FINAL EXAM MATH1050 (Functions-Differentiation-integration-sequences-matrices) chapters
AAM1994
practice-exam-3-sem-1-20202.pdf
Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
Part A
There are 20 marks in Part A. Each of the following questions is worth 2 marks. Write your answers in the space provided. Show full working. Part marks may be awarded for correct working.
1. Determine the domain and range of h(x) = √ x− 7.
2. Solve for x if 8
x > 2.
continued...
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
3. Determine dy
dx for y = esin x.
4. Determine
∫ (12x + 4)(3x2 + 2x)5 dx.
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
5. Determine the following limit, or show that it does not exist.
lim x→3
( x− 3 √ x− 2
)
6. Determine the following limit, or show that it does not exist.
lim x→4
( x2 −x− 12
x− 4
)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
7. The first three terms of an arithmetic sequence are −2, 3 and 8.
(a) Determine a closed form expression for the nth term. Simplify your answer.
(b) Determine the sum of the first 25 terms of the sequence.
8. Determine the 12th term of the geometric sequence −0.4, 0.8,−1.6, ...
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
9. Let A and B be the following matrices.
A =
3 0−1 2
1 1
, B = ( 1 4 2
3 1 5
) . Determine AB.
10. Let C =
( 1 2 4 3
) . Determine C−1.
End of Part A
continued...
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
Part B
There are 43 marks in Part B. Each of the following questions carries the stated number of marks. Write your answers in the space provided. Show full working. Part marks may be awarded for correct working.
1. Let A =
x2 1 xx 4 x− 2
x + 2 x2 3
.
(a) Determine det(A). (3 marks)
(b) Does the matrix have an inverse when x = 1? (2 marks)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
2. Let f(x) = 1
√ x− 1
and g(x) = x2.
(a) Determine the domain and range of f. (1 mark)
(b) Determine the domain and range of f. (1 mark)
(c) Determine g ·f and state the domain and range. (2 marks) (d) Is g a 1-1 function? Explain. (1 mark)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
3. The curve with equation y2 = x3 + 3x2 is called the Tschirnhausen Cubic (see graph below).
(a) Find dy
dx by implicit differentiation. (2 marks)
(b) Find the equation of the tangent line to this curve at the point (1,−2). (2 marks)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
4. (a) Determine
∫ 1 0
4x2
(x3 + 2)3 dx (3 marks)
(b) Determine
∫ 3 −3
(−|x| + 3) dx (4 marks)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
5. A function f has a derivative given by f ′(x) = 2 √ x +
c √ x
and passes through the points (0, 2)
and (1, 4).
(a) Determine c. (2 marks)
(b) Hence, explain why f has no stationary points. Carefully state your reasoning. (3 marks)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
6. Find the sum of all integers between 100 and 200 (both inclusive) that are NOT divisible by 4.
(5 marks)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
7. Use the principle of Mathematical Induction to prove that 24 divides 52n − 1, for all integers n ≥ 1.
(6 marks)
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
8. The graph of y = ae−x(a > 0) is shown below. P lies on the graph and the rectangle OAPB is drawn. As P moves along the curve the rectangle constantly changes. Determine the coordinates of P such that the rectangle OAPB has minimum perimeter. (6 marks)
End of exam
Formulae Sheet...
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Semester One Practice Final 3 Examination, 2020 MATH1050 Mathematical Foundations
Formulae Sheet
Cosine Rule: a2 = b2 + c2 − 2bc cos A
Sine Rule: sin A
a =
sin B
b =
sin C
c
sin(θ + φ) = sin θ cos φ + cos θ sin φ
sin(θ −φ) = sin θ cos φ− cos θ sin φ
cos(θ + φ) = cos θ cos φ− sin θ sin φ
cos(θ −φ) = cos θ cos φ + sin θ sin φ
cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ
sin 2θ = 2 sin θ cos θ
For an arithmetic sequence of the form {aj}nj=1, where aj = a + (j − 1)d,
Sn = n∑
i=1
ai = n
2 (2a + (n− 1)d).
For a geometric sequence of the form {aj}nj=1, where aj = arj−1,
Sn = n∑
i=1
ai = a(1 − rn)
1 − r = a(rn − 1) r − 1
.
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