Note: Please refer to questions in the final version. A few questions have been revised. Solutions are correct.
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
SECTION A: CALCULUS (50 marks)
1. Answer the following questions. Justify your answers.
(a) (2 marks) For what values of x does the series 1 + 2x + 3x + 4x + · · · + nx + · · · converge?
(b) (4 marks) Which of the following series diverge?
I.
∞
X
n=1
1
2
n +1
II.
∞
X
n=1
n
n+1
∞
X
(−1)n
√
III.
n+1
n=1
(c) (4 marks) Is the following series absolutely convergent, conditionally convergent or
divergent?
arctan((−1)n n)
arctan(−1) arctan(2) arctan(−3)
+
+
+ ... +
+ ...
2
4
8
2n
Page 3 of 13
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
2. (8 marks) Determine the radius and interval of convergence for the series
∞
X
(x − 2)n
n=1
n(−3)n
Page 3 of 12
.
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
√
3. Let f (x) = 1 + x.
(a) (4 marks) Construct the Maclaurin series for f (x) up to and including the term in x3 .
(b) (3 marks) Use (a) to determine the rst three non-zero terms of the Maclaurin series
√
for f (x) = 4 − x2 .
R 2√
4 − x2 dx.
0
R √
----------------------------------------------------------------------------------------(d) (4 marks) Use a trigonometric substitution to calculate 02 4 − x2 dx.
Calculate
R √
(d)-------------------------------------------------------------------(e) (2 marks) Check your answer for (d) by interpreting the denite integral 2 4 − x2 dx
(c) (2 marks) Use (b) to calculate an approximate value for
0
-------------as an area.
Page 5 of 13
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
√
3. Let f (x) = 1 + x.
(a) (5 marks) Construct the Maclaurin series for f (x) up to and including the term in x3 .
(b) (3 marks) Use (a) to determine the rst three non-zero terms of the Maclaurin series
√
for f (x) = 4 − x2 .
(c) (2 marks) Use (b) to calculate an approximate value for
Calculate
R √
R 2√
4 − x2 dx.
0
----------------------------------------------------------(d) (5
marks) Interpret the denite integral: 02 4 − x2 dx as
an area. Hence nd the exact
R 2√
--------------------------------------------------------------------------------------value of 0 4 − x2 dx. Compare your answer with your answer from (c).
Page 4 of 12
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
k
sec2 x
dx = ln 2, determine the value of k.
0 1 + tan x
(b) (4 marks) Determine the volume V of the solid of revolution obtained by rotating the
p
graph of f (x) = x ln(x) above the interval [1, e] about the x-axis.
Z ∞
1
(c) (4 marks) Evaluate
dx.
2
−1 x + 5x + 6
4. (a) (4 marks) If
Z
Page 5 of 12
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
4. The following integral function is used in the theory of the diraction of light waves:
Z
x
cos
C(x) =
0
1 2
πt dt,
2
for x ≥ 0.
(a) (2 marks) Determine the critical points of C(x).
any one of the critical points of C(x).
(b) (3 marks) Classify -------------------------------the critical points of C(x).
Only one required.
Page 6 of 12
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
SECTION B: LINEAR ALGEBRA (50 marks)
1. (10 marks) In each case, state whether the statement is true or false. No justication is
required.
Statement
(a) Let u and v be vectors in R3 . If u × v = 0, then either u = 0 or v = 0.
(b) Let u,v and w be vectors in R3 . Then (u · v) × w = u · (v × w)
(c) Let A and B be square matrices. If AB = AC , then B = C .
(d) A linear system of 3 equations in 4 unknowns never has any solutions as
there are insucient equations.
(e) If a system of linear equations has two dierent solutions, it must have
innitely many solutions.
(f) Let A =
1 −3
2 −6
. The vector
3
1
is in the null space of A.
(g) If A and B are n × n matrices, then (AB)T = AT B T .
(h) If A is a 3 × 3 matrix, then det (3A) = 3 det (A).
(i)
(j)
1 4
is an eigenvector of
.
2 3
  
