UnFoldPrep
Browse Archive
Gujarat Technological UniversitySummer 2023 Examination

GTU 3110015 Mathematics - 2 (Maths 2) Summer 2023 Paper Solution & PDF

B.E. · Common Engineering · Semester 1 · Subject Code: 3110015
Download Official GTU PDF
Share:
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 the directional derivative of f(x,y,z)=x2+5y2+3z2f(x, y, z) = x^2 + 5y^2 + 3z^2 at point (1,1,1)(1, 1, 1) in the direction 3i^+4j^+5k^3\hat{i} + 4\hat{j} + 5\hat{k}.
3 Marks
(b)
A vector field is given by F⃗=3x2yzi^+x3zj^+x3yk^\vec{F} = 3x^2 yz \hat{i} + x^3 z \hat{j} + x^3 y \hat{k}, show that ff is irrotational and find the scalar ϕ\phi such that F⃗=∇ϕ\vec{F} = \nabla \phi.
4 Marks
(c)
Verify the Greens theorem for F⃗=x2i^+xy2j^\vec{F} = x^2 \hat{i} + x y^2 \hat{j}, along the square bounded by x=0,x=1,y=0x = 0, x = 1, y = 0 and y=1y = 1.
7 Marks

Question 2

14 MarksMedium
(a)
Discuss about ordinary point, singular point, regular singular point and irregular singular point of differential equation (x2−1)y′′+3xy′+5x2y=0(x^2 - 1)y'' + 3xy' + 5x^2 y = 0.
3 Marks
(b)
Express f(x)=x3+2x2+3x−1f(x) = x^3 + 2x^2 + 3x - 1 in terms of Legendre’s polynomial.
4 Marks
(c)
Find a power series solution of y′′−25y=0y'' - 25y = 0 near an ordinary point x=0x = 0.
7 Marks
OR OPTION
(c)
Prove that Pn(x)=1n!2ndndxn(x2−1)nP_n(x) = \frac{1}{n! 2^n} \frac{d^n}{dx^n} (x^2 - 1)^n
7 Marks

Question 3

14 MarksMedium
(a)
Find the Laplace transform of:
ie3tt5e^{3t} t^5
iitcos⁡3tt \cos 3t
iiisin⁡25tt\frac{\sin^2 5t}{t}
3 Marks
(b)
Find the Inverse Laplace transform of:
i1(s−1)(s−2)(s−3)\frac{1}{(s-1)(s-2)(s-3)}
ii1s4−9s2\frac{1}{s^4 - 9s^2}
iiis+1s2+2s+10\frac{s+1}{s^2 + 2s + 10}
ivtan⁡−1(s4)\tan^{-1}\left(\frac{s}{4}\right)
4 Marks
(c)
Solve the initial value problem using Laplace transformation: y′′−5y′−6y=e3twith y(0)=3 and y′(0)=2y'' - 5y' - 6y = e^{3t} \quad \text{with } y(0) = 3 \text{ and } y'(0) = 2
7 Marks
OR OPTION
(a)
Find the Laplace transform of:
ie4tu(t−2)e^{4t} u(t - 2)
ii(t2+1)u(t−1)(t^2 + 1) u(t - 1)
iiisin⁡3t u(t−π)\sin 3t \, u(t - \pi)
3 Marks
(b)
Using convolution theorem find the Inverse Laplace transform of 1(s2−4)(s2−9)\frac{1}{(s^2 - 4)(s^2 - 9)}
4 Marks
(c)
Find the Fourier integral of f(x)={0∣x∣<55∣x∣>5f(x) = \begin{cases} 0 & |x| < 5 \\ 5 & |x| > 5 \end{cases}
7 Marks

Question 4

14 MarksMedium
(a)
Solve the Differential equation excos⁡y dx−exsin⁡y dy=0e^x \cos y \, dx - e^x \sin y \, dy = 0.
3 Marks
(b)
Solve d2ydx2+6dydx+9y=0,y(0)=1,  y′(0)=2\frac{d^2 y}{dx^2} + 6\frac{dy}{dx} + 9y = 0, \quad y(0) = 1, \; y'(0) = 2
4 Marks
(c)
Solve:
idydx+ytan⁡x=sin⁡2x,y(0)=2\frac{dy}{dx} + y \tan x = \sin 2x, \quad y(0) = 2
ii(x3+y3) dx−xy2 dy=0(x^3 + y^3) \, dx - x y^2 \, dy = 0
7 Marks
OR OPTION
(a)
Solve (x2−y2) dx+xy dy=0(x^2 - y^2) \, dx + xy \, dy = 0.
3 Marks
(b)
Solve d2ydx2−5dydx−14y=0,y(0)=3,  y′(0)=1\frac{d^2 y}{dx^2} - 5\frac{dy}{dx} - 14y = 0, \quad y(0) = 3, \; y'(0) = 1
4 Marks
(c)
Solve:
i(y−px)(p2+1)=tan⁡−1p(y - px)(p^2 + 1) = \tan^{-1} p
iip2x2=x2+p2p^2 x^2 = x^2 + p^2
7 Marks

Question 5

14 MarksMedium
(a)
Solve d2ydx2+dydx−12y=e−4x−12x\frac{d^2 y}{dx^2} + \frac{dy}{dx} - 12y = e^{-4x} - 12x
3 Marks
(b)
Using method of undetermined coefficient obtain the solution of d2ydx2+4y=sin⁡x\frac{d^2 y}{dx^2} + 4y = \sin x
4 Marks
(c)
Find the solution of differential equations:
1d2ydx2+dydx+y=x2ex\frac{d^2 y}{dx^2} + \frac{dy}{dx} + y = x^2 e^x
2x2d2ydx2−3xdydx+4y=cos⁡(log⁡x)x^2 \frac{d^2 y}{dx^2} - 3x \frac{dy}{dx} + 4y = \cos(\log x)
7 Marks
OR OPTION
(a)
Solve d4ydx4+8d2ydx2+16y=cos⁡2x\frac{d^4 y}{dx^4} + 8\frac{d^2 y}{dx^2} + 16y = \cos 2x
3 Marks
(b)
Using method of undetermined coefficient: d2ydx2+dydx−6y=x2\frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = x^2
4 Marks
(c)
iUsing method of variation of parameters find the solution of differential equation d2ydx2+16y=cot⁡4x\frac{d^2 y}{dx^2} + 16y = \cot 4x
iiFind the solution of differential equation x2d2ydx2−6xdydx+6y=x2+1x2x^2 \frac{d^2 y}{dx^2} - 6x \frac{dy}{dx} + 6y = x^2 + \frac{1}{x^2}
7 Marks
College Exam Groups

Studying for Mathematics - 2?

Circulate this solved paper with KaTeX formulas and 1-click AI step solvers to your batchmates on WhatsApp or Telegram.

Related papers

About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Mathematics - 2 (Maths 2) (Summer 2023, 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.

Download PDF