Gujarat Technological UniversitySummer 2023 examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Summer 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 2:30 PM – 5:00 PM
Questions
5 with internal choice
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2 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 marksSame type asked in 3 of 7 papers
Determine an analytic function whose real part is e2x(xcos⁡2y−ysin⁡2y)e^{2x}(x \cos 2y - y \sin 2y).

Also asked in Summer 2026 Q2c OR, Summer 2025 Q1c

(b)
4 marks
Solve the equation z2+(2i−3)z+5−i=0z^2 + (2i - 3)z + 5 - i = 0.
(c)
7 marksAsked in 3 of 7 papers
Show that u(x,y)=2x−x3+3xy2u(x, y) = 2x - x^3 + 3xy^2 is harmonic and find a harmonic conjugate of u(x,y)u(x, y).

Also asked in Summer 2026 Q1c, Summer 2025 Q2c OR

Q.2

14 Marks
(a)
3 marksSame type asked in 2 of 7 papers
Evaluate ∫C∣z∣2 dz\int_C |z|^2 \, dz around the square with vertices at (0,0),(1,0),(1,1),(0,1)(0, 0), (1, 0), (1, 1), (0, 1).

Also asked in Summer 2025 Q3c

(b)
4 marksSame type asked in 7 of 7 papers

Expand f(z)=1(z+2)(z+4)f(z) = \frac{1}{(z+2)(z+4)} valid for the following regions:

i∣z∣<2|z| < 2
ii2<∣z∣<42 < |z| < 4

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

(c)
7 marksSame type asked in 6 of 7 papers
iEvaluate ∫Cz dz(z−1)(z−2)\int_C \frac{z \, dz}{(z-1)(z-2)} where CC is the circle ∣z∣=12|z| = \frac{1}{2}.
iiEvaluate ∫Cdzz2−7z+12\int_C \frac{dz}{z^2 - 7z + 12} where CC is the circle ∣z∣=3.5|z| = 3.5.

Also asked in Summer 2026 Q5b, Summer 2025 Q3a, Winter 2024 Q3c, Winter 2024 Q3b OR, Summer 2024 Q3c, Winter 2023 Q3c, Winter 2023 Q4b OR

(c)
7 marksSame type asked in 7 of 7 papers
Define mobius transformation. Determine the mobius transformation which 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.

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

Q.3

14 Marks
(a)
3 marks
Find and plot the image of triangular region in the zz-plane with vertices (0,0),(1,0),(0,1)(0, 0), (1, 0), (0, 1) under the transformation w=(1−i)z+3w = (1 - i)z + 3.
(b)
4 marks
Find the values of aa and bb such that the function f(z)=x2+ay2−2xy+i(bx2−y2+2xy)f(z) = x^2 + ay^2 - 2xy + i(bx^2 - y^2 + 2xy) is analytic.
(c)
7 marksSame type asked in 6 of 7 papers
Determine the poles of the function f(z)=z2(z−1)2(z+2)f(z) = \frac{z^2}{(z-1)^2 (z+2)} and residue at each pole. Hence evaluate ∫Cf(z) dz\int_C f(z) \, dz where CC is the circle ∣z∣=3|z| = 3.

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

(a)
3 marksSame type asked in 7 of 7 papers
Expand f(z)=1−ezzf(z) = \frac{1 - e^z}{z} in Laurent’s series about z=0z = 0.

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

(b)
4 marksSame type asked in 4 of 7 papers
Find modulus and argument of:
i1+2i1−(1−i)2\frac{1 + 2i}{1 - (1-i)^2}
ii(1+i)21−i\frac{(1+i)^2}{1-i}

Also asked in Summer 2026 Q1a, Summer 2024 Q1a, Winter 2023 Q1a

(c)
7 marksSame type asked in 6 of 7 papers
iEvaluate ∫C3z2+7z+1z+1 dz\int_C \frac{3z^2 + 7z + 1}{z+1} \, dz where CC is ∣z∣=12|z| = \frac{1}{2}.
iiEvaluate ∫Cz2+1z2−1 dz\int_C \frac{z^2 + 1}{z^2 - 1} \, dz where CC is ∣z−1∣=1|z - 1| = 1.

Also asked in Summer 2026 Q5b, Summer 2025 Q3a, Winter 2024 Q3c, Winter 2024 Q3b OR, Summer 2024 Q3c, Winter 2023 Q3c, Winter 2023 Q4b OR

Q.4

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Solve yzp−xzq=xyyz p - xz q = xy.

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, Winter 2023 Q4c OR

(b)
4 marksSame type asked in 4 of 7 papers
Form partial differential equation by eliminating the arbitrary constants aa and bb from z=axey+12a2e2y+bz = a x e^y + \frac{1}{2} a^2 e^{2y} + b.

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

(c)
7 marks
iSolve 25r−40s+16t=025r - 40s + 16t = 0.
iiSolve p2+q2=x+yp^2 + q^2 = x + y.
(a)
3 marksSame type asked in 6 of 7 papers
Solve (mz−ny)p+(nx−lz)q=ly−mx(mz - ny)p + (nx - lz)q = ly - mx.

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, Winter 2023 Q4c OR

(b)
4 marksSame type asked in 6 of 7 papers
Form a partial differential equation by eliminating the arbitrary functions from f(x2−y2,xyz)=0f(x^2 - y^2, xyz) = 0.

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

(c)
7 marks
iSolve (D2−DD′+D′−1)z=cos⁡(x+2y)(D^2 - DD' + D' - 1)z = \cos(x + 2y).
iiSolve using Charpit’s Method: z2=pqxyz^2 = pqxy.

Q.5

14 Marks
(a)
3 marksSame type asked in 5 of 7 papers
Solve (D2+10DD′+25D′2)z=e3x+2y(D^2 + 10DD' + 25D'^2)z = e^{3x+2y}.

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

(b)
4 marks
Solve x∂u∂x−2y∂u∂y=0x \frac{\partial u}{\partial x} - 2y \frac{\partial u}{\partial y} = 0 using method of separation of variables.
(c)
7 marks
iSolve (D2−D′2)z=x−y(D^2 - D'^2)z = x - y.
iiSolve (2D2−5DD′+2D′2)z=sin⁡(2x+y)(2D^2 - 5DD' + 2D'^2)z = \sin(2x + y).
(a)
3 marksSame type asked in 6 of 7 papers
Solve (1−x)p+(z−y)q=3−z(1 - x)p + (z - y)q = 3 - z.

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, Winter 2023 Q4c OR

(b)
4 marksSame type asked in 2 of 7 papers
Solve 2∂u∂x=∂u∂t+u2\frac{\partial u}{\partial x} = \frac{\partial u}{\partial t} + u using method of separation of variables subject to the condition u(x,0)=4e−3xu(x, 0) = 4e^{-3x}.

Also asked in Summer 2024 Q5b

(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 Winter 2023 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
Summer 202570 marks · 5 questions
Winter 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) (Summer 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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