ENG1090 sample exam paper PDF

Preview

Summary

eng...

Description

Office Use Only

Monash University Sample Examination Paper Faculty of Engineering EXAM CODE:

ENG1090

TITLE OF PAPER:

FOUNDATION MATHEMATICS

EXAM DURATION:

3 hours writing time

READING TIME:

10 minutes

THIS PAPER IS FOR STUDENTS STUDYING AT: (tick where applicable)  Berwick  Clayton  Malaysia  Distributed Learning  Caulfield  Gippsland  Peninsula  Enhancement Studies  Pharmacy  Other (specify)

 Open Learning  South Africa

During an exam, you must not have in your possession, a book, notes, paper, calculator, pencil case, mobile phone or other material/item which has not been authorised for the exam or specifically permitted as noted below. Any material or item on your desk, chair or person will be deemed to be in your possession. You are reminded that possession of unauthorised materials in an exam is a discipline offence under Monash Statute 4.1. Students should ONLY enter their ID number and desk number on the examination script book, NOT their name. Please take care to ensure that the ID number and desk number are correct and are written legibly.

No examination papers are to be removed from the room. 1. All six questions may be attempted. All questions are worth 20 marks. A pass mark for this examination paper will be about 60 marks. 2. All questions consist of multiple parts, and each question should be read from the beginning. The marks awarded per part question are shown and they are awarded for key steps in the working, not only for the final answer. All working and explanations should be written clearly and concisely. 3. Your brief working and answers to the questions should be written on this paper, in the space provided. You may use a blank examination script book for additional working, but it will not be marked. 4. A brief formula sheet is provided on page 18, at the end of the examination paper. AUTHORISED MATERIALS CALCULATORS

 YES

 NO

OPEN BOOK

 YES

 NO

SPECIFICALLY PERMITTED ITEMS

 YES

 NO

Candidates must complete this section STUDENT ID

__ __ __ __ __ __ __ __

DESK NUMBER

__ __ __ __

EXAMINATION QUESTIONS BEGIN OVER THE PAGE…. Page 1 of 18

1.

(a)

2 Consider the two functions f ( x)  ln(1  x2 ) and g ( x)  1  x .

(i)

State the domains of both f and g , giving brief reasons for your answers.

X 2 marks

(ii)

Identify the domain of the composite function ( f  g) , giving a reason for your answer, and determine an expression for ( f  g ) as a function of x .

X 2 marks  (iii) Determine the inverse function f 1 ( x) which has negative values over its domain,  and identify both the domain and range of f 1 , justifying your answer.

X 3 marks Page 2 of 18

(b)

Consider the periodic function h( x )  sin(  x )  3 cos( x ) .

(i)

State the period of the function h( x ) , giving a brief reason for your answer.

X 1 mark

(ii)

Determine the amplitude A and phase angle  when the function h( x) is written as a single cosine function of that period. Show clearly all of the key steps in your working.

X 4 marks Page 3 of 18

(iii) Use your results to sketch the function h( x) over the domain 1  x 1 .

X 2 marks (c)

Sketch a graph of each of the following functions over the given domains: (i)

F (x ) 

x 1 for all x  1 x 1

X 2 marks

(ii)

G( x)  1  e x for all x  0 .

X 2 marks Page 4 of 18

(d)

Show that sec2 x  1  tan2 x for all values of x , stating clearly any other results used.

X 2 marks

Q1

2.

(a)

Consider the two complex numbers z1  cis (  34  ) and z2  1  3 i . (i)

Write z1 in Cartesian form.

X 2 marks

(ii)

Determine the modulus and principal argument of z2 and write it in polar form.

X 3 marks

(iii) Determine the real and imaginary parts of 2 / z2 .

X 3 marks Page 5 of 18

(iv) Use De Moivre’s Theorem to determine ( z 1)4 , and then write it in Cartesian form.

X 2 marks

(v)

Determine all distinct fourth roots of z2 , writing them in exponential form.

X 6 marks

(vi) Show your answers for ( z 1) 4 and ( z2 )1/ 4 on an Argand diagram.

X 3 marks Page 6 of 18

(b)

Sketch the set {z : Im z  0 and z  1} on an Argand diagram.

X 1 mark

Q2

3.

(a)

Consider the position vectors a  i  j  2 k and b  i  j  2 k . (i)

Determine a  b and a  2 b in terms of their Cartesian components.

X 2 marks

(ii)

Find b , the unit vector bˆ and identify all of the direction angles of b .

X 4 marks Page 7 of 18

(iii) Find the scalar resolute of a in the direction of b .

X 2 marks

(iv) Find the projection (vector resolute) of a in the direction of b , in terms of its Cartesian components.

