❉❛t❛ Pr♦✈✐❞❡❞✿ ❋♦r♠✉❧❛ ❙❤❡❡t
MAS241
SCHOOL OF MATHEMATICS AND STATISTICS August Supplementary
Exam Period 2019-2020
Mathematics II (Electrical) 1 hour (nominal)
❚❤✐s ✐s ❛♥ ♦♣❡♥ ❜♦♦❦ ❡①❛♠✳
❆♥s✇❡r ❛❧❧ q✉❡st✐♦♥s✳
❚❤❡ s✉❜♠✐ss✐♦♥ ❞❡❛❞❧✐♥❡ ✐s ✶✵ ❛♠ ✭❇❙❚✮✱ t✇❡♥t②✲❢♦✉r ❤♦✉rs ❛❢t❡r ✐t ✐s r❡❧❡❛s❡❞✳ ▲❛t❡ s✉❜✲
♠✐ss✐♦♥ ✇✐❧❧ ♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ✇✐t❤♦✉t ❡①t❡♥✉❛t✐♥❣ ❝✐r❝✉♠st❛♥❝❡s✳ ■t ✐s ❡①♣❡❝t❡❞ t❤❛t ②♦✉ ✇✐❧❧
❜❡ ❛❜❧❡ t♦ ❝♦♠♣❧❡t❡ t❤✐s ❡①❛♠ ✐♥ ❛♣♣r♦①✐♠❛t❡❧② ♦♥❡ ❤♦✉r ❛♥❞ ✐t ✐s r❡❝♦♠♠❡♥❞❡❞ t❤❛t ②♦✉
s✉❜♠✐t t❤❡ ✇♦r❦ ✇✐t❤✐♥ ❢♦✉r ❤♦✉rs✳ ❨♦✉ ✇✐❧❧ ♥♦t ❜❡ ♣❡♥❛❧✐s❡❞ ❢♦r t❛❦✐♥❣ ❧♦♥❣❡r✱ ❤♦✇❡✈❡r✳
❯♥❧❡ss ✐t ✐s ❡①♣❧✐❝✐t❧② st❛t❡❞ ♦t❤❡r✇✐s❡✱ ✐t ✐s ✐♥t❡♥❞❡❞ t❤❛t ❝❛❧❝✉❧❛t✐♦♥s ❛r❡ ♣❡r❢♦r♠❡❞ ❜② ❤❛♥❞
✭♣♦ss✐❜❧② ✇✐t❤ t❤❡ ❛✐❞ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r✮✳ ❚♦ ❣❛✐♥ ❢✉❧❧ ♠❛r❦s✱ ②♦✉ ✇✐❧❧ ♥❡❡❞ t♦ s❤♦✇ ②♦✉r ✇♦r❦✲
✐♥❣✳ ❨♦✉ ✇✐❧❧ ♥♦t ❣❡t ❢✉❧❧ ♠❛r❦s ✐❢ ②♦✉ s✐♠♣❧② ✇r✐t❡ ❞♦✇♥ ♦✉t♣✉t ❢r♦♠ ❛ ❝♦♠♣✉t❡r ♣❛❝❦❛❣❡✳
❇② ✉♣❧♦❛❞✐♥❣ ②♦✉r s♦❧✉t✐♦♥s ②♦✉ ❞❡❝❧❛r❡ t❤❛t ②♦✉r s✉❜♠✐ss✐♦♥ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ②♦✉r ♦✇♥
✇♦r❦✱ t❤❛t ❛♥② ✉s❡ ♦❢ s♦✉r❝❡s ♦r t♦♦❧s ♦t❤❡r t❤❛♥ ♠❛t❡r✐❛❧ ♣r♦✈✐❞❡❞ ❢♦r t❤✐s ♠♦❞✉❧❡ ✐s ❝✐t❡❞
❛♥❞ ❛❝❦♥♦✇❧❡❞❣❡❞ ❛♥❞ t❤❛t ♥♦ ✉♥❢❛✐r ♠❡❛♥s ❤❛✈❡ ❜❡❡♥ ✉s❡❞✳
❚♦t❛❧ ♠❛r❦s ✹✵✳
Please leave this exam paper on your desk
Do not remove it from the hall
❘❡❣✐str❛t✐♦♥ ♥✉♠❜❡r ❢r♦♠ ❯✲❈❛r❞ ✭✾ ❞✐❣✐ts✮
t♦ ❜❡ ❝♦♠♣❧❡t❡❞ ❜② st✉❞❡♥t
MAS241 1 Turn Over
MAS241
Blank
MAS241 2 Continued
MAS241
✶ ✭✐✮ ❚❤❡ ❝❤❛r❣❡ q(t) ♦❢ ❛ ❝✉rr❡♥t ✐♥ ❛ ❝✐r❝✉✐t ✐s ❞❡s❝r✐❜❡❞ ❜②
q′′(t) + 2q′(t) + 5q(t) = δ(t)
s✉❜❥❡❝t t♦ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s q(0) = q′(0) = 0✱ ✇❤❡r❡ δ(t) st❛♥❞s ❢♦r t❤❡
❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥✳ ❉❡t❡r♠✐♥❡ t❤❡ ❝❤❛r❣❡ q(t) ❛t t✐♠❡ t > 0✳ ✭✶✵ ♠❛r❦s✮
✭✐✐✮ ▲❡t f(t)←→
2e−jω
1 + ω2
❜❡ ❛ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♣❛✐r✳ ❋✐♥❞ f(t)✳ ✭✺ ♠❛r❦s✮
✭✐✐✐✮ ▲❡t f(t) = e−|t| ❛♥❞ g(t) = r❡❝t2(t)✳ ❋✐♥❞ F{f ∗ g(t)}✳ ✭✺ ♠❛r❦s✮
✷ ✭✐✮ ▲❡t D ⊂ R2 ❜❡ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s (0, 0)✱ (1, 0)✱
(1, 1)✳ ❋✐♥❞ ∫∫
D
ex
2
dA.
