Gujarat Technological UniversityWinter 2023 examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Winter 2023 Question Paper & PDF

B.E. · Common Engineering · Semester 3 · Subject code 3130005
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Total
70 marks
To pass
23 marks (33%)
Duration
2.5 Hours 10:30 AM – 1:00 PM
Questions
5 with internal choice
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2 of 25 questions on this paper were also asked in other years. 19 more are types of question GTU sets again with new data. Each is marked below with the years it appeared.See all 37 repeated 3130005 questions, most asked first

Q.1

14 Marks
(a)
3 marksSame type asked in 4 of 7 papers
Express (2+i3−i)2\left(\frac{2+i}{3-i}\right)^2 into Polar form.

Also asked in Summer 2026 Q1a, Summer 2024 Q1a, Summer 2023 Q3b OR

(b)
4 marksSame type asked in 3 of 7 papers
Find and plot the fourth roots of (−1)(-1).

Also asked in Summer 2026 Q3a, Winter 2024 Q1b

(c)
7 marksSame type asked in 3 of 7 papers
Solve (D2+DD′+D′−1)z=sin⁡(x+2y)(D^2 + DD' + D' - 1)z = \sin(x + 2y).

Also asked in Summer 2025 Q4b OR, Winter 2024 Q5c

Q.2

14 Marks
(a)
3 marks
Determine aa and bb such that u=ax3+bxyu = ax^3 + bxy is harmonic.
(b)
4 marks

Discuss the continuity of f(z)f(z) at the origin:

f(z)={zˉz,if z≠00,if z=0f(z) = \begin{cases} \frac{\bar{z}}{z}, & \text{if } z \ne 0 \\ 0, & \text{if } z = 0 \end{cases}

(c)
7 marksSame type asked in 3 of 7 papers
Show that the function u=excos⁡yu = e^x \cos y is harmonic. Find the conjugate function vv and express u+ivu + iv as an analytic function of zz.

Also asked in Winter 2024 Q3b, Summer 2024 Q1b

(c)
7 marksSame type asked in 7 of 7 papers
Find the bilinear transformation which maps the points 1,−1,∞1, -1, \infty onto the points 1+i,1−i,11 + i, 1 - i, 1 respectively. Also, find its fixed points.

Also asked in Summer 2026 Q2c, Winter 2025 Q1c, Summer 2025 Q2c, Winter 2024 Q1c, Summer 2024 Q1c, Summer 2023 Q2c OR

Q.3

14 Marks
(a)
3 marksSame type asked in 3 of 7 papers
Find the radius of convergence of the power series
∑n=1∞zn2n+1\sum_{n=1}^\infty \frac{z^n}{2^n + 1}

Also asked in Winter 2025 Q3b, Summer 2024 Q2b

(b)
4 marks
Separate (i)i(\sqrt{i})^{\sqrt{i}} into real and imaginary parts.
(c)
7 marksSame type asked in 6 of 7 papers
State Cauchy’s Integral Theorem and use it to find
∫Ce2zz2+1 dz,where C is ∣z∣=12\int_C \frac{e^{2z}}{z^2 + 1} \, dz, \quad \text{where } C \text{ is } |z| = \frac{1}{2}

Also asked in Summer 2026 Q5b, Summer 2025 Q3a, Winter 2024 Q3c, Winter 2024 Q3b OR, Summer 2024 Q3c, Summer 2023 Q2c, Summer 2023 Q3c OR

(a)
3 marksSame type asked in 4 of 7 papers
Evaluate ∫02+iz2 dz\int_0^{2+i} z^2 \, dz along the line y=x2y = \frac{x}{2}.

Also asked in Summer 2026 Q2b, Winter 2025 Q2b, Winter 2024 Q2b

(b)
4 marksSame type asked in 3 of 7 papers
Expand f(z)=cos⁡zf(z) = \cos z as a Taylor series about z=0z = 0.

Also asked in Summer 2026 Q3b OR, Winter 2025 Q3b OR

(c)
7 marksSame type asked in 4 of 7 papers
Write Cauchy’s Integral formula and hence evaluate:

∫Csin⁡πz2+cos⁡πz2(z−1)(z−2) dz,where C is ∣z∣=3\int_C \frac{\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)} \, dz, \quad \text{where } C \text{ is } |z| = 3

Also asked in Summer 2026 Q3b, Winter 2025 Q2c OR, Summer 2024 Q2c OR

Q.4

14 Marks
(a)
3 marksSame type asked in 3 of 7 papers
Classify the singular point z=0z = 0 for function f(z)=1z4−4z2f(z) = \frac{1}{z^4 - 4z^2}.

Also asked in Summer 2025 Q3a OR, Summer 2024 Q3a OR

(b)
4 marksSame type asked in 3 of 7 papers
Find the complete integral of p−3x2=q2−yp - 3x^2 = q^2 - y.

