Gujarat Technological UniversityWinter 2025 examination

GTU 2130002 Advance Engineering Mathematics (AEM) Winter 2025 Question Paper & PDF

B.E. · Common Engineering · Semester 3 · Subject code 2130002
Download official PDFView PDF here
Share with your class
Total
70 marks
To pass
23 marks (33%)
Duration
2.5 Hours 10:30 AM – 1:30 PM
Questions
5 with internal choice
No solutions on this paper yetNeed them for your exam? Ask once. Papers that students ask for are solved first, and we aim to add them within 24 hours.

Q.1

14 Marks
(a)
3 marks
Define Signum function, Dirac's Delta function and Beta function.
(b)
4 marks
Solve d3ydx3+dydx=cosec⁡x\frac{d^3 y}{dx^3} + \frac{dy}{dx} = \operatorname{cosec} x by method of variation of parameters.
(c)
7 marks
Obtain the Fourier series of f(x)=(π−x2)2f(x) = \left(\frac{\pi - x}{2}\right)^2 in the interval 0≤x≤2π0 \le x \le 2\pi. Hence deduce that π212=112−122+132−…\frac{\pi^2}{12} = \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \dots

Q.2

14 Marks
(a)
3 marks
Find half range cosine series for f(x)=x,0<x<3f(x) = x, 0 < x < 3.
(b)
4 marks
Solve d3ydx3−d2ydx2+3dydx+5y=excos⁡3x\frac{d^3 y}{dx^3} - \frac{d^2 y}{dx^2} + 3\frac{dy}{dx} + 5y = e^x \cos 3x.
(c)
7 marks
Find the series solution of y′′=2y′y'' = 2y' in powers of xx.
(c)
7 marks
Find the series solution of (1+x2)y′′+xy′−xy=0(1 + x^2)y'' + xy' - xy = 0.

Q.3

14 Marks
(a)
3 marks
Solve dydx+2ytan⁡x=sin⁡x\frac{dy}{dx} + 2y\tan x = \sin x.
(b)
4 marks
Solve (x3+3xy2)dx+(3x2y+y3)dy=0(x^3 + 3xy^2)dx + (3x^2 y + y^3)dy = 0.
(c)
7 marks
Solve (D3−D2−6D)y=x2+1(D^3 - D^2 - 6D)y = x^2 + 1.
(a)
3 marks
Solve dydx+ycos⁡x+sin⁡y+ysin⁡x+xcos⁡y+x=0\frac{dy}{dx} + \frac{y\cos x + \sin y + y}{\sin x + x\cos y + x} = 0.
(b)
4 marks
Solve (x2y2+2)y dx+(2−x2y2)x dy=0(x^2 y^2 + 2)y \, dx + (2 - x^2 y^2)x \, dy = 0.
(c)
7 marks
Solve (D4+2D3−3D2)y=x2+3e2x(D^4 + 2D^3 - 3D^2)y = x^2 + 3e^{2x}.

Q.4

14 Marks
(a)
3 marks
Find the Laplace Transform of e−2t(sin⁡4t+t2)e^{-2t}(\sin 4t + t^2).
(b)
4 marks
Find the Inverse Laplace Transform of log⁡(1+ω2s2)\log\left(1 + \frac{\omega^2}{s^2}\right).
(c)
7 marks
Find the Inverse Laplace transform of 5s+3(s−1)(s2+2s+5)\frac{5s + 3}{(s - 1)(s^2 + 2s + 5)} using partial fractions.
(a)
3 marks
Find the Laplace Transform of sin⁡2tsin⁡3t\sin 2t \sin 3t.
(b)
4 marks
Find the Laplace Transform of t2cosh⁡πtt^2 \cosh \pi t.
(c)
7 marks
Solve the initial value problem d2xdt2+2dxdt+5x=e−tsin⁡t\frac{d^2 x}{dt^2} + 2\frac{dx}{dt} + 5x = e^{-t}\sin t where x(0)=0x(0) = 0 and x′(0)=1x'(0) = 1 using Laplace transform.

Q.5

14 Marks
(a)
3 marks
Form a partial differential equation by eliminating the arbitrary constants for the equation z=(x−2)2+(y−3)2z = (x - 2)^2 + (y - 3)^2.
(b)
4 marks
Solve pz−qz=z2+(x+y)2pz - qz = z^2 + (x + y)^2 using Lagrange's method.
(c)
7 marks
Solve ∂2z∂x2−2∂z∂x+∂z∂y=0\frac{\partial^2 z}{\partial x^2} - 2\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 using method of separation of variables.
(a)
3 marks
Solve p(1+q)=qzp(1 + q) = qz.
(b)
4 marks
Solve the equation by method of direct integration: ∂2z∂x∂y=sin⁡xsin⁡y\frac{\partial^2 z}{\partial x \partial y} = \sin x \sin y, given that ∂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}.
(c)
7 marks
Solve ∂u∂x+∂u∂y=2(x+y)u\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 2(x + y)u using method of separation of variables.

More Advance Engineering Mathematics papers

Questions repeat across sessions. Working through two or three papers shows you which topics GTU asks most.

Summer 202570 marks · 5 questions
Winter 202470 marks · 5 questions
Summer 202470 marks · 5 questions

Other Semester 3 subjects

BE Common

2130003
Mechanics of Solids
3130005
Complex Variables and Partial Differential Equations
BE03000051
Professional Communication and Ethics
BE03000061
Indian Constitution

Preparing with friends?

Send this paper to your class group. Every question has its own link.

About this paper

Official Gujarat Technological University (GTU) examination paper for Advance Engineering Mathematics (AEM) (Winter 2025, 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.

Download official PDF All Advance Engineering Mathematics Papers (4) →Report a typo or correction
ViewDownload PDF