✭✶✵ ♠❛r❦s✮
✭✐✐✮ ▲❡t f(x, y) =
√
x2 + y2 ❛♥❞ D := {(x, y) : x2 + y2 ≤ 1}✳ ❋✐♥❞
∫∫
D
f(x, y)dA.
✭✶✵ ♠❛r❦s✮
End of Question Paper
MAS241 3
MAS241 FORMULA SHEET
Laplace transform:
The Laplace transform of a function f(t) is given by:
L{f(t)}(s) :=
∫ ∞
0
e−stf(t)dt.
Properties of the Laplace transform: L{f(t)} = F (s) in the following table.
L{af(t) + bg(t)} = aL{f(t)}+ bL{g(t)} linearity
L{f ′(t)} = sF (s)− f(0) differentiation w.r.t. t
L{f ′′(t)} = s2F (s)− sf(0)− f ′(0) second differentiation w.r.t. t
L{e−ktf(t)} = F (k + s) frequency shift
L{f(t− a)H(t− a)} = e−asF (s) (for a > 0) time shift
L{f(at)} = 1
a
F ( s
a
) (for a > 0) scaling
L{f ∗ g(t)} = L{f(t)}L{g(t)} (for f(t), g(t) causal) convolution
Table of standard Laplace transforms:
f(t) L{f(t)}(s) Region of validity
tn (for n ≥ 0) n!
sn+1
Re(s) > 0
sin(kt) k
s2+k2
Re(s) > 0
cos(kt) s
s2+k2
Re(s) > 0
H(t− T ) (for T ≥ 0) e
−sT
s
Re(s) > 0
δ(t− T ) (for T ≥ 0) e−sT s ∈ C
Fourier transform:
The Fourier transform and inverse Fourier transforms are given by:
F{f(t)}(ω) = F (ω) :=
∫ ∞
−∞
f(t)e−jωtdt, f(t) = F−1{F (ω)} =
1
2pi
∫ ∞
−∞
F (ω)ejωtdω.
Properties of the Fourier transform: F{f(t)} = F (ω) in the following table:
F{ejθtf(t)} = F (ω − θ) frequency shift
F{f(t− T )} = e−jωTF (ω) time shift
F{f (n)(t)} = (jω)nF (ω) differentiation
F{F (t)} = 2pif(−ω) symmetry
F{f(at)} = 1
|a|
F (ω
a
) scaling
F{f ∗ g(t)} = F{f(t)}F{g(t)} convolution
Table of standard Fourier transforms:
f(t) F{f(t)}(ω)
e−a|t| (for a > 0) 2a
a2+ω2
rectT (t) sinc(
Tω
2
)
1 2piδ(ω)
Fourier series:
The Fourier series of a periodic function f(t) with fundamental period T is given by
S[f ] =
a0
2
+
∞∑
n=1
(
an cos(ωnt) + bn sin(ωnt)
)
where
ωn =
2pin
T
, an =
2
T
∫ T/2
−T/2
f(t) cos(ωnt)dt, bn =
2
T
∫ T/2
−T/2
f(t) sin(ωnt)dt.
Coordinate systems:
Cylindrical polar coordinates
(x, y, z) = (r cos(θ), r sin(θ), z)
(r, θ, z) =
(√
x2 + y2, arctan( y
x
), z
)
dV = rdrdθdz.
Spherical polar coordinates
(x, y, z) = (ρ sin(φ) cos(θ), ρ sin(φ) sin(θ), ρ cos(φ))
(ρ, θ, φ) =
(√
x2 + y2 + z2, arctan( y
x
), arccos( z
ρ
)
)
dV = ρ2 sin(φ)dρdφdθ.