ST. PAUL’S COLLEGE
F.6 Internal Examination 2015 - 2016
Mathematics - Extended Part
Module 2 (Algebra and Calculus) Section A
A
Name:
Class:
Class Number:
Group:
Score of Section A:
/ 50
Time allowed : 2 hours 30 minutes
INSTRUCTIONS
1. This paper consists of Section A and Section B. Each section carries 50 marks.
2. Answer ALL questions in this paper.
3. Write your answers in the spaces provided in the corresponding Question-Answer
Book.
4. Graph paper and supplementary answer sheets will be supplied on request. Write
your name, class, class number and group number on each sheet and staple them
INSIDE the corresponding Question-Answer Book.
5. Unless otherwise specified, all working must be clearly shown.
6. Unless otherwise specified, numerical answers should be exact.
7. The diagrams in this paper are not necessarily drawn to scale.
8. Marks will be deducted for untidiness and poor presentation.
FORMULAS FOR REFERENCE
sin (A  B) = sin A cos B  cos A sin B
cos (A  B) = cos A cos B  sin A sin B
tan (A  B) =
tan A  tan B
1  tan A tan B
2 sin A cos B = sin (A + B) + sin (A − B)
A+ B
A− B
cos
2
2
A+ B
A− B
sin A − sin B = 2 cos
sin
2
2
A+ B
A− B
cos A + cos B = 2 cos
cos
2
2
A+ B
A− B
cos A − cos B = −2 sin
sin
2
2
sin A + sin B = 2 sin
2 cos A cos B = cos (A + B) + cos (A − B)
2 sin A sin B = cos (A − B) − cos (A + B)
SECTION A (50 marks)
1.
2
3
2𝑛−1
Show, by mathematical induction, that ∑𝑛𝑘=3 𝑘(𝑘−2) = 2 − 𝑛(𝑛−1) for any positive
integer n ≥3.
(4 marks)
2. (a) Express (√3 − 1)8 as 𝐴√3 + 𝐵 where A and B are integers.
1
(b) Rationalize (√3−1)8 .
(5 marks)
Answers written in the margins will not be marked.
Page total
F.6 Internal Exam (2015-2016)
Page 2 of 12
Mathematics Module 2
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
*******************************************************
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
F.6 Internal Exam (2015-2016)
Page 3 of 12
Mathematics Module 2
Page total
3.
1
It is given that lim (1 + )𝑛 = e.
𝑛→∞
1
𝑛
𝑛
1
4
Show that lim (1 − 𝑛−1) = 𝑒 and hence find lim (1 − 𝑛−1)𝑛 .
𝑛→∞
𝑛→∞
(5 marks)
4. (a) It is given that tan 𝑥 tan 𝑦 = 𝑘, where k is a constant.
(i)
Show that cos(𝑥 + 𝑦) = (1 − 𝑘) cos 𝑥 cos 𝑦.
(ii)
Hence show that (1 − 𝑘) cos(𝑥 − 𝑦) = (1 + 𝑘) cos(𝑥 + 𝑦).
(b) If θ is a solution of tan(𝜃 + 40°) tan(𝜃 − 20°) = 𝑘,
1−𝑘
show that cos(2𝜃 + 20°) = 2(1+𝑘).
Hence, find a solution to the equation tan(𝜃 + 40°) tan(𝜃 − 20°) = 2.
(Give your answer correct to the nearest 0.1 degree.)
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
(6 marks)
Answers written in the margins will not be marked.
Page total
F.6 Internal Exam (2015-2016)
Page 4 of 12
Mathematics Module 2
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
F.6 Internal Exam (2015-2016)
Page 5 of 12
Mathematics Module 2
Page total
5. A ladder AB is laid on the floor of a ╗- shape corridor.
The ladder makes an angle θ with one of the walls
of the corridor as shown in the figure. (0° < 𝜃 < 90°)
A

125 cm
216 cm
(a) Let the length of the ladder AB be L cm,
express L in terms of θ.
B
(b) Find the shortest possible length of AB and the
corresponding value of θ.
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
(7 marks)
Answers written in the margins will not be marked.
Page total
F.6 Internal Exam (2015-2016)
Page 6 of 12
Mathematics Module 2
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
F.6 Internal Exam (2015-2016)
Page 7 of 12
Mathematics Module 2
Page total
6.
Let 𝑓(𝑥) = 𝑥 √3𝑥 + 2.
(a) Find 𝑓′(𝑥).
(b) Hence, evaluate ∫
(ln 𝑥)(9𝑥+4)
√3𝑥+2
d𝑥.
(6 marks)
𝑎
𝑎
7. (a) If ∫0 𝑓(𝑥) d𝑥 = ∫0 𝑓(𝑎 − 𝑥) d𝑥,
𝑎
𝑎
𝑎
show that 2 ∫0 𝑓(𝑥) d𝑥 = ∫0 𝑓(𝑥) d𝑥 + ∫0 𝑓(𝑎 − 𝑥) d𝑥.
3
(b) Hence, evaluate ∫0
𝑥3
𝑥 3 −(𝑥−3)3
d𝑥.
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
(6 marks)
Answers written in the margins will not be marked.
Page total
F.6 Internal Exam (2015-2016)
Page 8 of 12
Mathematics Module 2
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
F.6 Internal Exam (2015-2016)
Page 9 of 12
Mathematics Module 2
Page total
𝑥1 2
8. (a) Let 𝐴 = (𝑥2 2
𝑥3 2
𝑥1
𝑥2
𝑥3
1
1), where 𝑥1 , 𝑥2 and 𝑥3 are real numbers.
1
(i) Factorize det(A).
(ii) Hence show that det(A)≠ 0 if 𝑥1 , 𝑥2 and 𝑥3 are distinct.
(iii) Find the inverse of A if 𝑥1 = −2, 𝑥2 = 0 and 𝑥3 = 2.
(b) Let 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 be a quadratic curve. If (−2,1), (0,2) and (2,2) are points
on the curve, find a, b and c.
(7 marks)
9.
Given that A, P and B are points with position vectors ⃑⃑⃑⃑⃑
𝑂𝐴 = 4𝒊 + 𝒋,
(a) Show that A, P and B are collinear.
(b) Find the ratio 𝐴𝑃: 𝑃𝐵.
(4 marks)
Answers written in the margins will not be marked.
Page total
F.6 Internal Exam (2015-2016)
Page 10 of 12
Mathematics Module 2
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
⃑⃑⃑⃑⃑ = 8𝒊 + 9𝒋 and 𝑂𝑃
⃑⃑⃑⃑⃑ = 13 𝒊 + 6𝒋 respectively.
𝑂𝐵
2
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
F.6 Internal Exam (2015-2016)
Page 11 of 12
Mathematics Module 2
Page total
Answers written in the margins will not be marked.
Answers written in the margins will not be marked.
End of Section A
Answers written in the margins will not be marked.
Page total
F.6 Internal Exam (2015-2016)
Page 12 of 12
Mathematics Module 2