Gujarat Technological UniversitySummer 2023 examination

GTU 3110014 Mathematics - 1 (Maths 1) Summer 2023 Question Paper & PDF

B.E. · Common Engineering · Semesters 1 and 2 · Subject code 3110014
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Total
70 marks
To pass
23 marks (33%)
Duration
2.5 Hours 10:30 AM – 1:30 PM
Questions
5 with internal choice
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1 of 25 questions on this paper were also asked in other years. 15 more are types of question GTU sets again with new data. Each is marked below with the years it appeared.See all 35 repeated 3110014 questions, most asked first

Q.1

14 Marks
(a)
3 marksSame type asked in 5 of 7 papers
Evaluate:
lim⁡x→0ex−e−x−2xx−sin⁡x\lim_{x \to 0} \frac{e^x - e^{-x} - 2x}{x - \sin x}

Also asked in Summer 2026 Q1a, Winter 2025 Q1a, Summer 2024 Q1a, Winter 2023 Q1a

(b)
4 marksSame type asked in 3 of 7 papers
Show that series
∑n=0∞(1n−1n+1)\sum_{n=0}^{\infty} \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \right)
is convergent. Hence find sum of the series.

Also asked in Summer 2026 Q1b, Winter 2025 Q1b

(c)
7 marksSame type asked in 5 of 7 papers
Define eigenvalue and eigen vector of a matrix. Find all eigenvalues and eigenvectors of matrix:
A=[1−333−536−64]A = \begin{bmatrix} 1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4 \end{bmatrix}

Also asked in Summer 2026 Q5c OR, Summer 2025 Q2c, Summer 2024 Q5c OR, Winter 2023 Q2c

Q.2

14 Marks
(a)
3 marksSame type asked in 4 of 7 papers
Evaluate:
∫01∫02x∫0y2xyz dz dy dx\int_0^1 \int_0^{2x} \int_0^y 2xyz \, dz \, dy \, dx

Also asked in Summer 2026 Q4c OR, Winter 2025 Q4b OR, Summer 2024 Q5a OR

(b)
4 marksSame type asked in 5 of 7 papers
Find equations of tangent plane and normal line to the surface
x22−y23−z=0at (2,3,−1)\frac{x^2}{2} - \frac{y^2}{3} - z = 0 \quad \text{at } (2, 3, -1)

Also asked in Winter 2025 Q3b, Summer 2025 Q3b, Winter 2024 Q1a, Winter 2023 Q3b OR

(c)
7 marksSame type asked in 5 of 7 papers
Find Fourier series of 2π2\pi-periodic function f(x)=x+x2;−π<x<πf(x) = x + x^2; \quad -\pi < x < \pi.

Also asked in Winter 2025 Q2c, Winter 2025 Q2c OR, Winter 2024 Q2c, Winter 2024 Q2c OR, Summer 2024 Q1c, Winter 2023 Q2c OR

(c)
7 marksSame type asked in 5 of 7 papers
Find the Fourier series for the periodic extension of:
f(x)={sin⁡x,0≤x≤π0,π≤x≤2πf(x) = \begin{cases} \sin x, & 0 \le x \le \pi \\ 0, & \pi \le x \le 2\pi \end{cases}

Also asked in Winter 2025 Q2c, Winter 2025 Q2c OR, Winter 2024 Q2c, Winter 2024 Q2c OR, Summer 2024 Q1c, Winter 2023 Q2c OR

Q.3

14 Marks
(a)
3 marks
Define Improper integrals of type-I. Evaluate:
∫1∞dtt2+5t+6\int_1^\infty \frac{dt}{t^2 + 5t + 6}
(b)
4 marks
Find the volume of the solid generated by revolving the region bounded by y=x2,y=0y = x^2, y = 0 and x=2x = 2 about the xx-axis.
(c)
7 marks
Answer the following:
i)Using ratio test discuss convergence of series
∑n=1∞(n+3)!3!⋅n!⋅3n\sum_{n=1}^\infty \frac{(n+3)!}{3! \cdot n! \cdot 3^n}
3 Marks
ii)Use Taylor series to represent function f(x)=2x3+x2+3x−8f(x) = 2x^3 + x^2 + 3x - 8 in powers of (x−1)(x - 1)4 Marks
(a)
3 marksSame type asked in 2 of 7 papers
Define Beta function. Evaluate:
∫01x4(1−x)5 dx\int_0^1 x^4 (1 - \sqrt{x})^5 \, dx

Also asked in Summer 2024 Q1b

(b)
4 marksSame type asked in 3 of 7 papers
Find the arc length of f(x)=ln⁡(cos⁡x),0≤x≤π4f(x) = \ln(\cos x), \quad 0 \le x \le \frac{\pi}{4}.

