UNIVERSITY OF SOUTHAMPTON
MATH1055W1
SEMESTER 1 EXAMINATION 2011/12
MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING
MOCK 1
Duration: 120 min
This paper has two parts, part A, consisting of 20 multiple choice questions worth 2 marks each, and part B, consisting of 5 longer questions,
worth 12 marks each.
Exactly one of the 5 options (a), (b), (c), (d), (e) in each of the 20 multiple
choice questions is correct. (Sometimes that is “(e) none of the above”!)
Correct answers will attract 2 marks, no answer 0 marks and wrong
answers −0.5 marks.
MATH1055 did not exist in 2011/12, this exam has been put together by
combining topics from the old exams for MATH1013 and MATH1017.
All questions are from the 2011-12 exams, unless specified otherwise.
Formula sheet FS/1054-55/12 will be provided.
Only University approved calculators may be used.
A foreign language dictionary (paper version) is permitted provided it contains no notes,
additions or annotations.
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PART A
From 1013:
A2. (2 marks) Given the function f (x, y) = x2 sin2 (y) , which of the following is true?
∂f
∂x
∂f
(b)
∂x
∂f
(c)
∂x
∂f
(d)
∂x
∂f
= 2x2 cos(y),
∂y
= 4x sin(y) cos(y) and
(a)
∂f
= x2 sin(2y),
∂y
∂f
= 2x cos(y) and
= 2x2 sin(y) cos(y),
∂y
∂f
= 2 sin(y) cos(x) and
= 2 sin(x) cos(y),
∂y
= 2x sin2 (y) and
(e) none of the above.
A4. (2 marks) The region R in the first quadrant is enclosed by the lines y = 1 and
π
y = sin x from x = 0 to x = . The volume of the solid obtained by revolving R
2
about the x-axis is given by
Z π
Z π
2
2
x sin x dx , (b) 2π
x cos x dx ,
(a) 2π
0
Z
(c) π
π
2
0
(1 − sin x)2 dx ,
0
Z
(d) π
π
2
sin2 x dx ,
0
(e) none of the above.
A8. (2 marks) The general solution of the differential equation
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= y sec2 x is
dx
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MATH1055W1
(b) y = tan x + Cex ,
(a) y = tan x + C ,
(c) y = etan x + C ,
(d) y = Cetan x
(e) none of the above.
Z
4
Z
1
1
(a) −6,
f (5 − x)dx ?
f (x)dx = 6, what is the value of
A10. (2 marks) If
(b) 3,
4
(c) 0,
(d) 6,
(e) none of the above.
π A12. (2 marks) The derivative of the function f (x) = ln cos
is
x
π π π −π
π
π
, (b) − tan
(a)
, (c) tan
, (d) 2 tan
,
x
x
x
x
x
x2 cos πx
(e) none of the above.
A14. (2 marks) The area of the region in the first quadrant, enclosed by the graph of the
curve y = x(1 − x) and the x-axis is
(a)
1
,
3
(b)
2
,
3
(c)
5
,
6
(d) 1,
(e) none of the above.
2
Z
A16. (2 marks) The improper integral
1
1
(a) not defined,
(b) equal to 2,
1
(x − 1) 3
(c) equal to
dx is
2
,
3
(d) equal to
3
,
2
(e) none of the above.
(Question A20. continued on next page)
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A20. (2 marks) Let A22 denote the cofactor of the matrix


−2 1 0
A= 1 3 2
3 4 1
in the second row and second column, and let det(A) denote the determinant of A.
Then
(a) A22 = 2 and det(A) = 15,
(b) A22 = 4 and det(A) = −15,
(c) A22 = −2 and det(A) = 5,
(d) A22 = −4 and det(A) = −5,
(e) none of the above.
From 1017:
A2. (2 marks)
The equation
(3x2 t2 + t cos x)
dx
+ 2x3 t = G(x)
dt
is exact if
(a) G(x) = t sin x,
(b) G(x) = cos x,
(c) G(x) = − sin x,
(d) G(x) = 6x2 t2 ,
(e) none of the above.
A4. (2 marks)
Which of the following ordinary differential equations is nonlinear?
3
d2 x
dx
d2 x
5
3 d x
(a)
− 3 + 6x = t (b) exp(t ) 3 − 5 2 + 6x = 0,
dt2
dt
dt
dt
d2 x
dx
dx t
(c)
−
5
sin(t)
+
x
=
cos(t)
,
(d)
− = 0,
dt2
dt
dt x
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(e) none of the above?
A6. (2 marks)
The locus of the point z that satisifies the equation |z − 1 + j| = 4 is
(a) a circle centred on (−1, 1) with radius 2,
(b) a circle centred on (1, −1) with radius 2,
(c) a circle centred on (−1, 1) with radius 4,
(d) a circle centred on (1, −1) with radius 4,
(e) none of the above.
A8. (2 marks)
The eigenvalues of the matrix
C=
5 3
3 5
are
(a) λ1 = 5 + 3j and λ3 = 5 − 3j ,
(b) λ1 = 2 and λ2 = 8 ,
(c)
λ1 = λ2 = 8, (d)λ1 = 1 and λ2 = 3, (e) none of the above.
A10. (2 marks) Given the matrix


