Gujarat Technological UniversityWinter 2024 examination

GTU BE01000041 Mathematics-I Winter 2024 Question Paper & PDF

B.E. · Common Engineering · Semesters 1 and 2 · Subject code BE01000041
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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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Q.1

14 Marks
(a)
3 marks
Find lim⁡x→0(1x2−1sin⁡2x)\lim\limits_{x \to 0} \left( \dfrac{1}{x^2} - \dfrac{1}{\sin^2 x} \right)
(b)
4 marks
Write second derivative test for local extrema and find the extrema of f(x)=x4−4x3+10f(x) = x^4 - 4x^3 + 10
(c)
7 marks
Expand exe^x into the power series of xx. Find number of terms required to calculate ee with an error less than 10−410^{-4}.

Q.2

14 Marks
(a)
3 marks
Define Beta and Gamma functions and evaluate ∫0∞x3(1+x2)(1+x)10 dx\displaystyle\int_0^\infty \frac{x^3 (1 + x^2)}{(1 + x)^{10}}\,dx
(b)
4 marks
Discuss the convergence of ∫−∞∞11+x2 dx\displaystyle\int_{-\infty}^{\infty} \frac{1}{1 + x^2}\,dx
(c)
7 marks
The circle x2+y2=a2x^2 + y^2 = a^2 is rotated about xx-axis to generate a sphere, find its volume and sketch the region.
(c)
7 marks
Find the surface area of a solid generated by the revolution of the circle r=2acos⁡θr = 2a\cos\theta about the initial line.

Q.3

14 Marks
(a)
3 marks
Determine whether lim⁡(x,y)→(0,0)x6−y2x3y\lim\limits_{(x,y) \to (0,0)} \dfrac{x^6 - y^2}{x^3 y} exists, find it if exists.
(b)
4 marks
Applying the chain rule find dwdt\dfrac{dw}{dt} if w=x2y−y2w = x^2 y - y^2, where x=sin⁡tx = \sin t and y=ety = e^t also find dwdt\dfrac{dw}{dt} at t=0t = 0
(c)
7 marks
Find the extreme value of the function f(x,y)=x3+y3+3x2−3y2f(x, y) = x^3 + y^3 + 3x^2 - 3y^2
(a)
3 marks
If u=tan⁡−1(yx)u = \tan^{-1}\left( \dfrac{y}{x} \right) prove that ∂2u∂x2+∂2u∂y2=0\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0
(b)
4 marks
Find the tangent plane and normal line of the surface f(x,y,z)=x2+y2+z−9=0f(x, y, z) = x^2 + y^2 + z - 9 = 0 at point (1,2,4)(1, 2, 4)
(c)
7 marks
Applying Lagrange multipliers to determine the dimension of a rectangular box, open at the top, having volume 32 ft332\ ft^3, and requiring the least amount of material for its construction.

Q.4

14 Marks
(a)
3 marks
State sandwich theorem for the sequence and discuss the convergence of the sequence {cos⁡nn}\left\{ \dfrac{\cos n}{n} \right\}
(b)
4 marks
Test the convergence of the series ∑n=0∞(4n+35n)\sum\limits_{n=0}^{\infty} \left( \dfrac{4^n + 3}{5^n} \right), also find the sum of the series if it is convergent.
(c)
7 marks
Applying appropriate test find the interval of convergence and radius of convergence of the series ∑n=0∞(x−5)nn2\sum\limits_{n=0}^{\infty} \dfrac{(x - 5)^n}{n^2}
(a)
3 marks
Investigate the convergence of the series ∑n=0∞4n n! n!(2n!)\sum\limits_{n=0}^{\infty} \dfrac{4^n\, n!\, n!}{(2n!)}
(b)
4 marks
Discuss the convergence of the series and find it’s sum ∑n=0∞1n(n+1)\sum\limits_{n=0}^{\infty} \dfrac{1}{n(n + 1)}
(c)
7 marks
Applying appropriate test find the interval of convergence and radius of convergence of the series ∑n=0∞(−1)nxn3n(n+1)\sum\limits_{n=0}^{\infty} \dfrac{(-1)^n x^n}{3^n (n + 1)}

Q.5

14 Marks
(a)
3 marks
Evaluate ∬Ry2x dA\displaystyle\iint_R y^2 x\,dA over the rectangular region R={(x,y):−3≤x≤2, 0≤y≤1}R = \{(x, y) : -3 \le x \le 2,\ 0 \le y \le 1\}
(b)
4 marks
Use double integration to determine the area of the region RR inclosed between the parabola y=12x2y = \dfrac{1}{2}x^2, and the line y=2xy = 2x, also sketch the region.
(c)
7 marks
Applying change of the order of the integration to evaluate ∫02∫y21ex2 dx dy\displaystyle\int_0^2 \int_{\frac{y}{2}}^{1} e^{x^2}\,dx\,dy, also sketch the region.
(a)
3 marks
Evaluate ∫01∫x1∫0y−xdz dy dx\displaystyle\int_0^1 \int_x^1 \int_0^{y - x} dz\,dy\,dx
(b)
4 marks
Evaluate the following integral by changing in to the polar co ordinate. ∬Rex2+y2 dA\displaystyle\iint_R e^{x^2 + y^2}\,dA where RR is the semi circular region bounded by the x-axis and the curve y=1−x2y = \sqrt{1 - x^2}
(c)
7 marks
A thin plate covers the triangular region bounded by the x-axis and the line x=1x = 1 and y=2xy = 2x in the first quadrant. The plate density at the point (x,y)(x, y) is δ(x,y)=6x+6y+6\delta(x, y) = 6x + 6y + 6 find the plate’s mass, first moments and the centre of mass about the coordinate axis.

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Official Gujarat Technological University (GTU) examination paper for Mathematics-I (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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