UNIVERSITY OF SOUTHAMPTON
MATH1055W1
SEMESTER 1 CLASS TEST 2013/14
MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING
Duration: 60 min
This is version F of the question paper. Write “Version F” on your
answer sheet with your name.
This paper consists of 25 multiple-choice questions, each worth
2 marks. Enter your answers into the designated answer sheet
(AS/MATH1055TEST/2014).
Exactly one of the 5 options (a), (b), (c), (d), (e) in each question is
correct. (Sometimes that is “(e) none of the above”!) Find that option
and blob it on the designated answer sheet. Correct answers will attract
2 marks, no answer 0 marks and wrong answers −0.5 marks.
Formula sheet FS/1054-55/14 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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MATH1055W1
1. The function f (x) = x − sin(x) has
(a) exactly two stationary points,
(c) no stationary points,
(b) infinitely many stationary points,
(d) exactly one stationary point,
(e) none of the above.
2. Euler’s formula states that
(a) sin(α + jβ) = cos(α) − j sin(β),
(c) ejα = cos(α) + j sin(α),
(b) ejα = sin(α) + j cos(α),
(d) cos(α + jβ) = cos(α) + j sin(β),
(e) none of the above.
3. The derivative of the function f (x) = 5x is
(a) ln(5x ),
(b) ln(5) ln(5x ),
(c) 5x ln(5),
(d) x5x−1 ,
(e) none of the above.
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4. The partial derivatives
MATH1055W1
∂f
∂f
and
of the function
∂x
∂y
f (x) = cos2 (x − y) are
(a)
∂f
∂f
= −2 sin(x − y) cos(x − y) and
= 2 sin(x − y) cos(x − y),
∂x
∂y
(b)
∂f
∂f
= 2 sin(x − y) cos(x) and
= −2 sin(x − y) cos(y),
∂x
∂y
(c)
∂f
∂f
= 2 sin(x) and
= −2 sin(y),
∂x
∂y
(d)
∂f
∂f
= 2x sin(x − y) cos(x − y) and
= −2y sin(x − y) cos(x − y),
∂x
∂y
(e) none of the above.
cos(x)
is
9 − x2
5. The domain of the function f (x) = √
(a) −3 < x < 3,
(b) −3 ≤ x ≤ 3,
(c) x 6= ±3,
(d) x ≥ 3 or x ≤ −3,
(e) none of the above.
Z
6. The improper integral
1
3
x− 4 dx is
0
(a) not defined,
(b) equal to 1,
(c) equal to 4,
(d) equal to
1
,
4
(e) none of the above.
TURN OVER
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MATH1055W1
7. The derivative of the function f (x) = ln(x2 ) cos(3x) is
(a)
cos(3x)
sin(3x)
2
+
2
ln(x
)
sin(3x)
,
(b)
,
x2
x2
(c) −2 ln(x2 ) sin(3x),
(d)
2 cos(3x)
− 3 ln(x2 ) sin(3x),
x
(e) none of the above.
8. If a = 3i − 6j + 2k then
(a) |a| = 7 and â = 3i − 6j + 2k,
(b) |a| = 49 and â =
(c) |a| = 49 and â = 3i − 6j + 2k,
(d) |a| = 7 and â =
1
(3i − 6j + 2k),
49
1
(3i − 6j + 2k),
7
(e) none of the above.
9. If z = 3 − j then
(a) z ∗ − 2z = −3 − j
and
(b) z ∗ − 2z = −3 + 3j
and
(c) z ∗ − 2z = −3 − j
(d) z ∗ − 2z = −3 + 3j
and
and
1
= 1,
z 10
1
= 1,
z 10
1
= √1 ,
z 10
1
= √1 ,
z 10
(e) none of the above.
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Z
10.
MATH1055W1
4
dx =
4 − x2
−1
2
(a) 4 ln |4 − x | + C ,
(b) tan
(c) ln |2 + x| − ln |2 − x| + C ,
x
2
+ C,
(d) 2 ln |2 − x| −
1
ln |2 + x| + C ,
2
(e) none of the above.
