Gujarat Technological UniversitySummer 2025 examination

GTU BE01000041 Mathematics-I Summer 2025 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
Investigate the convergence of ∫0111−x dx\displaystyle\int_0^1 \frac{1}{1 - x}\,dx.
(b)
4 marks
Define beta and gamma functions. What is the relationship between beta and gamma functions?
(c)
7 marks
The region between the curve y=xy = \sqrt{x}, 0≤x≤40 \le x \le 4 and the xx-axis is revolved about the xx-axis to generate a solid. Find its volume.

Q.2

14 Marks
(a)
3 marks
Evaluate lim⁡x→03x−sin⁡xx\lim\limits_{x \to 0} \dfrac{3x - \sin x}{x} using L’Hospital rule.
(b)
4 marks
Find the Taylor series generated by f(x)=exf(x) = e^x at x=2x = 2.
(c)
7 marks
Find the Maclaurin series for f(x)=(1+x)kf(x) = (1 + x)^k where kk is any real number. Using it, find the Maclaurin series for the function 11−x\dfrac{1}{1 - x}.
(c)
7 marks
Find the local extreme values of the function f(x)=3x4−2x3−6x2+6x+1f(x) = 3x^4 - 2x^3 - 6x^2 + 6x + 1.

Q.3

14 Marks
(a)
3 marks
Test the the series ∑n=1∞1n2+1\sum\limits_{n=1}^{\infty} \dfrac{1}{n^2 + 1} for convergence or divergence.
(b)
4 marks
Test the convergence of the series ∑n=1∞(−1)n+11n\sum\limits_{n=1}^{\infty} (-1)^{n+1} \dfrac{1}{n}.
(c)
7 marks
Define absolutely convergent series and conditionally convergent series. Investigate the convergence of the series ∑n=1∞sin⁡nn2\sum\limits_{n=1}^{\infty} \dfrac{\sin n}{n^2}.
(a)
3 marks
Investigate the convergence of the series ∑n=1∞2nn!\sum\limits_{n=1}^{\infty} \dfrac{2^n}{n!}.
(b)
4 marks
Find the sum of the series ∑n=1∞1n(n+1)\sum\limits_{n=1}^{\infty} \dfrac{1}{n(n + 1)}.
(c)
7 marks
For what values of x does the series ∑n=1∞(x−3)nn\sum\limits_{n=1}^{\infty} \dfrac{(x - 3)^n}{n} converge?

Q.4

14 Marks
(a)
3 marks
If f(x,y)=xy2x2+y4f(x, y) = \dfrac{x y^2}{x^2 + y^4}, does lim⁡(x,y)→(0,0)f(x,y)\lim\limits_{(x,y) \to (0,0)} f(x, y) exist?
(b)
4 marks
If z=x2y+3xy4z = x^2 y + 3x y^4, where x=sin⁡2tx = \sin 2t and y=cos⁡ty = \cos t, find dzdt\dfrac{dz}{dt} when t=0t = 0.
(c)
7 marks
Find the extreme values of the function f(x,y)=x2+2y2f(x, y) = x^2 + 2y^2 on the circle x2+y2=1x^2 + y^2 = 1.
(a)
3 marks
Find the equation of the tangent plane at the point (−2,1,−3)(-2, 1, -3) to the ellipsoid x24+y2+z29=3\dfrac{x^2}{4} + y^2 + \dfrac{z^2}{9} = 3.
(b)
4 marks
Find the directional derivative of the function f(x,y)=x2y3−4yf(x, y) = x^2 y^3 - 4y at the point (2,−1)(2, -1) in the direction of the vector v=2i+5j\mathbf{v} = 2\mathbf{i} + 5\mathbf{j}.
(c)
7 marks
Find the local extreme values of the function f(x,y)=xy−x2−y2−2x−2y+4f(x, y) = xy - x^2 - y^2 - 2x - 2y + 4.

Q.5

14 Marks
(a)
3 marks
Evaluate ∫03∫02(4−y2) dy dx\displaystyle\int_0^3 \int_0^2 (4 - y^2)\,dy\,dx
(b)
4 marks
Evaluate ∬R(3x+4y2) dA\displaystyle\iint_R (3x + 4y^2)\,dA where RR is the region in the upper half-plane bounded by the circles x2+y2=1x^2 + y^2 = 1 and x2+y2=4x^2 + y^2 = 4.
(c)
7 marks
Sketch the region of integration, reverse the order of integration and evaluate the integral ∫01∫x1sin⁡(y2) dy dx\displaystyle\int_0^1 \int_x^1 \sin(y^2)\,dy\,dx.
(a)
3 marks
Evaluate the triple integral ∭Bxyz2 dV\displaystyle\iiint_B x y z^2\,dV where BB is the rectangular box given by B={(x,y,z):0≤x≤1, −1≤y≤2, 0≤z≤3}B = \{(x, y, z) : 0 \le x \le 1,\ -1 \le y \le 2,\ 0 \le z \le 3\}.
(b)
4 marks
Find the area of the region RR bounded by y=2x2y = 2x^2 and y2=4xy^2 = 4x.
(c)
7 marks
Find the volume of the solid that lies under the paraboloid z=x2+y2z = x^2 + y^2 and above the region RR in the xyxy-plane bounded by the lines y=2xy = 2x and the parabola y=x2y = x^2.

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Official Gujarat Technological University (GTU) examination paper for Mathematics-I (Summer 2025, 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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