Gujarat Technological UniversitySummer 2025 examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Summer 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 2:30 PM – 5:00 PM
Questions
5 with internal choice
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1 of 25 questions on this paper were also asked in other years. 18 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
Find real and Imaginary parts of z=3i2+3iz = \frac{3i}{2+3i}.
(b)
4 marks
Find zz, if Arg(z+1)=π6\text{Arg}(z + 1) = \frac{\pi}{6} and Arg(z−1)=2π3\text{Arg}(z - 1) = \frac{2\pi}{3}.
(c)
7 marksSame type asked in 3 of 7 papers
Determine an analytic function whose real part is u(x,y)=e2x(xcos⁡2y−ysin⁡2y)u(x, y) = e^{2x}(x \cos 2y - y \sin 2y).

Also asked in Summer 2026 Q2c OR, Summer 2023 Q1a

Q.2

14 Marks
(a)
3 marks
Evaluate lim⁡Z→iZ−iZ2+1\lim_{Z \to i} \frac{Z - i}{Z^2 + 1}.
(b)
4 marks
Find log⁡(1−i)\log(1 - i) and log⁡(1+i)\log(1 + i).
(c)
7 marksSame type asked in 7 of 7 papers
Find the bilinear transformation that maps z1=0,z2=1,z3=∞z_1 = 0, z_2 = 1, z_3 = \infty onto w1=−1,w2=−i,w3=1w_1 = -1, w_2 = -i, w_3 = 1 respectively.

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

(c)
7 marksAsked in 3 of 7 papers
Verify that u=2x−x3+3xy2u = 2x - x^3 + 3xy^2 is harmonic in the whole complex plane and finds its harmonic conjugate function v(x,y)v(x, y).

Also asked in Summer 2026 Q1c, Summer 2023 Q1c

Q.3

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Evaluate ∫Cez(z−1)(z−3) dz\int_C \frac{e^z}{(z-1)(z-3)} \, dz counter clockwise around the circle ∣z∣=2|z| = 2.

Also asked in Summer 2026 Q5b, Winter 2024 Q3c, Winter 2024 Q3b OR, Summer 2024 Q3c, Winter 2023 Q3c, Winter 2023 Q4b OR, Summer 2023 Q2c, Summer 2023 Q3c OR

(b)
4 marksSame type asked in 6 of 7 papers
Determine the residues of the function f(z)=z+1(z2−16)(z+2)f(z) = \frac{z+1}{(z^2-16)(z+2)} at each of its poles.

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

(c)
7 marksSame type asked in 2 of 7 papers
Evaluate ∫CRe(z2) dz\int_C \text{Re}(z^2) \, dz, where CC is the boundary of the square with vertices 0,i,1+i,10, i, 1 + i, 1 in the clockwise wise direction.

Also asked in Summer 2023 Q2a

(a)
3 marksSame type asked in 3 of 7 papers
Classify the poles of f(z)=1z2−z6f(z) = \frac{1}{z^2 - z^6}.

Also asked in Summer 2024 Q3a OR, Winter 2023 Q4a

(b)
4 marksSame type asked in 7 of 7 papers
Find the Laurent's series of the function f(z)=1(z+2)(z+4)f(z) = \frac{1}{(z+2)(z+4)} for the region 2<∣z∣<42 < |z| < 4.

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

(c)
7 marksSame type asked in 2 of 7 papers
Using Residue theorem evaluate the integral ∫0π117−8cos⁡θ dθ\int_0^\pi \frac{1}{17 - 8 \cos \theta} \, d\theta.

Also asked in Summer 2024 Q3c OR

Q.4

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Determine the residues of the function f(z)=3z−4z(z−1)(z−2)f(z) = \frac{3z-4}{z(z-1)(z-2)} at each of its poles.

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

(b)
4 marksSame type asked in 2 of 7 papers
Solve ∂3z∂x2∂y=sin⁡(3x+2y)\frac{\partial^3 z}{\partial x^2 \partial y} = \sin(3x + 2y).

Also asked in Summer 2024 Q5a OR

(c)
7 marksSame type asked in 6 of 7 papers
Solve x2(y−z)p+y2(z−x)q=z2(x−y)x^2(y - z)p + y^2(z - x)q = z^2(x - y).

Also asked in Winter 2025 Q4b, Winter 2025 Q4c 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

(a)
3 marks
Find the fixed points of w=1+1zw = 1 + \frac{1}{z}.
(b)
4 marksSame type asked in 3 of 7 papers
Solve ∂2z∂x2+4∂2z∂x∂y−5∂2z∂y2=sin⁡(2x+3y)\frac{\partial^2 z}{\partial x^2} + 4 \frac{\partial^2 z}{\partial x \partial y} - 5 \frac{\partial^2 z}{\partial y^2} = \sin(2x + 3y).

Also asked in Winter 2024 Q5c, Winter 2023 Q1c

(c)
7 marksSame type asked in 2 of 7 papers
Solve: ∂2z∂x∂y=sin⁡xsin⁡y\frac{\partial^2 z}{\partial x \partial y} = \sin x \sin y, for which ∂z∂y=−2sin⁡y\frac{\partial z}{\partial y} = -2 \sin y when x=0x = 0 and z=0z = 0 when yy is an odd multiple of π2\frac{\pi}{2}.

Also asked in Summer 2024 Q5a OR

Q.5

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Form the partial differential equation by eliminating arbitrary function from z=f(x+at)+g(x−at)z = f(x + at) + g(x - at).

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

(b)
4 marksSame type asked in 3 of 7 papers
Solve p2+q2=x+yp^2 + q^2 = x + y.

Also asked in Winter 2024 Q5a, Winter 2023 Q4b

(c)
7 marksSame type asked in 2 of 7 papers
Using method of separation of variables solve 3∂u∂x+2∂u∂y=03 \frac{\partial u}{\partial x} + 2 \frac{\partial u}{\partial y} = 0, given that u(x,0)=4e−xu(x, 0) = 4e^{-x}.

Also asked in Summer 2026 Q5a

(a)
3 marks
Solve z=px+qy+1+p2+q2z = px + qy + \sqrt{1 + p^2 + q^2}.
(b)
4 marksSame type asked in 6 of 7 papers
Solve y2zxp+xzq=y2\frac{y^2 z}{x} p + xzq = y^2.

Also asked in Winter 2025 Q4b, Winter 2025 Q4c 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 3 of 7 papers
The base of a semi-infinite strip of a metal plate is 30 cm30\text{ cm} and it kept at 100∘C100^\circ\text{C}. The two long edges are at zero temperature. Find the temperature at any point 15 cm15\text{ cm} away from the base and situated midway between the long edges.

Also asked in Summer 2026 Q5c OR, Summer 2024 Q5c

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
Winter 202470 marks · 5 questions
Summer 202470 marks · 5 questions

Other Semester 3 subjects

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2130002
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2130003
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BE03000051
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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) (Summer 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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