Gujarat Technological UniversityWinter 2024 examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Winter 2024 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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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 the principal argument Arg(z)Arg(z) when z=−21+i3z = \frac{-2}{1+i\sqrt{3}}.
(b)
4 marksSame type asked in 3 of 7 papers
Find all the values of following: (1−i)23(1-i)^{\frac{2}{3}}.

Also asked in Summer 2026 Q3a, Winter 2023 Q1b

(c)
7 marksSame type asked in 7 of 7 papers
Find the bilinear transformation which maps the points z=0,−i,−1z = 0, -i, -1 into w=i,1,0w = i, 1, 0 respectively.

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

Q.2

14 Marks
(a)
3 marks
Show that f(z)=z3f(z) = z^3 is analytic everywhere.
(b)
4 marksSame type asked in 4 of 7 papers
Evaluate ∫(x2−iy2) dz\int (x^2 - iy^2) \, dz along the parabola y=2x2y = 2x^2 from (1,2)(1,2) to (2,8)(2,8).

Also asked in Summer 2026 Q2b, Winter 2025 Q2b, Winter 2023 Q3a OR

(c)
7 marksSame type asked in 7 of 7 papers
Find Laurent's series expansion in powers of zz that represent f(z)=1z2(1−z)f(z) = \frac{1}{z^2(1-z)} for the following domains: (i) ∣z∣<1|z| < 1 (ii) ∣z∣>1|z| > 1

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

(c)
7 marksSame type asked in 2 of 7 papers
Find the image of the half-plane x>cx > c, when c>0c > 0 under the transformation w=1zw = \frac{1}{z}. Show the regions graphically.

Also asked in Winter 2025 Q2c

Q.3

14 Marks
(a)
3 marks
If u+ivu + iv is analytic, show that v−iuv - iu and −v+iu-v + iu are also analytic.
(b)
4 marksSame type asked in 3 of 7 papers
Show that u(x,y)=x2−y2+xu(x,y) = x^2 - y^2 + x is harmonic. Find the corresponding analytic function f(z)=u+ivf(z) = u + iv.

Also asked in Summer 2024 Q1b, Winter 2023 Q2c

(c)
7 marksSame type asked in 6 of 7 papers
Evaluate ∫Ccos⁡πzz2−1 dz\int_C \frac{\cos \pi z}{z^2 - 1} \, dz, where CC is the rectangle whose vertices are 2±i,−2±i2 \pm i, -2 \pm i.

Also asked in Summer 2026 Q5b, Summer 2025 Q3a, Summer 2024 Q3c, Winter 2023 Q3c, Winter 2023 Q4b OR, Summer 2023 Q2c, Summer 2023 Q3c OR

(a)
3 marksSame type asked in 6 of 7 papers
Obtain the residue of f(z)=z−3(z+1)(z+2)f(z) = \frac{z - 3}{(z + 1)(z + 2)} at its poles.

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

(b)
4 marksSame type asked in 6 of 7 papers
Evaluate ∫Ce2z(z+1)4 dz\int_C \frac{e^{2z}}{(z + 1)^4} \, dz, where CC is the circle ∣z∣=2|z| = 2.

Also asked in Summer 2026 Q5b, Summer 2025 Q3a, Summer 2024 Q3c, Winter 2023 Q3c, Winter 2023 Q4b OR, Summer 2023 Q2c, Summer 2023 Q3c OR

(c)
7 marksSame type asked in 3 of 7 papers
Evaluate ∫0∞xsin⁡xx2+9 dx\int_0^{\infty} \frac{x \sin x}{x^2 + 9} \, dx using residue.

Also asked in Winter 2025 Q3c OR, Winter 2023 Q5a

Q.4

14 Marks
(a)
3 marksSame type asked in 4 of 7 papers
Form a partial differential equation for the equation (x−a)(y−b)−z2=x2+y2(x-a)(y-b) - z^2 = x^2 + y^2.

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

(b)
4 marksSame type asked in 6 of 7 papers
Solve: xp+yq=3zxp + yq = 3z.

Also asked in Winter 2025 Q4b, Winter 2025 Q4c OR, Summer 2025 Q4c, Summer 2025 Q5b 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 by Charpit's Method: p=(z+qy)2p = (z + qy)^2.

Also asked in Summer 2026 Q4c OR, Winter 2025 Q4c, Summer 2024 Q4c

(a)
3 marksSame type asked in 6 of 7 papers
Form a partial differential equation by eliminating the arbitrary function from z=f(x2−y2)z = f(x^2 - y^2).

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

(b)
4 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, Summer 2024 Q4b, Winter 2023 Q4c OR, Summer 2023 Q4a, Summer 2023 Q4a OR, Summer 2023 Q5a OR

(c)
7 marks
Solve (D2−2DD′+D′2)z=ex+2y+x3(D^2 - 2DD' + D'^2)z = e^{x+2y} + x^3.

Q.5

14 Marks
(a)
3 marksSame type asked in 3 of 7 papers
Solve: p−x2=q+y2p - x^2 = q + y^2.

Also asked in Summer 2025 Q5b, Winter 2023 Q4b

(b)
4 marks
Solve (D2−2DD′+D′2)z=tan⁡(x+y)(D^2 - 2DD' + D'^2)z = \tan(x + y).
(c)
7 marksSame type asked in 3 of 7 papers
Solve (D2+DD′−6D′2)z=sin⁡(2x+y)(D^2 + DD' - 6D'^2)z = \sin(2x + y).

Also asked in Summer 2025 Q4b OR, Winter 2023 Q1c

(a)
3 marks
Solve: (p−q)(z−px−qy)=1(p - q)(z - px - qy) = 1.
(b)
4 marksSame type asked in 3 of 7 papers
Solve: ∂3z∂x3−3∂3z∂x2∂y+2∂3z∂y3=0\frac{\partial^3 z}{\partial x^3} - 3 \frac{\partial^3 z}{\partial x^2 \partial y} + 2 \frac{\partial^3 z}{\partial y^3} = 0.

Also asked in Summer 2026 Q4b OR, Winter 2023 Q5a OR

(c)
7 marksAsked in 4 of 7 papers
Using the method of separation of variables, solve ∂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 2023 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
Summer 202470 marks · 5 questions

Other Semester 3 subjects

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2130002
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2130003
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About this paper

Official Gujarat Technological University (GTU) examination paper for Complex Variables and Partial Differential Equations (CVPD) (Winter 2024, 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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