Gujarat Technological UniversityWinter 2025 examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Winter 2025 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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4 of 25 questions on this paper were also asked in other years. 16 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 marks
Show that ∣z∣z∣−1∣≤∣arg⁡(z)∣|\frac{z}{|z|} - 1| \le |\arg(z)|.
(b)
4 marks
Prove that u=x2−y2u = x^2 - y^2 and v=yx2+y2v = \frac{y}{x^2+y^2} are harmonic functions of (x,y)(x, y) but are not harmonic conjugates.
(c)
7 marksSame type asked in 7 of 7 papers
Find the bilinear transformation which maps the points z=1,i,−1z = 1, i, -1 into w=i,0,−iw = i, 0, -i respectively. Find invariant points and image of ∣z∣<1|z| < 1 under this mapping.

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

Q.2

14 Marks
(a)
3 marksSame type asked in 2 of 7 papers
Evaluate the integral ∫Czˉ dz\int_C \bar{z} \, dz, where CC is right half of the circle z=2eiθ(−π2<θ<π2)z = 2e^{i\theta} \left(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\right).

Also asked in Summer 2024 Q2a

(b)
4 marksSame type asked in 4 of 7 papers
Evaluate ∫03+iz2 dz\int_0^{3+i} z^2 \, dz along the parabola x=3y2x = 3y^2.

Also asked in Summer 2026 Q2b, Winter 2024 Q2b, Winter 2023 Q3a OR

(c)
7 marksSame type asked in 2 of 7 papers
Explain inversion transformation. Find the image of 14≤y≤12\frac{1}{4} \le y \le \frac{1}{2} under mapping w=1/zw = 1/z. Show this translation on graph.

Also asked in Winter 2024 Q2c OR

(c)
7 marksSame type asked in 4 of 7 papers
State Cauchy integral theorem and Cauchy integral formula. Evaluate ∮csin⁡πz2+cos⁡πz2(z−1)(z−2) dz\oint_c \frac{\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)} \, dz, where c:∣z∣=3c : |z| = 3.

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

Q.3

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Determine the poles of the function and the residue at each pole: f(z)=z2(z−2)2(z−1)f(z) = \frac{z^2}{(z-2)^2(z-1)}.

Also asked in Summer 2026 Q3c OR, Summer 2025 Q3b, Summer 2025 Q4a, Winter 2024 Q3a OR, Summer 2024 Q3b OR, Summer 2023 Q3c

(b)
4 marksSame type asked in 3 of 7 papers
Find the radii of convergence and region of convergence of the following:
i∑n=1∞zn2n+1\sum_{n=1}^\infty \frac{z^n}{2^{n+1}}
ii∑n=1∞n!nnzn\sum_{n=1}^\infty \frac{n!}{n^n} z^n

Also asked in Summer 2024 Q2b, Winter 2023 Q3a

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

Find the Laurent series of 1z(z2−3z+2)\frac{1}{z(z^2 - 3z + 2)} for region:

i0<∣z∣<10 < |z| < 1
ii1<∣z∣<21 < |z| < 2
iii∣z∣>2|z| > 2

Also asked in Summer 2026 Q3c, Summer 2025 Q3b OR, Winter 2024 Q2c, Summer 2024 Q3b, Winter 2023 Q4c, Summer 2023 Q2b, Summer 2023 Q3a OR

(a)
3 marksAsked in 3 of 7 papers
Define: Singular point, Isolated singular point, Residue. Explain types of isolated singular points.

Also asked in Summer 2026 Q2a, Summer 2024 Q3a

(b)
4 marksSame type asked in 3 of 7 papers
Expand f(z)=z−1z+1f(z) = \frac{z-1}{z+1} as Taylor series about the following points: a) z=0z = 0, b) z=1z = 1.

Also asked in Summer 2026 Q3b OR, Winter 2023 Q3b OR

(c)
7 marksSame type asked in 3 of 7 papers
Evaluate following real integration using residue theorem: ∫−∞∞cos⁡xx2+1 dx\int_{-\infty}^\infty \frac{\cos x}{x^2 + 1} \, dx.

