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Gujarat Technological UniversitySummer 2024 Examination

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

Question 1

14 MarksMedium
(a)
Using L’ Hospital’s rule, evaluate lim⁡x→1x−xx1+log⁡x−x\lim_{x \to 1} \frac{x - x^x}{1 + \log x - x}
3 Marks
(b)
Define Beta function and evaluate ∫01x3(1−x)5 dx\int_0^1 x^3 (1 - \sqrt{x})^5 \, dx
4 Marks
(c)
Find the Fourier series of f(x)={π+x−π<x<0π−x0<x<πf(x) = \begin{cases} \pi + x & -\pi < x < 0 \\ \pi - x & 0 < x < \pi \end{cases}
7 Marks

Question 2

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)
Express f(x)=2x3+3x2−8x+7f(x) = 2x^3 + 3x^2 - 8x + 7 in terms of (x−2)(x - 2).
4 Marks
(c)
Find the area of the surface of revolution of a quadrant of a circular arc as obtained by revolving it about a tangent at one of its ends.
7 Marks
OR OPTION
(c)
Find the length of the loop of the curve 9ay2=(x−2a)(x−5a)29ay^2 = (x - 2a)(x - 5a)^2
7 Marks

Question 3

14 MarksMedium
(a)
Evaluate ∫0∞dv(1+v2)(1+tan⁡−1v)\int_0^\infty \frac{dv}{(1 + v^2)(1 + \tan^{-1} v)}
3 Marks
(b)
Test the convergence of the series ∑n=1∞2n+1n2(n+1)2\sum_{n=1}^\infty \frac{2n + 1}{n^2 (n+1)^2}
4 Marks
(c)
If θ=tne−r24t\theta = t^n e^{-\frac{r^2}{4t}} then find nn so that 1r2∂∂r(r2∂θ∂r)=∂θ∂t\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \theta}{\partial r} \right) = \frac{\partial \theta}{\partial t}
7 Marks
OR OPTION
(a)
Check the convergence of ∫051x2 dx\int_0^5 \frac{1}{x^2} \, dx
3 Marks
(b)
Evaluate ∫0∞3−4x2 dx\int_0^\infty 3^{-4x^2} \, dx
4 Marks
(c)
Find the Fourier series of f(x)=12(π−x)f(x) = \frac{1}{2}(\pi - x) in the interval (0,2π)(0, 2\pi). Hence, deduce that π4=1−13+15−17+…\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots
7 Marks

Question 4

14 MarksMedium
(a)
Prove that tan⁡−1x=x−x33+x55−x77+…\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots
3 Marks
(b)
Find the Fourier sine series of f(x)=exf(x) = e^x in 0<x<π0 < x < \pi.
4 Marks
(c)
Find the extreme values of the function f(x,y)=x3+y3−3x−12y+20f(x, y) = x^3 + y^3 - 3x - 12y + 20.
7 Marks
OR OPTION
(a)
Find the directional derivative of f(x,y,z)=x2yz+4xz2f(x, y, z) = x^2 yz + 4xz^2 at (1,−2,−1)(1, -2, -1) in the direction of 2i^−j^−2k^2\hat{i} - \hat{j} - 2\hat{k}.
3 Marks
(b)
Solve the following system by Gauss-Jordan method:

−2y+3z=13x+6y−3z=−26x+6y+3z=5\begin{aligned} -2y + 3z &= 1 \\ 3x + 6y - 3z &= -2 \\ 6x + 6y + 3z &= 5 \end{aligned}

4 Marks
(c)
Change the order of integration and evaluate: ∫01∫01−y2cos⁡−1x1−x21−x2−y2 dx dy\int_0^1 \int_0^{\sqrt{1-y^2}} \frac{\cos^{-1} x}{\sqrt{1-x^2}\sqrt{1-x^2-y^2}} \, dx \, dy
7 Marks

Question 5

14 MarksMedium
(a)
Evaluate ∫03∫01(x2+3y2) dy dx\int_0^3 \int_0^1 (x^2 + 3y^2) \, dy \, dx
3 Marks
(b)
Apply Cayley-Hamilton theorem to A=[122−1]A = \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix} and deduce that A8=625IA^8 = 625 I.
4 Marks
(c)
Evaluate ∫02∫02x−x2xx2+y2 dy dx\int_0^2 \int_0^{\sqrt{2x-x^2}} \frac{x}{x^2+y^2} \, dy \, dx by changing to polar coordinates.
7 Marks
OR OPTION
(a)
Evaluate ∫02∫12∫0yzxyz dx dy dz\int_0^2 \int_1^2 \int_0^{yz} xyz \, dx \, dy \, dz
3 Marks
(b)
Using Gauss Jordan method, find inverse of A=[234431124]A = \begin{bmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{bmatrix}
4 Marks
(c)
Find the eigen values and eigen vectors of the matrix A=[466132−1−4−3]A = \begin{bmatrix} 4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -4 & -3 \end{bmatrix}
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 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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