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Gujarat Technological UniversitySummer 2023 Examination

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

B.E. · Common Engineering · Semester 1 · Subject Code: 3110014
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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)
Evaluate: lim⁡x→0ex−e−x−2xx−sin⁡x\lim_{x \to 0} \frac{e^x - e^{-x} - 2x}{x - \sin x}
3 Marks
(b)
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.
4 Marks
(c)
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}
7 Marks

Question 2

14 MarksMedium
(a)
Evaluate: ∫01∫02x∫0y2xyz dz dy dx\int_0^1 \int_0^{2x} \int_0^y 2xyz \, dz \, dy \, dx
3 Marks
(b)
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)
4 Marks
(c)
Find Fourier series of 2π2\pi-periodic function f(x)=x+x2;−π<x<πf(x) = x + x^2; \quad -\pi < x < \pi.
7 Marks
OR OPTION
(c)
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}
7 Marks

Question 3

14 MarksMedium
(a)
Define Improper integrals of type-I. Evaluate: ∫1∞dtt2+5t+6\int_1^\infty \frac{dt}{t^2 + 5t + 6}
3 Marks
(b)
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.
4 Marks
(c)
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
7 Marks
OR OPTION
(a)
Define Beta function. Evaluate: ∫01x4(1−x)5 dx\int_0^1 x^4 (1 - \sqrt{x})^5 \, dx
3 Marks
(b)
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}.
4 Marks
(c)
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
7 Marks

Question 4

14 MarksMedium
(a)
Find the sum of series ∑n=0∞2n−13n\sum_{n=0}^\infty \frac{2^n - 1}{3^n}
3 Marks
(b)
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
4 Marks
(c)
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.
7 Marks
OR OPTION
(a)
Discuss Convergence of series ∑n=1∞ln⁡(n2n+1)\sum_{n=1}^\infty \ln\left( \frac{n}{2n + 1} \right)
3 Marks
(b)
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.}
4 Marks
(c)
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.
7 Marks

Question 5

14 MarksMedium
(a)
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.
3 Marks
(b)
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}
4 Marks
(c)
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
7 Marks
OR OPTION
(a)
Evaluate: ∫02∫0π/2xsin⁡y dy dx\int_0^2 \int_0^{\pi/2} x \sin y \, dy \, dx
3 Marks
(b)
State Cayley-Hamilton theorem. Verify it for matrix A=[1005]A = \begin{bmatrix} 1 & 0 \\ 0 & 5 \end{bmatrix}
4 Marks
(c)
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
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 - 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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