Gujarat Technological UniversityWinter 2025 examination

GTU BE02000011 Mathematics – 2 Winter 2025 Question Paper & PDF

B.E. · Common Engineering · Semesters 1 and 2 · Subject code BE02000011
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Total
70 marks
To pass
23 marks (33%)
Duration
2.5 Hours 2:30 PM – 5:30 PM
Questions
5 with internal choice
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Q.1

14 Marks
(a)
3 marks
Check whether the function ezˉe^{\bar{z}} analytic or not at any point ?
(b)
4 marks
Solve xlog⁡xdydx+y=2log⁡xx\log x\dfrac{dy}{dx} + y = 2\log x
(c)
7 marks
Diagonalize the matrix A=[161120003]A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}

Q.2

14 Marks
(a)
3 marks
State Cauchy integral formula and Evaluate ∮Cdzz(z+3)\oint_C \dfrac{dz}{z(z + 3)} Where C is the circle ∣z∣=1|z| = 1
(b)
4 marks
Solve y′′−4y=0,y(0)=2,y′(0)=−1y'' - 4y = 0, y(0) = 2, y'(0) = -1
(c)
7 marks
iEvaluate ∮C(x−y+ix2) dz\oint_C (x - y + ix^2)\,dz Where C is a straight line from z=0z=0 to z=1+iz = 1+i.
2If sin⁡(α+iβ)=x+iy\sin(\alpha + i\beta) = x + iy then prove that x2sech⁡2β+y2cosech⁡2β=1x^2 \operatorname{sech}^2\beta + y^2 \operatorname{cosech}^2\beta = 1
(c)
7 marks
iEvaluate ∮Cez(z−1)4 dz\oint_C \dfrac{e^z}{(z - 1)^4}\,dz where C is the circle ∣z∣=3|z| = 3
iiUsing CR equations show that z∣z∣z|z| is no where analytic.

Q.3

14 Marks
(a)
3 marks
Solve dydx+2xy=xe−x2\dfrac{dy}{dx} + 2xy = xe^{-x^2}
(b)
4 marks
Find the rank of the matrix A=[1−12−3410203140102]A = \begin{bmatrix} 1 & -1 & 2 & -3 \\ 4 & 1 & 0 & 2 \\ 0 & 3 & 1 & 4 \\ 0 & 1 & 0 & 2 \end{bmatrix}
(c)
7 marks
Check whether u=x2−y2−yu = x^2 - y^2 - y is harmonic or not. If so find the harmonic conjugate vv of uu.
(a)
3 marks
Test for exactness and solve (tan⁡y+x) dx+(xsec⁡2y−3y) dy=0(\tan y+x)\,dx+ (x\sec^2 y-3y)\,dy=0
(b)
4 marks

Solve the following linear system by Gauss elimination method. x+2y−z=2,x−y+z=4,2x+y−z=5x + 2y - z = 2, x - y + z = 4, 2x + y - z = 5

(c)
7 marks
Define Harmonic function. Check whether u=x3−3xy2u = x^3 - 3xy^2 is Harmonic or not if so construct the analytic function f(z)=u+ivf(z) = u + iv

Q.4

14 Marks
(a)
3 marks
Solve (xy2−x) dx=(y+x2y) dy(xy^2 - x)\,dx = (y + x^2 y)\,dy
(b)
4 marks
Find inverse of the matrix A=[122212221]A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} by Cayley Hamilton theorem.
(c)
7 marks

State Cauchy residue theorem and Evaluate ∮Cz3−z2+z−1z3+4z dz\oint_C \dfrac{z^3 - z^2 + z - 1}{z^3 + 4z}\,dz, Where C is

a∣z∣=1|z| = 1, (b) ∣z∣=3|z| = 3
(a)
3 marks
Solve xdydx+y2x=yx\dfrac{dy}{dx} + \dfrac{y^2}{x} = y
(b)
4 marks
Find the inverse of the matrix A=[012123311]A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix} using Gauss Jordan method.
(c)
7 marks

Expand f(z)=1(z+1)(z−2)f(z) = \dfrac{1}{(z + 1)(z - 2)} in Laurent's series in the region

1∣z∣<1|z| < 1 (2) 1<∣z∣<21 < |z| < 2 (3) ∣z∣>2|z| > 2

Q.5

14 Marks
(a)
3 marks
Solve (x2+y2+x) dx+xy dy=0(x^2 + y^2 + x)\,dx + xy\,dy = 0
(b)
4 marks
Solve y′′−2y′+y=4exx2y'' - 2y' + y = 4\dfrac{e^x}{x^2}
(c)
7 marks
Solve y′′−6y′+9y=e3xx2y'' - 6y' + 9y = \dfrac{e^{3x}}{x^2} by the method of variation of Parameters.
(a)
3 marks
Solve y=3x+log⁡py = 3x + \log p
(b)
4 marks
Solve y′′+y′−6y=6x+3x2−6x3y'' + y' - 6y = 6x + 3x^2 - 6x^3 by undetermined coefficient method
(c)
7 marks
Find the power series solution of y′′+x2y=0y'' + x^2 y = 0 about x=0x = 0.

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About this paper

Official Gujarat Technological University (GTU) examination paper for Mathematics – 2 (Winter 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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