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Mid Semester Quiz
Undergraduate
BSc Eng Hons
Eng-MATH
In20-S3-MA2024 (116515)
Mid Semester Quiz
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Started on Thursday, 1 December 2022, 8:15 PM
State Finished
Completed on Thursday, 1 December 2022, 10:04 PM
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Time taken 1 hour 49 mins
Grade 30.0000 out of 30.0000 (100%)
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Question 1
The flux of the vector field F(x, y, z)
Complete
sphere x
2
+ y
2
+ z
2
= 1
= zi + yj + xk
across the unit
is...
Mark 1.0000
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out of 1.0000
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a.
0
b.
π
c.
4π/3
d.
π/2
e.
3π/4
Question 2
Using a suitable coordinate system find the volume of the solid
Complete
formed above the cone z
Mark 1.0000
out of 1.0000
Question 3
x
2
+ y
2
+ z
2
−−−−
−
−
2
2
= √x + y
and below the sphere
= z.
Hint: First figure out the center and radius of the sphere.
a.
π/4
b.
5π/6
c.
π/8
d.
4π/3
e.
2π/3
Evaluate the integral
Complete
Mark 2.0000
out of 2.0000
The improper triple integral is defined as the limit of the triple integral
over a solid finite sphere having radius a as a tends to infinity.
a.
π/2
b. 0
c.
2π
d. Not integrable
e.

∞
Question 4

Complete
Suppose that a bounded plane region D on the xy-plane, its boundary
curve C and the component functions of the vector field F := P i + Qj
Mark 2.0000
satisfy the hypotheses of Green's theorem. Then
out of 2.0000
∮
C
F ⋅ dr =
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a.
∬
b.
∬
c.
∬
D
D
D
(divF) dA
(curlF) ⋅ k dA
(curlF) dA
d. 0
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e.
∬
D
(divF) ⋅ k dA
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Question 5
Define T
: x = u
2
2
− v , y = 2uv
; for u
≥ 0, v ≥ 0 .
Complete
Using T evaluate the integral ∬ R y dA ; where R is the region bounded
Mark 2.0000
by the x axis and the parabolas y 2
y ≥ 0.
out of 2.0000
= 4 − 4x
and y 2
= 4 + 4x
with
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
a. 0
b. 2
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c.
π
d. 1
e. 5
Question 6
Complete
Mark 1.0000
out of 1.0000
Using Stokes' theorem evaluate the integral ∬ S (curlF) ⋅ n dS ; where
F(x, y, z) = xzi + yzj + xyk
x
2
+ y
2
+ z
the xy-plane.
a. 2
b. 1
c.
2π
d. 0
e.

2
π/2
= 4
and S is the part of the sphere
that lies directly above the unit disk x2
+ y
2
= 1
on
Question 7
Complete
Mark 6.0000

out of 6.0000
Evaluate ∬ S
F ⋅ n dS
F(x, y, z) = xyi + (y
; where
2
+ e
xz
2
)j + sin(xy)k
and S is the surface of the 3D region E bounded by the parabolic
cylinder z
= 1 − x
2
and the planes z
= 0
,y
= 0
and y + z
= 2.
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
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a.
184/35
b.
184π/35
c.
184π/33
d. 0
e.
Question 8
Complete
Mark 1.0000
out of 1.0000
2π
A solid E lies within the cylinder x2
+ y
2
= 1,
below the plane z
= 4
and above the paraboloid z = 1 − x − y . The density at any point is
proportional to its distance from the axis of the cylinder with
2
2
proportionality constant K . The mass of E is...
a.
12πK /5
b.
13πK /6
c.
11πK /6
d.
∞
e. None of the above
Question 9
Complete
Mark 6.0000
out of 6.0000
Let f (z)
:= iz/3
For every ϵ
> 0,
; for every z
∈ C.
choose an appropriate δ
> 0
such that for every
z ∈ C,
the statement " 0
< |z − 1/2| < δ
⟹
|f (z) − i/6| < ϵ "
holds.
a. No such δ exists as the corresponding limit does not exist.

b.
δ := 6ϵ
c.
δ := 2ϵ
d.
δ := 6iϵ
e.
δ := 3iϵ
Question 10
Complete
Evaluate ∬ S
y dS
; where S is the surface z
= x + y
2
on
(x, y) ∈ [0, 1] × [0, 2].
Mark 1.0000

out of 1.0000
a.
13√5
3
b.

13√2
11
c.

13√2
3
d.

e.
13
3√5
13√2
7
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

Question 11
A point is expressed in rectangular coordinates as (0, 2√3, −2). The
Complete
point can be expressed in spherical coordinates (ρ, θ, ϕ) as...
–
Mark 1.0000
out of 1.0000
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
a.
(2, π/2, π/3)
b.
(2, π/2, π/6)
c.
(4, −3π/2, 2π/3)
d.
(4, π/2, 2π/3)
e.
(4, 3π/2, 2π/3)
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Question 12
Complete
If C is a smooth curve given by a vector valued function r defined on
[a, b] ,
Mark 3.0000
∫
out of 3.0000
C
r ⋅ dr =
a.
1
b.
1
2
2
2
[|r(b)|
2
[(r(b))
c.
|r(b)|
d.
(r(b))
2
2
− |r(a)| ]
2
− (r(a)) ]
2
− |r(a)|
2
2
− (r(a))
e. None of the above
Question 13
Complete
The work done by the field F(x, y, z)
mM G
= −
2
2
( x +y +z
Mark 3.0000
out of 3.0000
2
3/2
(xi + yj + xk)
)
in moving a particle from point (3, 4, 12) to the point (2, 2, 0) along a
piecewise-smooth curve C is
a.
mG(
1
2√3
−
1
13
)
b. mMG( 1 − 1 )
13
2 √2
c. \(mMG(\frac{3}{2\sqrt{2}}-\frac{1}{15})\)
d. \(mMG(\frac{1}{2\sqrt{2}}-\frac{1}{15})\)
e. \(mMG(\frac{1}{2\sqrt{3}}-\frac{1}{13})\)
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