FINAL EXAM MATH1050 (Functions-Differentiation-integration-sequences-matrices) chapters

Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

Part A

There are 20 marks in Part A. Each of the following questions carries the stated number of marks. Write your answers in the space provided. Show full working. Part marks will be awarded for correct working.

1. Let A,B, and C be the following matrices.

A =

  3 0−1 2

1 1

 , B = ( 1 4 2

3 1 5

) and C =

( 4 2 1 3

) .

(a) If AT −B = (

2 d −1 e 1 f

) , find the values of d,e and f. (1 mark)

(b) Determine C−1. (2 marks)

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

2. Let y = cos(4x3 + 2x). Determine dy

dx . (2 marks)

3. Let y = x2 ln x. Determine dy

dx . (2 marks)

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

4. Determine the following integrals.

(a)

∫ 3x4(x5 + 7)2 dx. (2 marks)

(b)

∫ x2 − 1 x

dx. (2 marks)

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

5. The first three terms of a finite arithmetic sequence are −5,−1 and 3.

(a) Find an expression for the nth term. Simplify your answer. (1 mark)

(b) Find the sum of the first 50 terms of the sequence. (1 mark)

6. The first three terms of a finite geometric sequence are 2, 3 and 4.5. Find the sum of the first 19 terms of the sequence. Round your answer to the nearest integer. (2 marks)

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

7. Determine lim x→0

x3 − 8x x

. (2 marks)

8. Determine lim x→−2

3x2 + 5x− 2 x2 − 2x− 8

. (2 marks)

9. Let f(x) = 2

√ x− 1

. Determine the domain of f(x). (1 mark)

End of Part A

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

Part B

There are 44 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 will be awarded for correct working.

1. Let A =

  4 2 12 2 3

5 1 2

  and B = α

  1 −3 411 3 −10 −8 6 4

 . If B is the inverse of A, determine the

value of α. (3 marks)

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

2. Roughly but clearly sketch the graph of the function f(x) with the following given properties.

f(−2) = 3, f(−1) = 0, f(0) = −2 f ′(2) = 0, f ′(x) < 0 whenever x < 2, f ′(x) > 0 whenever x > 2, lim

x→∞ f(x) = 0. (4 marks)

y

x

continued...

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Semester One Practice Final 2 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)

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

4. Determine the following integrals.

(a)

∫ ( 1

x √ x

+ xex 2

) dx. (3 marks)

(b)

∫ π 3

0

sin x

cos x dx. (3 marks)

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

(c)

∫ e x−1 x

x2 dx. (5 marks)

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

5. Infinitely many rectangles can be inscribed under the curve y = e−2x. (Part of the curve is shown below.)

(a) Determine the coordinates of A such that the area of rectangle OBAC is a maximum.

(5 marks)

(b) What is the maximum area? (1 mark)

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

6. The functions f(x) = x20 − 1 and g(x) = x2 −x6 form the face of a cat as seen below.

(a) Verify that x = ±1 are solutions of f(x) = 0 and g(x) = 0. (1 mark)

(b) Find the area of the cat’s face. (4 marks)

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

7. Use the principle of Mathematical Induction to prove that 6n−1 is divisible by 5, for all integers n ≥ 1. (6 marks)

continued...

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Semester One Practice Final 2 Examination, 2020 MATH1050 Mathematical Foundations

8. The sum, Sn, of the first n terms of a geometric sequence, where a > 0, is given by

Sn = 7n −an

7n .

Determine the common ratio, r, of the sequence. (5 marks)

End of exam

Formulae Sheet...

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Semester One Practice Final 2 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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