X 2 marks

(v)

Determine the remaining part a of a and show that it is perpendicular to b .

X 2 marks

Page 8 of 18

(b)

A force of F   i  k Newtons acts on a body as it moves along a straight line from the point A at (2, 4, 1) to the point B at (3, 2,1) , measured in metres. (i)

Find the work done as the particle undergoes a displacement d from A to B.

X 4 marks

(ii)

Find the angle between the force F and the displacement d from A to B.

X 4 marks

Q3 Page 9 of 18

4.

(a)

(i)

Determine both the left and right-hand limits as x  1 for the function  x  f ( x)   k  2  x2 

for for for

x1 x 1 x1

and identify whether there is a value of k for which f ( x) is continuous at x  1 , justifying your answer.

X 3 marks

(ii)

Evaluate lim

x 

x ( x  1) , justifying the key steps of your answer. 2 x 2 1

X 2 marks

Page 10 of 18

(b)

Differentiate each of the following functions with respect to x , stating the rules used, and simplify the answer: (i)

g (x )  e x ln(x  1)

X 2 marks

(ii)

h( x) 

tan x x

X 2 marks

(iii)

F ( x )  exp( cos( x 2 ))

X 2 marks Page 11 of 18

(iv)

G ( x )  tan 1 ( x) (for example using that tan(tan 1 (x ))  x )

X 3 marks

(c)

A cylindrical block of metal of a given volume V , with a circular base, is to be manufactured so that it has the smallest surface area S for that shape. The volume of a cylinder with base radius r  0 and height h is V   r 2 h , and it has surface area is

S  2 r (r  h) , so the surface area for a fixed volume V is S (r )  2 r 2  (i)

2V . r

Find the critical value of r at which S (r ) may have a local minimum or maximum.

X 2 marks

(ii)

Use the second derivative test to show that S has a minimum at that value of r .

X 2 marks Page 12 of 18

(iii) Evaluate r (in cm) when V  16  cm3 and also determine the corresponding cylinder height h (in cm) at that value of the base radius r .

X 2 marks

Q4

5.

(a)

Determine the indefinite integral of each of the following functions with respect to x : (i)

1 f (x )  x  1   sin x  e x x

X 3 marks

(ii)

g ( x)  4 x (1 2 x2 )3 (for example using a suitable substitution)

X 3 marks Page 13 of 18

(iii)

h( x) 

1 (for example using partial fractions) x 2 ( x  1)

X 5 marks (b)

Evaluate



1 0

x 1  x2 dx , showing the key steps in your calculation.

X 4 marks Page 14 of 18

(c)

Consider the region A between the parabola y  x2 , the y -axis and the line y  1 . (i)

Determine the area of the region A .

X 2 marks

(ii)

If this region is rotated about the y -axis, determine the resulting volume V . [Hint: not about the x -axis.]

X 3 marks

Q5

6.

(a)

Determine the mean value f of the function f ( x)  sin x over the interval 0,   .

X 2 marks Page 15 of 18

(b)

For an RC electrical circuit, the charge Q(t ) on the capacitor satisfies ln 2  ln (2  Q)  (i)

t . 4

Determine Q as a function of t .

X 2 marks

(ii)

Find the value of Q as t approaches infinity, giving reasons for your answer.

X 2 marks

(iii) At which time t , if any, does Q  1 ?

X 1 mark (b)

A curve ( x, y) is determined implicitly through the equation y2  x3  x y  1 . (i)

Show that the point ( x, y)  (1,1) lies on this curve.

X 1 mark Page 16 of 18

(ii)

Find the slope dy / dx of the curve at the point ( x, y )  (1,1) .

X 4 marks

(iii) State the equation of the tangent line to the curve at ( x, y )  (1,1) .

X 2 marks

(c)

Consider the differential equation (i)

dy y   for x  1 dx x

Solve for the function y ( x) in terms of an unknown constant.

X 4 marks

(ii)

Determine the solution of the initial-value problem for which y  1 at x 1 .

X 2 marks

END OF EXAMINATION QUESTIONS

Q6 Page 17 of 18

SOME USEFUL FORMULAE sin( x  y )  sin x cos y  cos x sin y cos(x  y )  cos x cos y  sin x sin y sin 2 x  2 sin x cos x

cos 2x  cos x  sin x  2 cos x  1 1 2 sin x 2

2

2

sec x 

1 cos x

cosec x  cot x 

2

1 sin x

cos x sin x

e i   cos  i sin  sin  

e i   e i  2i

cos 

ei   e i  2

END OF EXAMINATION PAPER

Page 18 of 18...