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Gujarat Technological UniversitySummer 2025 Examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Summer 2025 Paper Solution & PDF

B.E. · Common Engineering · Semester 2 · Subject Code: 3130005
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Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours2:30 PM – 5:00 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Find real and Imaginary parts of z=3i2+3iz = \frac{3i}{2+3i}.
3 Marks
(b)
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}.
4 Marks
(c)
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).
7 Marks

Question 2

14 MarksMedium
(a)
Evaluate lim⁡Z→iZ−iZ2+1\lim_{Z \to i} \frac{Z - i}{Z^2 + 1}.
3 Marks
(b)
Find log⁡(1−i)\log(1 - i) and log⁡(1+i)\log(1 + i).
4 Marks
(c)
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.
7 Marks
OR OPTION
(c)
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).
7 Marks

Question 3

14 MarksMedium
(a)
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.
3 Marks
(b)
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.
4 Marks
(c)
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.
7 Marks
OR OPTION
(a)
Classify the poles of f(z)=1z2−z6f(z) = \frac{1}{z^2 - z^6}.
3 Marks
(b)
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.
4 Marks
(c)
Using Residue theorem evaluate the integral ∫0π117−8cos⁡θ dθ\int_0^\pi \frac{1}{17 - 8 \cos \theta} \, d\theta.
7 Marks

Question 4

14 MarksMedium
(a)
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.
3 Marks
(b)
Solve ∂3z∂x2∂y=sin⁡(3x+2y)\frac{\partial^3 z}{\partial x^2 \partial y} = \sin(3x + 2y).
4 Marks
(c)
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).
7 Marks
OR OPTION
(a)
Find the fixed points of w=1+1zw = 1 + \frac{1}{z}.
3 Marks
(b)
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).
4 Marks
(c)
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}.
7 Marks

Question 5

14 MarksMedium
(a)
Form the partial differential equation by eliminating arbitrary function from z=f(x+at)+g(x−at)z = f(x + at) + g(x - at).
3 Marks
(b)
Solve p2+q2=x+yp^2 + q^2 = x + y.
4 Marks
(c)
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}.
7 Marks
OR OPTION
(a)
Solve z=px+qy+1+p2+q2z = px + qy + \sqrt{1 + p^2 + q^2}.
3 Marks
(b)
Solve y2zxp+xzq=y2\frac{y^2 z}{x} p + xzq = y^2.
4 Marks
(c)
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.
7 Marks
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About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Complex Variables and Partial Differential Equations (CVPD) (Summer 2025, B.E. · Common Engineering, Sem 2). 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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