University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
1
Part 1
In this part, there are five possible answers, but only one is correct. Correct answer will receive 2
marks. Write the corresponding letter(A, B etc.) inside the box given at the right end.
1. After the elementary row operations, the augmented matrix [A, b] of a linear system Ax = b
has been transformed into the form below:


1 −2
1
3
 0
2 −1 −4  .
0
0
0
3
Which of the following statement is correct?
A. The system has unique solution.
B. Column vectors of the coefficient matrix A are linearly independent.
C. The system is inconsistent.
D. The vector b can be written as a linear combination of the column vectors of A.
E. The system has infinitely many solutions.
Answer:
2. Let A be an invertible square matrix of order 3 and B be an 3 × 2 matrix. Find the incorrect
statement from the following:
A. The product (ABT ) is defined.
B. (AT )−1 = (A−1 )T .
C. (AB)−1 = B−1 A−1 .
D. (AB)T = BT AT .
E. The matrix B + AB is a 3 × 2 matrix.
Answer:
3. Let u, v be two nonzero, non-parallel vectors. Find one incorrect statement from the following:
A.
u
|u|
is a unit vector parallel to u.
B. If another nonzero vector w lies in the same plane as u and v then u · v × w = 0.
C. If u and v are perpendicular to each other, then u · v = 0.
D. If u and v are perpendicular to each other, then u × v = 0.
E. The vector component of v parallel to the vector u is equal to
v·u
u.
|u|2
Answer:
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
2
4. In the following statements, only one statement is wrong. Find which one:
A. A force-couple system in a plane can be reduced into a single equivalent force acting on
the same plane.
B. The moment of a force F about a point P that lies on the line of action of F
is given by r × F(6= 0), where r is a position vector from P to any point on the line of
action of F.
C. A force-couple system in three dimension can be reduced into a wrench.
D. There is no moment equilibrium equation for a system of forces acting at a point.
E. In the trusses, it is assumed that all loadings are applied at the joints.
Answer:
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
3
Part 2
For each question in this part, there are five possible answers given, but only two answers are
correct; first correct answer will receive 2 marks, and if both correct answers are found 3 marks
will be credited. Write the corresponding letter(A, B etc.) inside the box given at the right end.
1. After few row operations to the augmented matrix [A, b] of the system Ax = b is


2 −1
1
2
 0
0
2
6 .
0
0 −5 −15
Find two incorrect statements about the system
A. A is an invertible matrix.
B. The solution is given by x1 =
t−1
,
2
x2 = t, x3 = 3, where t is a parameter.
C. Columns of A are not linearly independent.
D. The solution of the homogeneous system Ax = 0 is a straight line passing through the
origin, parallel to the vector (1, 2, 0)T .
E. The system is inconsistent.
Answer:


2 −2 −1
2. The inverse of a 3 × 3 matrix A is given by A−1 =  1 −2 1  .
−1 1
1
Find two incorrect statements from the following:




2
1 −1
−2 2
1
A. (AT )−1 = −2 −2 1 
B. adj(A) = −1 2 −1 .
−1 1
1
1 −1 −1
C. We cannot use Cramer’s rule to find the solution to the system of equations Ax = b,
where nonzero vector b ∈ R3 .
D. det(A) = −1.
E. The cofactor of a22 is equal to −2.
Answer:
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
4
3. The system consists of two forces and couple
acting on the beam. Two statements in the
following are correct. Find those two:
A. The system is equivalent to a clockwise couple of −40 N-m.
B. The system is equivalent to a force F = 60i + 20j and a clockwise couple of 80 N-m at
the origin.
C. The system is equivalent to a force F = 60i + 20j and an anti-clockwise couple of 640
N-m at the origin.
D. The system is equivalent to a single force F = 60i+20j and its line action passes through
the point 4i.
E. The system is equivalent to a force F = 60i + 20j and an anti-clockwise couple of 300
N-m at the mid-point.
Answer:
4. The loading is shown to the upper joints of a Pratt roof truss. Two statements about the
truss are not correct. Find those two:
A. GC is a zero force member.
B. Support reaction force acting at A is equal
to 2.5j.
C. Support reaction force acting at E is equal
to −2.5j.
D. BH is a zero force member.
E. Axial force in the member AB is a compressive force.
Answer:
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
5
Part 3: Answer all questions.
Question 1.
18 points




1 0 1
1
1.1 Let A =  1 1 0 , and b =  5 . Denote the columns of A by u1 , u2 , u3 , and
0 1 1
−2
let W = Span{u1 , u2 , u3 }.
(a) Is b in W ? (you must show all the steps).
 
