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Question 1: 1 Mark
Which of the following is a linear equation in x; y and z?
16
MAT1503/101/0/2022
1. −x−1 + e−
√
2y = 3z, where e = 2.71828 ....
2. 2π ln(e−1
z ) − 2y + z = ln(3) − x.
3.
p
y2 + 4y − 2z = 7x.
4. y + 4y − 2z = 7x−2.
Question 2: 1 Mark
Which of the following is a nonlinear equation in x; y and z?
1. −2y − 4y − z = 0.
2. π ln e2z − 1
2y − 2z = ln(e3) − x.
3. x + 4y − 2z = 0.
4. 3x + xy = 3z, where e = 2.71828 ....
In the following three questions, draw a table of logical operation in order to boil down statements into
digestible operations through the corresponding logical formulas.
Question 3: 1 Mark
Which of the following is the solution of the equation below?
0x + 0y = 0.
1. (0, 0, 0).
2. (1, 0, 0).
3. No such solution exists.
4. Infinitely many solution or (−1, 2, 1).
Question 4:
Consider the system of equations represented by the augmented matrix below.
2 2 2 10
2 0 −1 2
Which of the following is not a solution to this system?
1. (−1, 2, 4) or (1, 1, 0).
2. (1, 0, 0).
3. No such solution exists.
4. Infinitely many solution or (−1, 2, 1).
17
Question 5:
Determine which of the following is the solution set of the linear equations below.
3x − y + z = 2
2x − z = 2
1. {(x, y, z) : x = 1
3 (t − s + 2), y = t, z = s with s, t ∈ R}
2. {(x, y, z) : x = 1
3 (t + s + 2), y = t, z = s with s, t ∈ R}
3. {(x, y, z) : x = 1
3 (t − s − 2), y = t, z = s with s, t ∈ R}
4. {(x, y, z) : x = −13
(2 + t − s), y = t, z = s with s, t ∈ R}
6. {(x, y, z) : x = t, y = 5t + 4, z = 2t − 2 with t ∈ R}
Question 6:
Consider the system obtained from the augmented matrix below.
a b c
d e f
Choose the correct statement(s):
1. The system has no solution if ae = bd.
2. The system has exactly one solution whenever ae ̸= db.
3. The system is inconsistent for af ̸= ab.
4. The system has infinitely many solution if b
e = cf
.
5. The system has no solution if ae = bf or impossible.
Question 7: 1 Mark
Consider given the system obtained from the augmented matrix below.
a b c
d e f
Choose the correct option:
1. The system has infinitely many solution if ac
= b
d = ef
.
2. The system has exactly one solution whenever af ̸= db.
3. The system is inconsistent for af ̸= ab.
4. The system has no solution if ae = bd.
5. The system has no solution if ae = bf .
18
MAT1503/101/0/2022
Question 8: 1 Mark
Solve for X from the matrix equation below. Here I is the identity matrix and det(A) ̸= 0.
A2X + I = AB
Choose the correct option:
1. X is the identity matrix.
2. X = A−1B − I.
3. X = A−1B + A.
4. X = A−1B − A.
5. X = AB + B.
Question 9: 1 Mark
Solve for X from the matrix equation below. Here I is the identity matrix and det(B) ̸= 0 and det(A) ̸= 0.
BXA + A = B
Choose the correct option:
1. No such matrix X.
2. X = A−1B − A.
3. X = A−1 − B + A.
4. X = A−1 + B − A.
5. X = −A−1 + B−1.
6. X = −(A−1 + B−1).
Question 10: 1 Mark
Solve for X from the matrix equation below. Here I is the identity matrix and det(B) ̸= 0 and det(A) ̸= 0.
B(X − I)A + B = A
Choose the correct option:
1. No such matrix X.
2. X = A−1B − A.
3. X = A−1 − B + A.
4. X = A−1 + B − A.
5. X = −A−1 + B−1 + I.
6. X = −(A−1 + B−1).
19
Question 11: 1 Mark
Consider the following linear system:
(
2x −3y = −1
2x −3y = 1
.
1. x = 0 and y = 0 satisfy the system.
2. The system has exactly one solution.
3. The system is inconsistent.
4. The system has infinitely many solution.
Question 12: 1 Mark
Consider the system of linear equations given by
x − y + 3z = 1
2x − y + z = −1
x −3y − z = 2
Which of the following is the the augmented matrix for the above given system?
1 −1 3 1
2 −1 1 −1
1 3 −1 3
4.
1 −1 3 1
2 −1 1 −1
1 3 −1 0
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MAT1503/101/0/2022
5.
1 −1 3 1
2 −1 1 −1
1 −3 −1 2
Assume that we are given a coefficient matrix A of the system of linear equations AX = b, where b =
124
,
X =
0 1 2
2 5 6
1 0 0
Solve for the variable x2 and x3 in the above system.
1. The system has infinitely many solution.
2. The system has exactly one solution.
3. The system is inconsistent
4. x2 = 0 and x3 = 1 satisfy the system.
5. x2 = 1 = x3.
Use the Gauss-Jordan process to determine for which value (s) of λ will the following system have no
solutions?
1 2 −3 4
3 −1 5 2
4 1 (λ2 − 14) λ + 2
1. λ = 4.
2. λ = 8.
3. λ = −2.
4. λ = −4.
21
What conditions must the constants α, β, and γ satisfy for the following system
x + 2y + 3z = α
2x + 5y + 3z = β
2x + 7y − 3z = α + γ
to be consistent?
1. α = β and any value for γ.
2. α = −γ and any value for β.
3. The value for x is not unique and α = γ − β.