Also asked in Winter 2024 Q4c, Summer 2024 Q2c OR

(c)
7 marks
Answer the following:
i)Using Sandwich theorem find limit of sequence an=(cos⁡nn)a_n = \left( \frac{\cos n}{n} \right)3 Marks
ii)Define Radius of Convergence of power series. Find it for power series
∑n=1∞(x−1)nn3⋅3n\sum_{n=1}^\infty \frac{(x - 1)^n}{n^3 \cdot 3^n}
4 Marks

Q.4

14 Marks
(a)
3 marksSame type asked in 4 of 7 papers
Find the sum of series
∑n=0∞2n−13n\sum_{n=0}^\infty \frac{2^n - 1}{3^n}

Also asked in Summer 2026 Q2b, Winter 2025 Q2b, Winter 2023 Q5b OR

(b)
4 marks
If u=log⁡(x2+y2)u = \log(x^2 + y^2), show that
∂2u∂x2+∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
(c)
7 marksSame type asked in 7 of 7 papers
Find all local maxima, local minima and saddle points of the function f(x,y)=x3+3xy2−15x+y3−15yf(x, y) = x^3 + 3xy^2 - 15x + y^3 - 15y.

Also asked in Summer 2026 Q3b OR, Winter 2025 Q3c OR, Summer 2025 Q3c, Winter 2024 Q3c OR, Summer 2024 Q4c, Winter 2023 Q3c

(a)
3 marks
Discuss Convergence of series
∑n=1∞ln⁡(n2n+1)\sum_{n=1}^\infty \ln\left( \frac{n}{2n + 1} \right)
(b)
4 marks
Show that
lim⁡(x,y)→(0,0)x4−y2x4+y2does not exist.\lim_{(x, y) \to (0, 0)} \frac{x^4 - y^2}{x^4 + y^2} \quad \text{does not exist.}
(c)
7 marksAsked in 2 of 7 papers
A rectangular box without a lid is to be made from 12 m212\text{ m}^2 of cardboard. Find the maximum volume of such a box.

Also asked in Winter 2024 Q3c

Q.5

14 Marks
(a)
3 marks
Find Jacobian ∂(x,y)∂(r,θ)\frac{\partial(x, y)}{\partial(r, \theta)} for x=rcos⁡θ,y=rsin⁡θx = r \cos\theta, \quad y = r \sin\theta.
(b)
4 marksSame type asked in 6 of 7 papers
Solve the given linear system using Gauss-Elimination method:
x1+x2−2x3=12x1−3x2+x3=−83x1+x2+4x3=7\begin{aligned} x_1 + x_2 - 2x_3 &= 1 \\ 2x_1 - 3x_2 + x_3 &= -8 \\ 3x_1 + x_2 + 4x_3 &= 7 \end{aligned}

Also asked in Summer 2026 Q5c, Winter 2025 Q5c, Winter 2024 Q1c, Summer 2024 Q4b OR, Winter 2023 Q1c

(c)
7 marksSame type asked in 7 of 7 papers
Sketch the region of integration, change the order of integration and evaluate:
∫04∫x21y3+1 dy dx\int_0^4 \int_{\sqrt{x}}^2 \frac{1}{y^3 + 1} \, dy \, dx

Also asked in Summer 2026 Q4b, Winter 2025 Q4c, Summer 2025 Q4c, Summer 2025 Q4b OR, Winter 2024 Q4c OR, Summer 2024 Q4c OR, Winter 2023 Q4b

(a)
3 marks
Evaluate:
∫02∫0π/2xsin⁡y dy dx\int_0^2 \int_0^{\pi/2} x \sin y \, dy \, dx
(b)
4 marksSame type asked in 2 of 7 papers
State Cayley-Hamilton theorem. Verify it for matrix
A=[1005]A = \begin{bmatrix} 1 & 0 \\ 0 & 5 \end{bmatrix}

Also asked in Summer 2024 Q5b

(c)
7 marksSame type asked in 2 of 7 papers
Use the transformation x=u2−v2,y=2uvx = u^2 - v^2, \quad y = 2uv to evaluate the integral:
∫01∫021−xx2+y2 dy dx\int_0^1 \int_0^{2\sqrt{1-x}} \sqrt{x^2 + y^2} \, dy \, dx

Also asked in Winter 2025 Q4c OR

More Mathematics - 1 papers

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

Summer 202670 marks · 5 questions · Solved
Winter 202570 marks · 5 questions · Solved
Summer 202570 marks · 5 questions · Solved
Winter 202470 marks · 5 questions

Other Semester 1 subjects

BE Common

3110001
Chemistry
3110002
English
3110003
Programming for Problem Solving
3110004
Basic Civil Engineering

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About this paper

Official Gujarat Technological University (GTU) examination paper for Mathematics - 1 (Maths 1) (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.

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