Also asked in Winter 2024 Q3c OR, Winter 2023 Q5a

Q.4

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Eliminate the arbitrary function from the equation z=xy+f(x2+y2)z = xy + f(x^2 + y^2).

Also asked in Summer 2026 Q4a, Summer 2025 Q5a, Winter 2024 Q4a OR, Summer 2024 Q4a, Summer 2023 Q4b OR

(b)
4 marksSame type asked in 6 of 7 papers
Solve y2p−xyq=x(z−2y)y^2 p - xyq = x(z - 2y).

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

(c)
7 marksSame type asked in 4 of 7 papers
Solve following non-linear partial differential equations using Charpit's method: px+qy=pqpx + qy = pq.

Also asked in Summer 2026 Q4c OR, Winter 2024 Q4c, Summer 2024 Q4c

(a)
3 marks
Solve: ∂2z∂x2+z=0\frac{\partial^2 z}{\partial x^2} + z = 0 given that z=eyz = e^y and ∂z∂x=1\frac{\partial z}{\partial x} = 1 when x=0x = 0.
(b)
4 marksAsked in 2 of 7 papers
Solve following non-linear partial differential equations:
ap(1+q)=qzp(1 + q) = qz
bp2−q2=x−yp^2 - q^2 = x - y

Also asked in Summer 2024 Q4b OR

(c)
7 marksSame type asked in 6 of 7 papers
Solve the following Partial differential Equation by Lagrange's Method: (x2−yz)p+(y2−zx)q=(z2−xy)(x^2 - yz)p + (y^2 - zx)q = (z^2 - xy).

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

Q.5

14 Marks
(a)
3 marksSame type asked in 5 of 7 papers
Find the Particular integral of 4r+12s+9t=e3x−2y4r + 12s + 9t = e^{3x - 2y}.

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

(b)
4 marksAsked in 4 of 7 papers
Using method of separation of variables solve ∂u∂x=2∂u∂t+u\frac{\partial u}{\partial x} = 2 \frac{\partial u}{\partial t} + u given that u(x,0)=6e−3xu(x, 0) = 6e^{-3x}.

Also asked in Summer 2026 Q5a OR, Winter 2024 Q5c OR, Winter 2023 Q5c OR

(c)
7 marks
Show that u=sin⁡9tsin⁡(x/4)u = \sin 9t \sin(x/4) is a solution of a one dimensional wave equation.
(a)
3 marksSame type asked in 3 of 7 papers
Classify following second order homogeneous partial differential equations as elliptic, parabolic or hyperbolic:
a∂2u∂t2+4∂2u∂x∂t+4∂2u∂x2=0\frac{\partial^2 u}{\partial t^2} + 4 \frac{\partial^2 u}{\partial x \partial t} + 4 \frac{\partial^2 u}{\partial x^2} = 0
b2∂2u∂t2+4∂2u∂x∂t+3∂2u∂x2=02 \frac{\partial^2 u}{\partial t^2} + 4 \frac{\partial^2 u}{\partial x \partial t} + 3 \frac{\partial^2 u}{\partial x^2} = 0

Also asked in Summer 2026 Q4b, Summer 2024 Q5a

(b)
4 marks
Solve ∂2z∂x2+3∂2z∂x∂y+2∂2z∂y2=x+y\frac{\partial^2 z}{\partial x^2} + 3 \frac{\partial^2 z}{\partial x \partial y} + 2 \frac{\partial^2 z}{\partial y^2} = x + y.
(c)
7 marksAsked in 2 of 7 papers
Derive the solutions of one dimensional wave equation.

Also asked in Summer 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
Summer 202570 marks · 5 questions
Winter 202470 marks · 5 questions
Summer 202470 marks · 5 questions

Other Semester 3 subjects

BE Common

2130002
Advance Engineering Mathematics
2130003
Mechanics of Solids
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 2025, 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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