[pic]
[pic] 1M
[pic]
[pic] 1M
By comparing coefficient of xn ( 1,
[pic] 1M
( [pic] 1
END
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