1

α  is a linear combination of u1 and u2 ?
(b) For what value(s) of α is
5
(3)
(2)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
(c) Show that every vector in R3 is a linear combination of u1 , u2 and u3 .
6
(3)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
7




1 −1 3 2
1
1 5 1  and b =  5  .
1.2 Let A =  −2
4 −3 1 3
−3
(a) Solve the system Ax = b, and write the solution in parametric vector form.
(4)
(b) Using the result from Part (a), write the solution to the homogeneous system
Ax = 0 in the parametric vector form.
(1)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
8


1 1 h
1.3 Given A =  1 h 1 ,
h 1 1
(a) Determine the value(s) of h for which the column vectors of A form a linearly
independent set ?


−1
1
(b) If h = −1 compute the product AB where B =  3 −3  .
2
2
(3)
(2)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
Question 2.
9
18 points


1 −1
0
0 −1  .
2.1 Let A =  1
−6
2
3
(a) Find the inverse of the matrix A using elementary row operation
(Gauss-Jordan reduction).
(5)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
10
(b) Using part (a), determine (AT )−1 .
(1)
(c) Compute the solution of the linear system of equations, using the part (b).
(3)
x +y −6z = −1
−x
+2z =
1
−y +3z = 2.
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
11
2.2 Find the following determinants:
(a) Using the properties with as few operations as possible. Give explanation.
(i)
−1
0
0 0
3 −7
0 0
5
2 −3 0
1
2
3 5
(1)
(ii)
2
6 10
−3 −9 8
4 12 9
(1)
(b) By performing the row operations and cofactor expansion with the properties
of the determinant, find
0
1
2 −1
2
5 −7
3
.
0
3
6
2
−2 −5
4 −2
(3)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
12
2.3 Find the value of x2 that satisfy the following system of linear equations using Cramer’s rule.
Express the value of x1 and x3 in terms of the determinants (do not compute).
(4)
2x1 +x2 −x3 = 5
3x1 +2x2 −2x3 = 9
3x2 −x3 = 7
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
Question 3.
13
23 points
3.1 At the instant shown, the Harrier’s thrust is T = 17500i + 70000j − 9000k N and its
velocity is v = 7.5i + 1.9j − 0.8k m/s.
(a) Find the vector components of T parallel to the velocity vector v.
(3)
(b) The power currently being transferred to the airplane by the engine is the quantity
P = |Tp ||v|, where Tp is the component of T parallel to v. Compute the power P
of the engine at that instant.
(1)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
14
3.2 The 8 kg lamp is suspended in the position shown. The unstrechted length of the spring
AB is `0 = 0.4 m and the spring constant is kAB = 300 N/m.
(5)
(a) Find the tension in the cable AC and the spring force when the lamp is at the rest.
(b) Determine the required length of the cable AC.
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
15
3.3 Force of magnitude F = 600 N is applied on the collar B as shown. The collar is located
3 m along the rod from end C.
(a) Express the force vector F in terms of its x, y, z components.
(3)
(b) Determine the components of F that act parallel to, and perpendicular to the rod
AC.
(4)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
16
3.4 The bar AB is supported by a ball and socket
joint at A and the two cables as shown. If the
mass of the bar AB is negligible compared to the
mass of the suspended object E, the bar exerts a
force on the ball at B that points from A toward
B. The weight of the object E is 2000 N.
(a) Draw the free body diagram of the ball B, showing all the forces acting on it
(neglect the weight of the ball).
(1)
(b) Determine the magnitude of the force exerted on the bar by the ball at B and the
tensions in the cable DB and CB.
(5)
(c) Deduce the support reaction at the ball and the socket.
(1)
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
Question 4.
4.1 The bracket is subjected to three forces and
a couple.
(a) If you represent this system by a force
FR applied at O and a couple M , what
are FR and M ?
(3)
(b) If you represent the system by the force
FR , where does its line of action intersect the x axis.
(3)
17
12 points
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
4.2 The horizontal bar of weight 2 kN is supported by
the cable and by a pin support at A as shown.
(a) Determine the tension in the cable and the
reactions at A.
(4)
(b) The cable will safely support a tension of
6 kN. Based on this criterion, what is the
largest safe value of the weight W ?
(2)
18
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
Question 5.
The truss supports two horizontal loads F1 and F2 .
Let F1 = F2 = 10 kN.
(a) Find the reactions at the supports A and G.
(2)
(b) Can you identify any zero-force member ? If
so, which one. Explain.
(2)
(c) Determine the force on members BD, BE and
BG of the truss and state if the members are
in tension or compression.
(Hint: Use method of section )
(4)
(d) Using the above results, find the axial forces
in the remaining members.
(1)
19
9 points
University of KwaZulu-Natal Main Examination: 30 May 2018
Applied Mathematics 1A
20