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Gujarat Technological UniversityWinter 2024 Examination

GTU 3110015 Mathematics - 2 (Maths 2) Winter 2024 Paper Solution & PDF

B.E. · Common Engineering · Semester 1 · Subject Code: 3110015
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Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours10:30 AM – 1:30 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Find aa such that (x+3y)i^+(y−2z)j^+(x+az)k^(x+3y)\hat{i} + (y-2z)\hat{j} + (x+az)\hat{k} is solenoidal.
3 Marks
(b)
Solve yex dx+(2y+ex) dy=0y e^x \, dx + (2y + e^x) \, dy = 0.
4 Marks
(c)
Verify Green’s theorem in a plane for the integral ∫C[(x−2y) dx+x dy]\int_C [(x - 2y) \, dx + x \, dy] taken around the circle x2+y2=4x^2 + y^2 = 4.
7 Marks

Question 2

14 MarksMedium
(a)
Find the Laplace transform of (sin⁡2t−cos⁡2t)2(\sin 2t - \cos 2t)^2.
3 Marks
(b)
Find the Fourier sine integral of f(x)=e−bxf(x) = e^{-bx}.
4 Marks
(c)
Using the Frobenius method, obtain the series solution for 2x(1−x)y′′+(1−x)y′+3y=0about x0=02x(1-x)y'' + (1-x)y' + 3y = 0 \quad \text{about } x_0 = 0
7 Marks
OR OPTION
(c)
Using the method of undetermined coefficients, solve (D2−9)y=x+e2x−sin⁡2x(D^2 - 9)y = x + e^{2x} - \sin 2x
7 Marks

Question 3

14 MarksMedium
(a)
Find the arc length of the curve r⃗(t)=t2i^+t3j^\vec{r}(t) = t^2 \hat{i} + t^3 \hat{j} between (1,1)(1, 1) and (4,8)(4, 8).
3 Marks
(b)
Find the Laplace transform of e−tsin⁡tt\frac{e^{-t} \sin t}{t}
4 Marks
(c)
State the convolution theorem and verify it for f(t)=tf(t) = t and g(t)=e2tg(t) = e^{2t}.
7 Marks
OR OPTION
(a)
Find the inverse Laplace transform of tan⁡−1s\tan^{-1} s.
3 Marks
(b)
Solve x2p2+3xyp+2y2=0x^2 p^2 + 3xyp + 2y^2 = 0.
4 Marks
(c)
Solve the initial value problem using Laplace transformation: y′′−3y′+2y=4twith y(0)=1,  y′(0)=−1y'' - 3y' + 2y = 4t \quad \text{with } y(0) = 1, \; y'(0) = -1
7 Marks

Question 4

14 MarksMedium
(a)
Find the inverse Laplace transform of e−πss2−2s+2\frac{e^{-\pi s}}{s^2 - 2s + 2}
3 Marks
(b)
Solve (D2+1)y=e−x(D^2 + 1)y = e^{-x}.
4 Marks
(c)
Using method of variation of parameters, solve (D2−2D+2)y=extan⁡x(D^2 - 2D + 2)y = e^x \tan x.
7 Marks
OR OPTION
(a)
Classify the singular points of the equation x3(x−2)y′′+x3y′+6y=0x^3 (x - 2) y'' + x^3 y' + 6y = 0.
3 Marks
(b)
Find the Laplace transform of sin⁡t\sin \sqrt{t}.
4 Marks
(c)
Find the series solution of (1+x2)y′′+xy′−9y=0(1 + x^2) y'' + xy' - 9y = 0.
7 Marks

Question 5

14 MarksMedium
(a)
Solve dr+(2rcot⁡θ+sin⁡2θ) dθ=0dr + (2r \cot \theta + \sin 2\theta) \, d\theta = 0.
3 Marks
(b)
If y1=sin⁡xxy_1 = \frac{\sin x}{x} is one of the solution of xy′′+2y′+xy=0x y'' + 2y' + xy = 0, find the second solution.
4 Marks
(c)
Show that:
iJ12(x)=2πxsin⁡xJ_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \sin x
iiJ−12(x)=2πxcos⁡xJ_{-\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \cos x
7 Marks
OR OPTION
(a)
Find the Laplace transform of ∫0t∫0tsin⁡at dt dt\int_0^t \int_0^t \sin at \, dt \, dt
3 Marks
(b)
Solve (x2D2−xD+2)y=6(x^2 D^2 - xD + 2)y = 6.
4 Marks
(c)
Prove that J02(x)+2[J12(x)+J22(x)+J32(x)+… ]=1J_0^2(x) + 2\left[ J_1^2(x) + J_2^2(x) + J_3^2(x) + \dots \right] = 1
7 Marks
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About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Mathematics - 2 (Maths 2) (Winter 2024, B.E. · Common Engineering, Sem 1). 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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