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Exam of 23 pages for the course Linear Algebra I at Unisa (mat1503 exam pack)
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MAT1503Ass 1 2021
Memorandum
QUESTION 1
Which of the following is a linear equation in x, y, and z?
1. x − e2 y = 3z
1
2.
2π ln (ez ) − 2y + z = ln(3) − x
3. √y −2 + 4y − 2z = 7x
4. y + 4y − 2z = 7x 3
Solution
Linear Equations: Properties of natural logs: Exponent(s) laws:
Highest power of variables is 1. ln(x y ) = yln x 1 1
1. a−m = m
or m = a−m
one. a a
2. ln(ex ) = x
No products or roots of
variables. 3. ln(e) = 1
Note: I only chose properties
4. eln(x) = x of natural logs and exponents
Variables do not appear as
arguments of trig, logarithmic laws relevant to the concepts
or exponential functions. in question.
.
SOLUTION
1. x − e2 y = 3z ⇒ x − 7.38 … y = 3z
1
2.
2π ln (ez ) − 2y + z = ln(3) − x ⇒ 2πz −1 − 2y + z = ln(3)
1 1 1 1
Note: ln (ez ) = = z −1 or ln(e) = = z −1 ∵ ln(e) = 1and ln(3) = 1.098..
z z z
3.
√y −2 + 4y − 2z = 7x
4. y + 4y − 2z = 7x 3
QUESTION 2
© 2021
JTS Maths Tutoring
Keeping you mathematically informed
, -2- MAT1503
Ass 1 2021
Memorandum
Which of the following is a linear equation in x, y, and z?
1. x 2 − e2 y = 3z
2. 2π ln(ez ) − 2y + z = ln(3) − x
3. √y −1 + 4y − 2z = 7x
4. y + 4y − 2z = 7x 3
SOLUTION
1. x2 −e2 y = 3z
2. 2π ln(ez ) − 2y + z = ln(3) − x ⇒ 2πz − 2y + y = ln(3) − x
3. √y −1 + 4y − 2z = 7x
4. y + 4y − 2z = 7x 3
QUESTION 3
Which of the following is a nonlinear equation in x, y, and z?
1. y + 4y − 2z = 7x
2. 2π ln(e−z ) − 2y + z = ln(3) − x ⇒ −2πz − 2y + z = ln(3) − x
3. 7x − 4y − 2z = 0
4. x − exy = 3z
SOLUTION
1. y + 4y − 2z = 7x
2. 2π ln(e−z ) − 2y + z = ln(3) − x ⇒ −2πz − 2y + z = ln(3) − x
3. 7x − 4y − 2z = 0 exy
4. x− exy = 3z
© 2021
JTS Maths Tutoring
Keeping you mathematically informed
, -3- MAT1503
Ass 1 2021
Memorandum
In the following three questions, draw a table of logical operations in order to boil down statements
into digestible operations through the corresponding logical formulas.
QUESTION 4
Any homogeneous linear system has one solution or no solution.
1. T ∧ F, where T stands for True and F stands for False
2. (F ∨ F ) ∧ ⌝ T, where ⌝ is the symbol for logical negation
3. F ∧ (⌝ F ∨ T)
4. (T ∨ F ) ∧ F
5. (T ∨ F ) ∨ F
SOLUTION
Logical Connectives
Logical connectives are operators that are used to join one or more prepositions (propositions are
declarative statements that are either true or false but can’t be both).
Connective Connective Word Symbol
Conjunction And ∧
Disjunction Or ∨
Conditional If--then →
(implication)
Biconditional If and only if (iff) ↔
Negation Not ⌝ or ~ or ‘ or -
It is not the case that
© 2021
JTS Maths Tutoring
Keeping you mathematically informed
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