CITY UNIVERSITY OF HONG KONG
Course code & title :
GE1359 / MA1502 Algebra
Session
:
Semester A, 2020-21
Time allowed
:
Two hours
This paper has FOUR pages (including this cover page).
The trigonometric identities are provided on page 4.
Instructions to candidates:
1.
This paper consists of SIX questions.
2.
Answer ALL questions.
3.
Start each question on a new page.
4.
Show ALL steps clearly.
This is a closed-book examination.
Candidates are allowed to use the following materials/aids:
Approved calculators
Materials/aids other than those stated above are not permitted. Candidates will be
subject to disciplinary action if any unauthorized materials or aids are found on them.
NOT TO BE TAKEN AWAY
- 2 -
Question 1
Consider the function
𝑓(𝑥) = 2 + √
𝑥−1
.
𝑥+3
Find the largest possible domain and largest possible range of 𝑓(𝑥).
[11 marks]
Question 2
3
5
𝜋
(a) Given that cos 𝛼 = 5 and cos 𝛽 = − 13, where − 2 < 𝛼 < 0 and
sec(𝛼 − 𝛽), tan(𝛼 − 𝛽), and the quadrant containing 𝛼 − 𝛽.
(b)
𝜋
2
< 𝛽 < 𝜋. Find
[11 marks]
Find the general solution (in radians) of the equation
sin(5𝑥) + sin 𝑥 = cos(2𝑥).
[11 marks]
Question 3
(a)
Given that 3 − √2 is a root of the equation 𝑥 4 − 4𝑥 3 − 7𝑥 2 + 26𝑥 − 14 = 0, find all
the other roots.
(b)
[7 marks]
Consider the following system of linear equations.
𝑥 − 𝑦 + 2𝑧 = 1
{ −2𝑥 + 𝑦 + 𝑎𝑧 = 3
𝑥 + 𝑎𝑦 − 2𝑎𝑧 = 𝑎2
where 𝑎 is a real number.
Using Gaussian elimination, find all possible values of 𝑎 such that the system has
(i)
no solution;
(ii)
a unique solution;
(iii)
infinitely many solutions.
[10 marks]
- 3 -
Question 4
If 𝛼, 𝛽, 𝛾 are the roots of the equation 𝑥 3 + 5𝑥 2 + 𝑎𝑥 + 𝑏 = 0, where 𝑎 and 𝑏 are real
numbers, find a cubic equation (in terms of 𝑎 and 𝑏) whose roots are 𝛼 2 , 𝛽 2 , 𝛾 2 .
[11 marks]
Question 5
(a)
Prove by mathematical induction that 7𝑛 − 2𝑛 is divisible by 5 for all positive integers
𝑛.
(b)
[9 marks]
A sequence {𝑎𝑛 } is defined by 𝑎1 = 1 , 𝑎2 = 3 and 𝑎𝑛+2 = 2𝑎𝑛+1 + 𝑎𝑛 for
𝑛 = 1, 2, 3, ….
Prove by mathematical induction that
𝑛
(1 + √2) + (1 − √2)
𝑎𝑛 =
2
for 𝑛 = 1, 2, 3, ….
𝑛
[12 marks]
Question 6
(a) How many distinguishable permutations can be obtained from the letters of the word
MISSISSIPPI?
[4 marks]
(b) Use the binomial theorem to evaluate (0.99)7 correctly to six decimal places. [5 marks]
(c) How many different arrangements of six-digit numbers can be obtained from the numbers
1, 2, 3, 4, 5 and 6 such that the first digit is 1, 2 or 3, and the last digit is 3, 4, 5 or 6?
Assume that no digit can be used more than once.
- END -
[9 marks]
- 4 -
Trigonometric identities - Formula sheet
➢
Compound Angle Formulae
sin(𝐴 + 𝐵) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 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
tan(𝐴 − 𝐵) =
tan 𝐴 − tan 𝐵
1 + tan 𝐴 tan 𝐵
Double Angle Formulae
sin 2𝐴 = 2 sin 𝐴 cos 𝐴
cos 2𝐴 = cos 2 𝐴 − sin2 𝐴
➢
Half Angle Formulae
cos 2 𝐴 =
➢
1
(1 + cos 2𝐴),
2
1
sin2 𝐴 = (1 − cos 2𝐴)
2
Product-to-Sum Formulae
1
sin 𝐴 cos 𝐵 = [sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)]
2
1
cos 𝐴 sin 𝐵 = [sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵)]
2
1
cos 𝐴 cos 𝐵 = [cos(𝐴 + 𝐵) + cos(𝐴 − 𝐵)]
2
1
sin 𝐴 sin 𝐵 = − [cos(𝐴 + 𝐵) − cos(𝐴 − 𝐵)]
2
➢
Sum-to-Product Formulae
𝑥+𝑦
𝑥−𝑦
) cos (
sin 𝑥 + sin 𝑦 = 2 sin (
)
2
2
sin 𝑥 − sin 𝑦 = 2 cos (
𝑥+𝑦
𝑥−𝑦
) sin (
)
2
2
𝑥+𝑦
𝑥−𝑦
) cos (
cos 𝑥 + cos 𝑦 = 2 cos (
)
2
2
cos 𝑥 − cos 𝑦 = −2 sin (
𝑥+𝑦
𝑥−𝑦
) sin (
)
2
2