HG1M11-E1

The University of Nottingham

SCHOOL OF MATHEMATICAL SCIENCES

A LEVEL 1 MODULE, AUTUMN 20092010

ENGINEERING MATHEMATICS 1

Time allowed TWO Hours

Candidates may complete the front cover of their answer book and sign their desk card but mustNOT write anything else until the start of the examination period is announced.

This paper has TWO sections which carry equal marks.

Section A comprises TWELVE multiple-choice questions. Responsesmust be made on the response sheet provided.

Section B comprises THREE questions. Full marks may be obtained for TWO completeanswers. Credit will be given for the best TWO answers.

An indication is given of the approximate weighting of each section of a question by means of a figureenclosed by square brackets, eg [12], immediately following that section.

Only silent, self-contained calculators with a Single-line Displayor Dual-line Display are permitted in this examination.

Dictionaries are not allowed with one exception. Those whose first language is notEnglish may use a standard translation dictionary to translate between that language andEnglish provided that neither language is the subject of this examination. Subject specific

translation dictionaries are not permitted.

No electronic devices capable of storing and retrieving text, including electronicdictionaries, may be used.

Do NOT turn examination paper over until instructed to do so.

ADDITIONAL MATERIAL: Formula SheetMultiple choice answer sheets

INFORMATION FOR INVIGILATORS: Please collect the multiple-choice answer sheetsand scripts separately at the end of the exam.

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Instructions for answering the multiple-choice questions

(a) Responses will be read by a machine. You MUST NOT mark the response sheet in anyway other than as indicated on the response sheet.

(b) All rough work should be within the examination book; rough work will not be used forassessment.

(c) You MUST record exactly one response for each question; choose E if you wish to abstain.(Each response is marked +3 if correct, 1 if incorrect, and 0 for abstain. The total isscaled.)

(d) On the response sheet:

Please use an HB pencil.

Mark your answer with a single horizontal line.

If you make a mistake, erase it completely.

Do not mark with ticks, crosses or circles.

Do not forget to write your NAME and MODULE details.

Do not forget to enter and code your 7 digit STUDENT ID.

Mark the box corresponding to your School in the section headed Other Informationas follows:

School Code School Code

SChEME A Civil Engineering D

Electrical and Electronic B Mechanical, Materials, EEngineering Manufacturing, Engineering

and ManagementBuilt Environment C Other Courses F

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SECTION A

1 The complex number1 + 2i

4 i is equal to

(a)1

17(2 9i) ; (b) 1

17(2 + 9i) ; (c)

1

9(2 + 5i) ; (d) 1

9(2 5i) .

2 The complex number12(1 + i) is equal to

(a) e3ipi/4; (b) eipi/4; (c) eipi/4; (d) e3ipi/4.

3 The principal argument of3 i is

(a) 5pi6; (b) pi

6; (c)

pi

6; (d)

11pi

6.

4 The complex number z = 2e11ipi/6 is a solution of

(a) z3 = 8; (b) z3 = 6; (c) z3 = 6i; (d) z3 = 8i.

5 The inverse of the matrix(1 24 1

)is

(a) 19

( 1 24 1

); (b)

1

9

( 1 42 1

);

(c)1

9

( 1 24 1

); (d) 1

9

( 1 42 1

).

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6 If

A =

(2 14 5

),

then ATA is the matrix

(a)

(0 127 21

); (b)

(26 1212 18

);

(c)

(20 1218 26

); (d)

(20 1818 26

).

7 The eigenvalues of the matrix(8 46 2

)are

(a) 2 and 4; (b) 2 and 4; (c) 2 and 4; (d) 2 and 4;

8 The matrix 8 0 41 2 12 a 1

has no inverse if

(a) a =4

3; (b) a =

8

3; (c) a = 8

3; (d) none of these.

9 The derivative of y = cos (e cosx) with respect to x is

(a) cos(x)e cosx cos(e cosx

); (b) cos(x)e cosx cos (e cosx) ;

(c) sin(x)e cosx sin (e cosx) ; (d) sin(x)e cosx sin (e cosx) .10 The indefinite integral

x cos (x2) dx is

(a)1

2sin(x2)+C; (b) 1

2sin(x2)+C; (c) 2 sin

(x2)+C; (d) 2 sin (x2)+C;

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11 The value of the definite integral pi0

x sinxdx

is

(a) 1; (b) 1; (c) pi; (d) pi.

12 The Taylor series of cosx sinx about x = 0 starts with the terms

(a) 1 + x+1

2x2;

(b) 1 + x 12x2;

(c) 1 x+ 12x2;

(d) 1 x 12x2.

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SECTION B

13 (a) Write

z =(2 + 4i)(1 + 3i)

(1 i)(2 i)in the form z = x+ iy, where x and y are real numbers. Find the modulus of z. [5]

(b) Find all four solutions of the equation

z5 = 42 (1 + i) z2.

Write your results in exponential form and plot them on an Argand diagram. [12]

(c) Find the product of the three non-zero roots you have found in part (b). [2]

(d) Find two arguments of 42 (1 i). [2]

(e) Use de Moivres theorem to show that

cos (4) = cos4 6 cos2 sin2 + sin4 and

sin (4) = 4 cos sin (cos2 sin2 ) . [4]

14 (a) Write the system

x1 x3 = 13x1 +4x2 x3 = 2x1 +x2 +3x3 = 1

in the form Ax = b, stating clearly what A, x and b are. [3]

(b) Show that in the case of = 4 the equations in part (a) are inconsistent. Find a newright hand side for the third equation that makes the equations consistent for = 4. [9]

(c) Find all eigenvalues of the matrix A from part (a) in the case = 1, and confirmthat = 1 is one of them. Find an eigenvector of A (for = 1) correspondingto the eigenvalue = 1. [13]

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15 (a) Find and classify the stationary points of

f(x) =x2

x4 + 1.

Sketch the curve y = f(x). [11]

(b) Find the area between the curves y = g(x) and y = xex, where

g(x) = x2ex. [4]

(c) Show that the Taylor series of g about x = 0 starts with the term

g(x) = x2 + . . . . [4]

(d) Find the solution of

y = 2y2

e2x

satisfying the initial condition

y(0) = 1. [6]

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