11. The magnitude of the area enclosed by the curve y = 4x3 − 3x2 , the x-axis and the
vertical lines x = 1 and x = 2 is
(a) 8,
(b) 22,
(c) 24,
(d) 6,
(e) none of the above.
12. The values of x satisfying |x − 3| ≤ 1 are
(a) x ≤ −4 or x ≥ −2 ,
(c) 2 ≤ x or x ≥ 4,
(b) 2 ≤ x ≤ 4,
(d) −4 ≤ x ≤ −2,
(e) none of the above.
13. If a = 2i − j + k , b = 2i − k and c = 3j − 2k then a · b × c =
(a) −4,
(b) (6, −4, 6),
(c) 1,
(d) (−3, 8, 12),
(e) none of the above.
TURN OVER
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MATH1055W1
14. The equation of the plane perpendicular to the vector a = 4i − 3j + 2k and
through the point (1, −1, −2) is
(a) 4x − 3y + 2z = 0,
(c) x − y − 2z = 3,
(b) r = (1, −1, −2) + s(4, −3, 2),
(d) 4x − 3y + 2z = 3,
(e) none of the above.
15. If −π < x ≤ π , the values of x satisfying cos x = −
(a)
2π
2π
,− ,
3
3
(b) −
2π π
, ,
3 3
(c)
2π 4π
, ,
3 3
1
are
2
(d)
π
5π
,− ,
3
6
(e) none of the above.
Z
16.
dx
=
1 + 3x2
√
1
6x
(a) √ tan( 3x) + C , (b) −
+ C,
1 + 3x2
3
√
1
1
(c) √ tan−1 ( 3x) + C , (d) tan−1 (3x) + C ,
3
3
(e) none of the above.
17. If y 2 − 2xy = 16 , then
(a) −
y
,
y+x
(b)
dy
=
dx
x
x
, (c)
,
2y + x
x−y
(d)
y
,
y−x
(e) none of the above.
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18. Using partial fractions,
MATH1055W1
x
is equal to
(x + 1)2
(a)
1
1
1
x
+
,
(b)
−
,
x + 1 (x + 1)2
(x + 1)2 x + 1
(c)
1
x
1
1
−
,
(d)
+
,
x + 1 (x + 1)2
x + 1 (x + 1)2
(e) none of the above.
19. The principal value of the argument of −4 + 3j is
4
3
(a) −π − tan−1
, (b) −π + tan−1
,
3
4
4
3
(c) π + tan−1
, (d) π − tan−1
,
3
4
(e) none of the above.
20.
Z 1 √
3
x5
√
4
− x dx =
0
(a)
7
17
, (b) − ,
13
40
(c)
3
,
8
4
5
(d) − ,
(e) none of the above.
21. For which value of k will the function f (x) = kx −
x = −2?
(a) −2 ,
(b) −4 ,
1
4
(c) − ,
(d)
1
have a local maximum at
x
1
,
2
(e) none of the above.
TURN OVER
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Z
MATH1055W1
(x2 − 1)3 dx =
22.
4
x7 3x5
1 x3
3
(a)
−
+ x − x + C , (b)
− x + C,
7
5
4 3
1
3x 2
(c) (x2 − 1)4 + C , (d)
(x − 1)4 + C
4
2
(e) none of the above.
23. The derivative of the function f (x) = sin−1 (x2 − x) is
1
(a) p
1 − (x2 − x)2
2x − 1
,
cos2 (x2 − x)
(c)
,
(b)
(d) √
1
,
cos2 (x2 − x)
2x − 1
,
1 − x4 + 2x3 − x2
(e) none of the above.
Z
π
cos3 (x) sin(x) dx =
24.
π/2
(a) −1,
(b) 3,
π3
(d)
,
5
1
(c) − ,
4
(e) none of the above.
Z
π
Z
1
r2 sin2 θdrdθ , is equal to
25. The double integral
θ=0
(a)
π
,
6
(b)
2
,
3π
r=0
(c)
2π
,
3
(d) −
π
,
6
(e) none of the above.
END OF PAPER
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