1 0 γ
A=γ 1 1 ,
5 1 5
which of the following conditions on γ ensures that the equation AX = 0 has a
unique solution X?
(a) γ = 4,
(b) γ = 1,
(c) γ = 4 and γ = 1
(d) γ < 0,
(Question A12. continued on next page)
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A12. (2 marks)
Given the functions f (t) = t3 cos t and g(t) = sin t2 , which of the following is
true?
(a) g is neither odd nor even and f is even,
(b) g is odd and f is even,
(d) f is odd and g is even,
(c) both are odd,
(e) none of the above.
A20. (2 marks)
Which of the following functions f (t) has a Fourier series of the form
f (t) =
∞
X
an cos(nt)?
n=1
(a) f (t) = 1 + t2
(b) f (t) = t2
(c) f (t) = t
in
−1<t<1
in
in
−π <t<π
−π <t<π
(d) f (t) = t cos t
in
and
f (t + 2π) = f (t),
and
f (t + 2) = f (t),
and
f (t + 2π) = f (t),
−1<t<1
and
f (t + 2) = f (t),
(e) none of the above.
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from MATH1017, 2010-11:
A1. Which of the following statements about an n × n matrix A, where n > 1, is always
true?
(a) If det(A) = 0 then A has at least one row of zeroes,
(b) If a matrix B is obtained from A by exchanging two rows of A , then
det(A) =det(B),
(c) If r > 0, then det(rA) = r det(A),
(d) det(A + A) = 2det(A),
(e) none of the above.
A3. (2 marks) Which of the following is an integrating factor for the differential equation
−t2
(a) −t3 ,
2
(b) et ,
1
(c) 3 ,
t
dx
+ 3tx = t3 ?
dt
3
t
(d) − ,
(e) none of the above.
A9. (2 marks) Given that
2
f (x, y) = ex y ,
∂ 2f
which of the following is
?
∂y∂x
(a) e2x ,
2
2
(b) x2 yex y + 2xex y ,
2
(c) 2xex ,
2
2
(d) 2yex y + 2x2 ex y ,
(e) none of the above.
(Question A11. continued on next page)
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A11. (2 marks) A Fourier series is obtained for the function
(
f (t) =
t+3
t−3
0 6 t < 3,
3 6 t < 6,
if
if
and
f (t + 6) = f (t) for all t .
Which of the following is the value of the series at t = 3 ?
1
(a) 0, (b)
f1 (t) − f2 (t) , (c) 6, (d) 3,
2
(e) none of the above.
Z
A15. (2 marks)
(a)
p
√
2x
dx =
x2 − 1
x2 − 1 + C ,
(b) x2 cosh−1 (x) + C ,
(c) ln |x2 − 1| + C ,
(d) 2
p
x2 − 1 + C ,
(e) none of the above.
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PART B
From 1013:
B2. (a) (6 marks) Given that


2 1 3
A =  −1 3 1 
2 −2 1
determine
(i) A − AT ,
adj(A).
(ii)
(b) (6 marks) (from MATH1017, 2007-08)
Find the general solution of the differential equation
t2
2
2xe + cos t
− cos x
dx
dt
2
= 2xt sin t2 − sec2 t − 2tx2 et .
B4. (a) (6 marks) Find the solution of the differential equation
dy
= −xy sin x
dx
which satisfies y = 1 when x = π .
(b) (6 marks) from MATH1015 2011/12
If f is a function of x and y , where x = r cos θ and y = r sin θ ,
use the chain rule to obtain expressions for
∂f
∂θ
∂ 2f
and
.
∂θ2
in terms of x and y , and derivatives of f with respect to x and y .
TURN OVER
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MATH1055W1
From 1017:
B2. (a) (4 marks)
Show that λ = 2 is an eigenvalue of the matrix


2 0 1
A=0 2 1
1 1 2
and find the other eigenvalues.
(b) (5 marks) Find three independent eigenvectors of A.
(c) (3 marks) Show that the eigenvectors of A are all mutually orthogonal (i.e
perpendicular to each other). Explain why you should expect this.
B4. (a) (6 marks)
Find the solution of differential equation
t2
dx
= x2 + 3xt − 3t2 ,
dt
which satisfies the initial condition x = 1 when t = 1.
(b) (6 marks)
Obtain the general solution of the second order differential equation
dy
d2 y
+
5
+ 6y = e−2x .
2
dx
dx
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B1/3. (from MATH1017 2010-11)
(a) (6 marks) Using the Formula Sheet or the Data Book, derive the Fourier series of
the periodic function f (t) given by
(
−1 if − 2 6 t < 0 ,
f (t) =
1
if
0 6 t < 2,
and
f (t + 4) = f (t) for all t.
and write out the first four terms of the series.
(b) (7 marks) When a mass m oscillates at the end of a spring of modulus λ and
natural length L, the angular frequency ω is given by
ω2 =
λ
.
mL
Estimate the percentage error in the period T , defined by the formula
T =
2π
,
ω
if λ is underestimated by 2%, m is overestimated by 1% and L is underestimated
by E %. Discuss briefly the estimated period resulting from different values of E .
END OF PAPER
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