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Exam 2012, questions
Engineering Mathematics (HG1MM1)
University of Nottingham
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HG1M11-E
The University of Nottingham
SCHOOL OF MATHEMATICAL SCIENCES
A LEVEL 1 MODULE, AUTUMN 2012–
ENGINEERING MATHEMATICS 1
Time allowed TWO Hours
Candidates may complete the front cover of their answer bookand sign their desk card but must NOT 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. Responses must be made on the response sheet provided. Section B comprises THREE questions. Full marks may be obtained for TWO complete answers. 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 figure enclosed by square brackets, eg[12], immediately following that section. Only silent, self-contained calculators with a Single-line Display or Dual-line Display are permitted in this examination.
Dictionaries are not allowed with one exception. Those whose first language is not English may use a standard translation dictionary to translate between that language and English 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 electronic dictionaries, may be used.
Do NOT turn examination paper over until instructed to do so.
ADDITIONAL MATERIAL: Formula Sheet Multiple-Choice Answer Sheets
INFORMATION FOR INVIGILATORS: Please collect the multiple-choice answer sheets and scripts separately at the end of the exam.
HG1M11-E1 Turn over
Instructions for answering the multiple-choice questions
(a) Responses will be read by a machine. You MUST NOT mark the response sheet in any way other than as indicated on the response sheet.
(b) All rough work should be within the examination book; rough work will not be used for assessment.
(c) You MUST record exactly one response for each question; no response will be recorded as an abstain. (Each response is marked+4if correct,− 1 if incorrect, and 0 for abstain. The total is scaled.) (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 Information’ as follows:
Major Code
Chemical Engineering 000
EE & Mechatronic Engineering 001
Mechanical Engineering 002
Civil Engineering 003
HG1M11-E
5 Given the matrix
A=
1 2 3
4 5 6
7 8 9
,
2 AT−Ais the matrix
(a)
1 6 11
0 5 10
1 4 9
;
(b)
1 6 11
0 5 10
− 1 −4 9
;
(c)
1 −6 11
0 5 10
−1 4 9
;
(d)
1 6 11
0 5 − 10
−1 4 9
;
(e) none of these.
HG1M11-E
6 Given
B=
1 4
−2 3
0 − 1
, C=
(
2 0 − 1
3 1 1
)
,
the productBC
(a) does not exist; (b) is
14 4 3
5 3 5
− 3 − 1 − 1
;
(c) is
14 5 − 3
4 3 − 1
3 5 − 1
;
(d) is ( 2 9 1 14
)
;
(e) none of these.
7 The matrix
0 1 0
0 0 1
1 0 0
is
(a) singular; (b) symmetric; (c) antisymmetric; (d) orthogonal; (e) none of these.
8 The eigenvalues of the matrix ( 3 1 −2 6
)
are
(a)− 4 and− 5 ; (b)− 4 and 5 ; (c) 4 and− 5 ; (d) 4 and 5 ; (e) none of these.
HG1M11-E1 Turn over
Section B
13 (a) Express the complex number
(1 + 2i)(2 + 18i) (1−i)(2 + 4i)
in the formx+iywith realxandy, and find its modulus. [5]
(b) Find all solutions of the equation
z 4 = 128(1−
√
3 i).
Leave your solutions in exponential form, and plot them on anArgand diagram. [11]
(c) Use de Moivre’s theorem to show that
sin(3α) =−sin 3 α+ 3 sinαcos 2 α,
and find a similar expression forcos(3α). [4]
(d) Letnbe a positive integer, andw=−1 +i. Use Euler’s formula to show that
wn+wn= 2
n 2 + cos
(
3 nπ 4
)
.
Find a similar expression for
wn−wn. [5]
HG1M11-E1 Turn over
14
(a) Consider the matrix
A=
−2 2 − 3
−3 5 − 3
− 2 α − 1
.
(a) For which value or values ofαis the matrixAnon-singular? [3]
(i) In the caseα= 5, use the Gauss-Jordan method to findA− 1 , and hence find the solution of
Ax=
− 5
− 2
3
.
[12]
(ii) In the caseα= 4, confirm thatλ=− 1 is an eigenvalue ofA, and find the other eigenvalues ofA. Find all eigenvectors forλ=− 1. [8]
(iii) LetBandC be square matrices of the same order. Given thatBis symmetric and Cis antisymmetric, show that
BC−CB
is symmetric. You can use statements from the lectures concerning the transpose of a matrix without proving them. [2]
HG1M11-E
Exam 2012, questions
Module: Engineering Mathematics (HG1MM1)
University: University of Nottingham
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