Gujarat Technological UniversitySummer 2024 examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Summer 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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3 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
Represent (1+i)21−i\frac{(1+i)^2}{1-i} in a+iba+ib or u+ivu+iv form and find its modulus and argument.

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

(b)
4 marksSame type asked in 3 of 7 papers
Define a harmonic function. Show that u(x,y)=x2−y2u(x, y) = x^2 - y^2 is harmonic and find the corresponding analytic function f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y).

Also asked in Winter 2024 Q3b, Winter 2023 Q2c

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

Also asked in Summer 2026 Q2c, Winter 2025 Q1c, Summer 2025 Q2c, Winter 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 ∫04+2izˉ dz\int_0^{4+2i} \bar{z} \, dz along the curve z=t2+itz = t^2 + it.

Also asked in Winter 2025 Q2a

(b)
4 marksSame type asked in 3 of 7 papers
Find the centre and radius of convergence of the given power series:
∑n=0∞(2n)!(n!)2(z−3i)n\sum_{n=0}^\infty \frac{(2n)!}{(n!)^2}(z - 3i)^n

Also asked in Winter 2025 Q3b, Winter 2023 Q3a

(c)
7 marks
Define analytic function and If f(z)f(z) is analytic and ∣f(z)∣=c|f(z)| = c then show that f(z)f(z) is constant.
(c)
7 marksSame type asked in 4 of 7 papers
Evaluate ∮Ccos⁡πz2(z−1)(z−2) dz\oint_C \frac{\cos \pi z^2}{(z-1)(z-2)} \, dz; CC is ∣z∣=3|z| = 3.

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

Q.3

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

Also asked in Summer 2026 Q2a, Winter 2025 Q3a OR

(b)
4 marksSame type asked in 7 of 7 papers
Find the Laurent series of
7z−2(z+1)z(z−2);1<∣z+1∣<3\frac{7z - 2}{(z+1)z(z-2)}; \quad 1 < |z+1| < 3

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

(c)
7 marksSame type asked in 6 of 7 papers
Evaluate using Cauchy residue theorem ∮C2z+6z2+4 dz\oint_C \frac{2z + 6}{z^2 + 4} \, dz where C:∣z−i∣=2C: |z - i| = 2.

Also asked in Summer 2026 Q5b, Summer 2025 Q3a, Winter 2024 Q3c, Winter 2024 Q3b OR, Winter 2023 Q3c, Winter 2023 Q4b OR, Summer 2023 Q2c, Summer 2023 Q3c OR

(a)
3 marksSame type asked in 3 of 7 papers
Find the pole and its order of following functions:
1f(z)=sin⁡zz4f(z) = \frac{\sin z}{z^4}
2f(z)=1(z−5)3(z2−4)f(z) = \frac{1}{(z-5)^3 (z^2 - 4)}

Also asked in Summer 2025 Q3a OR, Winter 2023 Q4a

(b)
4 marksSame type asked in 6 of 7 papers
Find the residues at singular points of
f(z)=z2(z−1)2(z+2)f(z) = \frac{z^2}{(z-1)^2 (z+2)}

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

(c)
7 marksSame type asked in 2 of 7 papers
Evaluate following real integration using residue theorem:
∫02π1(2+cos⁡θ)2 dθ\int_0^{2\pi} \frac{1}{(2 + \cos \theta)^2} \, d\theta

Also asked in Summer 2025 Q3c OR

Q.4

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Form PDE by eliminating arbitrary Functions F(x+y+z,x2+y2−z2)=0F(x + y + z, x^2 + y^2 - z^2) = 0.

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

(b)
4 marksSame type asked in 6 of 7 papers
Solve following Linear Partial Differential Equations:
1xp+yq=x−yxp + yq = x - y
2(z−y)p+(x−z)q=y−x(z - y)p + (x - z)q = y - x

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

(a)
3 marks
Find the order of the following PDE (1 to 3). And check whether the equations are linear, quasilinear or nonlinear:
1∂u∂x−∂u∂t=0\frac{\partial u}{\partial x} - \frac{\partial u}{\partial t} = 0
2∂2u∂x2−e2x∂2u∂t2=u3\frac{\partial^2 u}{\partial x^2} - e^{2x}\frac{\partial^2 u}{\partial t^2} = u^3
3uyuyy+(ux)2=0u_y u_{yy} + (u_x)^2 = 0
(b)
4 marksAsked in 2 of 7 papers
Solve following Non-linear partial differential equations:
1p2−q2=x−yp^2 - q^2 = x - y
2p(1+q)=qzp(1 + q) = qz

Also asked in Winter 2025 Q4b OR

(c)
7 marks
Solve following Partial Differential Equations:
1(x2−y2−z2)p+2xyq=2xz(x^2 - y^2 - z^2)p + 2xyq = 2xz
2∂2z∂x2−2∂2z∂x∂y−8∂2z∂y2=0\frac{\partial^2 z}{\partial x^2} - 2\frac{\partial^2 z}{\partial x \partial y} - 8\frac{\partial^2 z}{\partial y^2} = 0

Q.5

14 Marks
(a)
3 marksSame type asked in 3 of 7 papers
Classify second order homogeneous partial differential equations as elliptic, parabolic or hyperbolic:
1∂2z∂x2−∂2z∂x∂y−12∂2z∂y2=0\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} - 12\frac{\partial^2 z}{\partial y^2} = 0
2∂2z∂x2−∂2z∂x∂y=0\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} = 0

Also asked in Summer 2026 Q4b, Winter 2025 Q5a OR

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

Also asked in Summer 2023 Q5b OR

(c)
7 marksSame type asked in 3 of 7 papers
Determine solution of two-dimensional Laplace equation ∂2u∂x2+∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 which satisfy the condition u(0,y)=u(l,y)=u(x,0)u(0, y) = u(l, y) = u(x, 0) and u(x,b)=sin⁡nπxlu(x, b) = \sin \frac{n\pi x}{l}.

Also asked in Summer 2026 Q5c OR, Summer 2025 Q5c OR

(a)
3 marksSame type asked in 2 of 7 papers
Solve partial differential equations by direct integration method: ∂2z∂x2=sin⁡x\frac{\partial^2 z}{\partial x^2} = \sin x.

Also asked in Summer 2025 Q4b, Summer 2025 Q4c OR

(b)
4 marksSame type asked in 5 of 7 papers
Solve second order homogeneous PDEs:
∂2z∂x2−∂2z∂x∂y−12∂2z∂y2=3e2x−3y\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} - 12\frac{\partial^2 z}{\partial y^2} = 3e^{2x - 3y}

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

(c)
7 marksAsked in 2 of 7 papers
Find the solution of One-Dimensional Wave Equation.

Also asked in Winter 2025 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

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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 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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