What is the exam pattern and passing mark for GTU Mathematics 3 (Mechanical) (BE03000211)?

The GTU theory examination carries 70 total marks with a duration of 2.5 Hours. Minimum passing criteria is 23 marks (33%). The paper consists of 5 main questions with official marking breakdown and internal choices.

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Gujarat Technological UniversityWinter 2025 examination

GTU BE03000211 Mathematics 3 (Mechanical) Winter 2025 Question Paper & PDF

B.E. · Mechanical Engineering · Semester 3 · Subject code BE03000211

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Total: 70 marks
To pass: 23 marks (33%)
Duration: 2.5 Hours 10:30 AM – 1:00 PM
Questions: 5 with internal choice

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Q.1

14 Marks

(a)

3 marks

1State Dirichlet's conditions for convergence of Fourier series.

2Shw that

1 + \Delta = E

3What is the order of error in the trapezoidal rule ?

(b)

4 marks

1Show that

u = e^x\cos y

is a solution of Laplace equation.

2Form PDE by eliminating arbitrary constants

a

and

b

from

z = (x + a)(y + b)

(c)

7 marks

1Solve

\dfrac{dy}{dx} = 3 + 2xy

where

y(0) = 1

for

x = 0.1

by Picard's method.

2Find the Laplace transform of f(t) defined as

f(t) = \begin{cases} t, & \text{if } 0 < t < 4 \\ 5, & \text{if } t > 4. \end{cases}

Q.2

14 Marks

(a)

3 marks

Solve:

(y + z)p + (z + x)q = x + y

(b)

4 marks

Solve:

\dfrac{\partial^2 z}{\partial x^2} + 3\dfrac{\partial^2 z}{\partial x\partial y} + 2\dfrac{\partial^2 z}{\partial y^2} = x + y

(c)

7 marks

Find the Fourier series of

f(x) = x^2

when

-\pi \le x \le \pi

and

f(x + 2\pi) = f(x)

(c)

7 marks

Find the Fourier series of

f(x) = \begin{cases} 8 & \text{if } 0 < x < 2, \\ -8 & \text{if } 2 < x < 4, \end{cases}

and

f(x + 4) = f(x)

Q.3

14 Marks

(a)

3 marks

Using Newton's backward interpolation formula, find

y(300)

using following observations.

$x$	50	100	150	200	250
$y$	618	724	805	906	1032

(b)

4 marks

Find the Lagrange's interpolation polynomial from the following data.

$x$	0	1	4	5
$f(x)$	1	3	24	39

(c)

7 marks

Solve by the method of separation of variables

4\dfrac{\partial u}{\partial x} + \dfrac{\partial u}{\partial y} = 3u

given that

u = 3e^{-y} - e^{-5y}

when

x = 0

(a)

3 marks

Solve:

p^2 + q^2 = x^2 + y^2

(b)

4 marks

Solve:

2\dfrac{\partial^2 z}{\partial x^2} - 5\dfrac{\partial^2 z}{\partial x\partial y} + 2\dfrac{\partial^2 z}{\partial y^2} = 5\sin(2x + y)

(c)

7 marks

Compute

f(9.2)

from the following values using Newton's divided difference formula.

$x$	8	9	9.5	11
$f(x)$	2.079442	2.197225	2.251292	2.397895

Q.4

14 Marks

(a)

3 marks

Given that Find $\dfrac{dy}{dx}$ and $\dfrac{d^2 y}{dx^2}$ at $x = 1.1$ using method of numerical differentiation.

$x$	1.0	1.1	1.2	1.3	1.4	1.5	1.6
$y$	7.989	8.403	8.781	9.129	9.451	9.750	10.031

(b)

4 marks

Find

\displaystyle\int_{10}^{16} y\,dx

by Simpson's

\dfrac{1}{3}

rule for the following values.

$x$	10	11	12	13	14	15	16
$y$	1.02	0.94	0.89	0.79	0.71	0.62	0.55

(c)

7 marks

A rod of length L with insulated side is initially at uniform temperature 100 C. Its ends are suddenly cooled at 0 C and kept at that temperature. Find the temperature

u(x, t)

(a)

3 marks

Evaluate

\displaystyle\int_0^1 \dfrac{dx}{1 + x^2}

using trapezoidal rule with

h = 0.2

(b)

4 marks

Evaluate

\displaystyle\int_0^3 \dfrac{dx}{1 + x}

with

n = 6

using Simpson's

\dfrac{3}{8}

rule.

(c)

7 marks

Given

\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{10}

y(0) = 1

, using fourth order Runge-Kutta method, find

y(0.2)

with

h = 0.1

Q.5

14 Marks

(a)

3 marks

Using Euler's method, find

y(0.2)

for given

\dfrac{dy}{dx} = y - \dfrac{2x}{y}

y(0) = 1

in two steps.

(b)

4 marks

Find:

L[2t^3 + e^{-2t} + t^{\frac{4}{3}}]

L[e^{-3t}\sin^2 t]

(c)

7 marks

Find:

L^{-1}\left[ \dfrac{s^3}{s^4 - 81} \right]

(a)

3 marks

Apply Runge-Kutta method of second order to find an approximate value of

y

x = 0.01

given that

\dfrac{dy}{dx} = x^2 + y

y(0) = 1

with

h = 0.01

(b)

4 marks

Find:

L[t\cos(4t + 3)]

L\left[ \dfrac{e^{-at} - e^{-bt}}{t} \right]

(c)

7 marks

Using Laplace transforms solve the initial value problem

\dfrac{d^2 y}{dt^2} + y = \sin 2t

y(0) = 2

\left( \dfrac{dy}{dt} \right)_{t=0} = 1

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About this paper

Official Gujarat Technological University (GTU) examination paper for Mathematics 3 (Mechanical) (Winter 2025, B.E. · Mechanical 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.

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