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Gujarat Technological UniversityWinter 2023 Examination

GTU 3110014 Mathematics - 1 (Maths 1) Winter 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 Hours2:30 PM – 5:30 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Evaluate lim⁡x→0+xln⁡x\lim_{x \to 0^+} x \ln x
3 Marks
(b)
Define beta and gamma functions. What is the relationship between beta and gamma functions?
4 Marks
(c)
Solve the following system of linear equations using Gauss-Jordan elimination:

x3+x4+x5=0−x1−x2+2x3−3x4+x5=0x1+x2−2x3−x5=02x1+2x2−x3+x5=0\begin{aligned} x_3 + x_4 + x_5 &= 0 \\ -x_1 - x_2 + 2x_3 - 3x_4 + x_5 &= 0 \\ x_1 + x_2 - 2x_3 - x_5 &= 0 \\ 2x_1 + 2x_2 - x_3 + x_5 &= 0 \end{aligned}

7 Marks

Question 2

14 MarksMedium
(a)
Define rank of the matrix. Find rank(A)\text{rank}(A) if A=[1224]A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}
3 Marks
(b)
Test the convergence of the series ∑n=1∞(11+n)n\sum_{n=1}^\infty \left(\frac{1}{1+n}\right)^n
4 Marks
(c)
Find eigenvalues and eigenvectors of the matrix A=[−522−2]A = \begin{bmatrix} -5 & 2 \\ 2 & -2 \end{bmatrix}
7 Marks
OR OPTION
(c)
Find the Fourier series of the function f(x)=x+πf(x) = x + \pi if −π<x<π-\pi < x < \pi and f(x+2π)=f(x)f(x + 2\pi) = f(x).
7 Marks

Question 3

14 MarksMedium
(a)
If w=x2+y2,x=r−s,y=r+sw = x^2 + y^2, x = r - s, y = r + s, using chain rule, prove that ∂w∂s=4s\frac{\partial w}{\partial s} = 4s.
3 Marks
(b)
Find the directional derivative of f(x,y,z)=2x2+3y2+z2f(x, y, z) = 2x^2 + 3y^2 + z^2 at point (2,1,3)(2, 1, 3) in the direction of the vector i−2k\mathbf{i} - 2\mathbf{k}.
4 Marks
(c)
Find local extreme values of the function f(x,y)=4x2+9y2+8x−36y+24f(x, y) = 4x^2 + 9y^2 + 8x - 36y + 24.
7 Marks
OR OPTION
(a)
Determine whether lim⁡(x,y)→(0,0)2x2yx3+y3\lim_{(x, y) \to (0, 0)} \frac{2x^2 y}{x^3 + y^3} exists and find it if exists.
3 Marks
(b)
Find the equation of the tangent plane to z=3x2−xyz = 3x^2 - xy at the point (1,2,1)(1, 2, 1).
4 Marks
(c)
Find the maximum and minimum values of the function f(x,y)=3x+4yf(x, y) = 3x + 4y on the circle x2+y2=1x^2 + y^2 = 1.
7 Marks

Question 4

14 MarksMedium
(a)
Calculate ∬Rf(x,y) dA\iint_R f(x, y) \, dA for f(x,y)=100−6x2yf(x, y) = 100 - 6x^2 y and R:0≤x≤2,−1≤y≤1R: 0 \le x \le 2, -1 \le y \le 1.
3 Marks
(b)
Sketch the region of integration for the integral ∫02∫x22x(4x+2) dy dx\int_0^2 \int_{x^2}^{2x} (4x + 2) \, dy \, dx and write an equivalent integral with the order of integration reversed.
4 Marks
(c)
Calculate ∬Rsin⁡xx dA\iint_R \frac{\sin x}{x} \, dA where RR is the triangle in the xyxy-plane bounded by the xx-axis, the line y=xy = x and the line x=1x = 1.
7 Marks
OR OPTION
(a)
Evaluate ∫1ln⁡8∫0ln⁡yex+y dx dy\int_1^{\ln 8} \int_0^{\ln y} e^{x+y} \, dx \, dy
3 Marks
(b)
Find the area of the region RR bounded by y=xy = x and y=x2y = x^2 in the first quadrant.
4 Marks
(c)
Evaluate ∫01∫01−x2(x2+y2) dy dx\int_0^1 \int_0^{\sqrt{1-x^2}} (x^2 + y^2) \, dy \, dx by changing into polar coordinates.
7 Marks

Question 5

14 MarksMedium
(a)
Find the Maclaurin series for cos⁡x\cos x.
3 Marks
(b)
Determine the convergence or divergence of the series ∑n=1∞ne−n2\sum_{n=1}^\infty n e^{-n^2}
4 Marks
(c)
Test the convergence of the series ∑n=1∞(−1)n−1n2\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2}
7 Marks
OR OPTION
(a)
Define monotonic sequence. Is the sequence {1n2}\{\frac{1}{n^2}\} monotonic?
3 Marks
(b)
Investigate the convergence of the series ∑n=0∞2n+53n\sum_{n=0}^\infty \frac{2^n + 5}{3^n}
4 Marks
(c)
Find the interval of convergence of the series x−x22+x33−…x - \frac{x^2}{2} + \frac{x^3}{3} - \dots
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 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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