Also asked in Summer 2025 Q5b, Winter 2024 Q5a

(c)
7 marksSame type asked in 7 of 7 papers

Obtain the Laurent’s series for the function f(z)=1z(1−z)f(z) = \frac{1}{z(1-z)} in the regions:

i∣z+1∣<1|z + 1| < 1
ii1<∣z+1∣<21 < |z + 1| < 2
iii∣z+1∣>2|z + 1| > 2

Also asked in Summer 2026 Q3c, Winter 2025 Q3c, Summer 2025 Q3b OR, Winter 2024 Q2c, Summer 2024 Q3b, Summer 2023 Q2b, Summer 2023 Q3a OR

(a)
3 marksSame type asked in 4 of 7 papers
Derive partial differential equation by eliminating aa and bb from z=(x−a)2+(y−b)2z = (x - a)^2 + (y - b)^2.

Also asked in Summer 2026 Q4a OR, Winter 2024 Q4a, Summer 2023 Q4b

(b)
4 marksSame type asked in 6 of 7 papers
Using Cauchy’s residue theorem, evaluate
∫C5z−2z(z−1) dz,∣z∣=2\int_C \frac{5z - 2}{z(z - 1)} \, dz, \quad |z| = 2

Also asked in Summer 2026 Q5b, Summer 2025 Q3a, Winter 2024 Q3c, Winter 2024 Q3b OR, Summer 2024 Q3c, Summer 2023 Q2c, Summer 2023 Q3c OR

(c)
7 marksSame type asked in 6 of 7 papers
Solve x(y−z)p+y(z−x)q=z(x−y)x(y - z)p + y(z - x)q = z(x - y).

Also asked in Winter 2025 Q4b, Winter 2025 Q4c OR, Summer 2025 Q4c, Summer 2025 Q5b OR, Winter 2024 Q4b, Winter 2024 Q4b OR, Summer 2024 Q4b, Summer 2023 Q4a, Summer 2023 Q4a OR, Summer 2023 Q5a OR

Q.5

14 Marks
(a)
3 marksSame type asked in 3 of 7 papers
Evaluate
∫0∞11+x2 dx\int_0^\infty \frac{1}{1 + x^2} \, dx
using contour integration.

Also asked in Winter 2025 Q3c OR, Winter 2024 Q3c OR

(b)
4 marksSame type asked in 5 of 7 papers
Solve (D2−4DD′+4D′2)z=e2x+3y(D^2 - 4DD' + 4D'^2)z = e^{2x + 3y}.

Also asked in Summer 2026 Q1b, Winter 2025 Q5a, Summer 2024 Q5b OR, Summer 2023 Q5a

(c)
7 marksAsked in 2 of 7 papers
Find the solution of the wave equation ∂2y∂t2=c2∂2y∂x2\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} such that y=acos⁡pty = a \cos pt when x=lx = l, and y=0y = 0 when x=0x = 0.

Also asked in Summer 2023 Q5c OR

(a)
3 marksSame type asked in 3 of 7 papers
Solve (2D2+5DD′+2D′2)z=0(2D^2 + 5DD' + 2D'^2)z = 0.

Also asked in Summer 2026 Q4b OR, Winter 2024 Q5b OR

(b)
4 marks
Solve p(1+q)=qzp(1 + q) = qz.
(c)
7 marksAsked in 4 of 7 papers
Using the method of separation of variable, find the solution of ∂u∂x=2∂u∂t+u\frac{\partial u}{\partial x} = 2\frac{\partial u}{\partial t} + u, u(x,0)=6e−3xu(x, 0) = 6e^{-3x}.

Also asked in Summer 2026 Q5a OR, Winter 2025 Q5b, Winter 2024 Q5c OR

More Complex Variables and Partial Differential Equations papers

Questions repeat across sessions. Working through two or three papers shows you which topics GTU asks most.

Summer 202670 marks · 5 questions
Winter 202570 marks · 5 questions
Summer 202570 marks · 5 questions
Winter 202470 marks · 5 questions

Other Semester 3 subjects

BE Common

2130002
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2130003
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BE03000051
Professional Communication and Ethics
BE03000061
Indian Constitution

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About this paper

Official Gujarat Technological University (GTU) examination paper for Complex Variables and Partial Differential Equations (CVPD) (Winter 2023, B.E. · Common Engineering, Sem 3). Features complete 70-mark regular & remedial examination pattern, official marking distribution across all 5 questions, and direct 1-click official PDF download.

Transcribed for student exam preparation from Gujarat Technological University official examination archives. Questions, syllabus guidelines, and curriculum marking schemes remain the intellectual property of Gujarat Technological University.

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