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Gujarat Technological UniversityWinter 2024 Examination

GTU 3110014 Mathematics - 1 (Maths 1) Winter 2024 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)
Find the equation of tangent plane and normal line to the surface xyz=6xyz = 6 at (1,2,3)(1, 2, 3).
3 Marks
(b)
Evaluate lim⁡x→0(1x2−1sin⁡2x)\lim_{x \to 0} \left( \frac{1}{x^2} - \frac{1}{\sin^2 x} \right)
4 Marks
(c)
Solve the following system by Gauss Elimination method:

x+y+2z=92x+4y−3z=13x+6y−5z=0\begin{aligned} x + y + 2z &= 9 \\ 2x + 4y - 3z &= 1 \\ 3x + 6y - 5z &= 0 \end{aligned}

7 Marks

Question 2

14 MarksMedium
(a)
Find the directional derivative of f(x,y,z)=xy2+yz3f(x, y, z) = xy^2 + yz^3 at the point (2,−1,1)(2, -1, 1) in the direction i^+2j^+2k^\hat{i} + 2\hat{j} + 2\hat{k}.
3 Marks
(b)
Expand x4−3x3+2x2−x+1x^4 - 3x^3 + 2x^2 - x + 1 in powers of (x−3)(x - 3).
4 Marks
(c)
Find the Fourier series of f(x)={x20<x<π0π<x<2πf(x) = \begin{cases} x^2 & 0 < x < \pi \\ 0 & \pi < x < 2\pi \end{cases}
7 Marks
OR OPTION
(c)
Find the Fourier series of f(x)=x+∣x∣f(x) = x + |x| in the interval −π<x<π-\pi < x < \pi.
7 Marks

Question 3

14 MarksMedium
(a)
Show that the sequence {un}\{u_n\}, where un=sin⁡nnu_n = \frac{\sin n}{n} converges to zero.
3 Marks
(b)
Evaluate ∫03dx(x−1)23\int_0^3 \frac{dx}{(x - 1)^{\frac{2}{3}}}
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 box.
7 Marks
OR OPTION
(a)
Evaluate ∫01(log⁡x)5 dx\int_0^1 (\log x)^5 \, dx
3 Marks
(b)
Find the cosine series for f(x)=π−xf(x) = \pi - x in the interval 0<x<π0 < x < \pi.
4 Marks
(c)
Find the extreme values of the function x3+xy2+21x−12x2−2y2x^3 + xy^2 + 21x - 12x^2 - 2y^2.
7 Marks

Question 4

14 MarksMedium
(a)
Test the convergence of the series ∑n=1∞nn2+1\sum_{n=1}^\infty \frac{\sqrt{n}}{n^2 + 1}
3 Marks
(b)
Evaluate the integral ∫0π2∫01−sin⁡θr2cos⁡θ dr dθ\int_0^{\frac{\pi}{2}} \int_0^{1-\sin \theta} r^2 \cos \theta \, dr \, d\theta
4 Marks
(c)
Find the length of the parabola x2=4yx^2 = 4y which lies inside the circle x2+y2=6yx^2 + y^2 = 6y.
7 Marks
OR OPTION
(a)
Evaluate ∫02∫1z∫0yzxyz dx dy dz\int_0^2 \int_1^z \int_0^{yz} xyz \, dx \, dy \, dz
3 Marks
(b)
Test the convergence of the series ∑(n+1)nxnnn+1\sum \frac{(n+1)^n x^n}{n^{n+1}}
4 Marks
(c)
Change the order of integration and evaluate: ∫0∞∫x∞e−yy dy dx\int_0^\infty \int_x^\infty \frac{e^{-y}}{y} \, dy \, dx
7 Marks

Question 5

14 MarksMedium
(a)
Test the convergence of the series ∑n=1∞n2e−n3\sum_{n=1}^\infty n^2 e^{-n^3}
3 Marks
(b)
Find aa and bb such that A=[a41b]A = \begin{bmatrix} a & 4 \\ 1 & b \end{bmatrix} has 33 and −2-2 as eigenvalues.
4 Marks
(c)
If u=f(y−xxy,z−xxz)u = f\left(\frac{y-x}{xy}, \frac{z-x}{xz}\right), show that x2∂u∂x+y2∂u∂y+z2∂u∂z=0x^2 \frac{\partial u}{\partial x} + y^2 \frac{\partial u}{\partial y} + z^2 \frac{\partial u}{\partial z} = 0
7 Marks
OR OPTION
(a)
Evaluate ∫0∞dv(1+v2)(1+tan⁡−1v)\int_0^\infty \frac{dv}{(1+v^2)(1+\tan^{-1} v)}
3 Marks
(b)
Using Gauss Jordan method find inverse of A=[843211121]A = \begin{bmatrix} 8 & 4 & 3 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix}
4 Marks
(c)
Show that the matrix A=[1−20122123]A = \begin{bmatrix} 1 & -2 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{bmatrix} is not diagonalizable.
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) (Winter 2024, 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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