  
−1
0 
 1
The set  2  ,  2  ,  0  is a linearly independent set of


3
4
0
1
1
vectors.
Page 7 of 12
True/False
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
2. (a) The linear system
a1 x + b 1 y + c1 z = 0
a2 x + b2 y + cz z = 0
a3 x + b 3 y + c3 z = 0


0

has only the zero solution 
 0 .
0
i. (3 marks) What can be said about the solutions of the following system? Explain.
a1 x + b1 y + c1 z = 3
a2 x + b 2 y + cz z = 7
a3 x + b3 y + c3 z = 11

 
 



a
b
c
1
1
1



 
 

ii. (3 marks) Is the set W =  a2  ,  b2  ,  c2  linearly independent?




a3
b3
c3
Explain.
(b) (4 marks) The augmented matrix for a linear system has been reduced by row operations
to the given row echelon form. Solve the system.


1 0 8 −5 6


 0 1 4 −9 3 
0 0 1 1 2
Page 8 of 12
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
3. (6 marks) The volume of a tetrahedron whose sides are the vectors a,b, and c is given by
the formula
1
|a · (b × c)|.
6

 
  

0
1
3
−1
 
  

Determine the volume of the testrahedron with vertices 
 0  ,  2  , 4 , −3  .
0
−1
0
4
Page 9 of 12
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
4. (a) (6 marks) Let A be a invertible matrix. Let λ be an eigenvalue of A, with corresponding
eigenvector x.
i. Show that λ 6= 0.
ii. Show that 1/λ is an eigenvalue of A−1 with corresponding eigenvector x.
(b) (8 marks) Let A =
−2 −7
1
2
!
.
i. Find the eigenvalues of A.
ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector.
iii. Make use of part (a)
! to determine an eigenvalue and a corresponding eigenvector
of
2/3
7/3
−1/3 −2/3
.
Page 10 of 12
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
5. (a) (7 marks) Explain why the following sets of vectors are linearly dependent.

 



−1
5



 

i.  2  ,  −10 




4
−20
(
!
!
!)
3
4
1
ii.
,
,
−1
5
2
      

1
0
1 





    

 0   1   1 
    
iii. 
 0  ,  0  ,  0 


     




 0
0
0 
Give a geometrical description of the vector space spanned by the vectors in (i). What
is the dimension of this vector space?
 



a


 
(b) (3 marks) Is the set W =  b  a ≥ 2c a vector space?




c
Remember to justify your answers.
continued...
Page 12 of 13
MATH1051 - Calculus and Linear Algebra I
Semester One Final Examinations, 2020
Formula Sheet
tan θ =
sin θ
cos θ
sin2 θ + cos2 θ = 1
cot θ =
cos θ
sin θ
sec θ =
sec2 θ = tan2 θ + 1
1
cos θ
csc θ =
1
sin θ
csc2 θ = cot2 θ + 1 sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1
= 1 − 2 sin2 θ
Dierentiation rules (appropriate domains assumed):
d
dx
d x
e
dx
d
dx
d
dx
sin x = cos x
d
dx
= ex
arcsin x = √ 1
1−x2
d
dx
d
dx
cos x = − sin x
ln x =
1
x
tan x = 1 + tan2 x
d a
x
dx
arccos x = − √ 1
1−x2
d
dx
= axa−1 , a 6= 0
arctan x =
1
1+x2
If f is continuous and non-negative on [a, b] and the graph of y = f (x) above the interval [a, b] is
rotated about the x-axis, the resulting solid has volume
Z
V =π
b
[f (x)]2 dx.
a
The sum to n terms of a geometric series with rst term a and common ratio, r 6= 1 is
n−1
X
arj =
j=0
a(1 − rn )
.
1−r
The Taylor series of a function f about x = a is given by
∞
X
f (n) (a)
n=0
n!
(x − a)n ,
(provided this exists).
Page 12 of 12