Australian
National
University
Student Number:
u
Mathematical Sciences Institute
EXAMINATION: Semester 1 — Mid Exam, 2022
MATH1014: Mathematics and Applications 2, Semester 1
Exam Duration: 120 minutes.
Reading Time: 15 minutes.
Materials Permitted In The Exam Venue:
• Unmarked English-to-foreign-language dictionary (no approval required).
• One A4 page with hand written notes on both sides.
Materials To Be Supplied To Students:
• None
Instructions To Students:
• You must justify your answers and show your work. Please be neat.
Q1
10
Q2
10
Q3
10
Q4
10
Q5
10
Q6
20
Q7
10
Total / 80
Question 1 10 marks
(a) Determine the angle between the following vectors in R3 :
v1 =

2
1
3
 , v2 =

4
1
−3
 . (1)
(5 marks)
(b) Consider the plane inside R3 with normal vector n =

1
1
1
 which passes through the
origin, and the line L() =

−2
0
−4
 +

3
1
2
 , where ∈ R . Find the point of intersection,
or determine that they do not intersect. (5 marks)
Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2,
Page 2 of 8
Question 2 10 marks
Recall that the space 2×2(R) of 2 × 2 matrices with real number entries forms a vector
space. Determine whether or not the set
=
{[
1 0
0 1
]
,
[
1 0
0 −1
]
,
[
0 1
0 0
]
,
[
0 0
−1 0
]}
(2)
is a basis for 2×2(R) .
Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2,
Page 3 of 8
Question 3 10 marks
The sets 1 = {1, , 2} and 2 = {1 + , 1 − , 2} are both bases for the vector space P2
of polynomials with degree at most 2 . Construct the change of coordinates map
2←1
from the basis 1 to the basis 2 , and use it to write the coordinate vector [p()]2 , where
p() = 32 + 2 − 4 .
Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2,
Page 4 of 8
Question 4 10 marks
Consider the following parameter-dependent 3 × 3 matrix, where the parameter ∈ R :
() =

2 8 4
1 2
0 0 3
 . (3)
(a) Determine the rank of the matrix . (6 marks)
(b) Determine the dimension of the null space of , i.e. calculate dimnul() . (2 marks)
(c) Is the matrix invertible? Justify your answer. (2 marks)
Your answers to the above questions may depend on the value of the parameter .
Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2,
Page 5 of 8
Question 5 10 marks
Determine whether the following series converge or diverge.
(a)
∞∑︁
=1
1√
3 + 1
, (5 marks)
(b)
∞∑︁
=1
(!)25
(2)! (5 marks)
Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2,
Page 6 of 8
Question 6 20 marks
Consider the power series
− ln 2 +
∞∑︁
=1

2
Answer the following questions:
(a) (i) Find the domain of convergence. (4 marks)
(ii) For which does the series converge conditionally? (2 marks)
(b) For the values of determined in part (a), define
() = − ln 2 +
∞∑︁
=1

2 .
Check that ′() = 12 − . (7 marks)
(c) Apply the fundamental theorem of calculus
() − (0) =
∫
0
′()
to check that ln 23 =
∞∑︁
=1
(−1)
2 . (7 marks)
Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2,
Page 7 of 8
Question 7 10 marks
Sketch the polar curves = 2 cos and = 2 sin and find the area of the region where the
interiors of the two polar curves overlap.
Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2,
Page 8 of 8