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Gujarat Technological UniversitySummer 2025 Examination

GTU 3110014 Mathematics - 1 (Maths 1) Summer 2025 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)
Evaluate: lim⁡x→0[x−2−csc⁡2x]\lim_{x \to 0} [x^{-2} - \csc^2 x]
3 Marks
(b)
Check the consistency of the system of linear equations. Solve it if consistent:

3x+y−3z=132x−3y+7z=52x+19y−47z=32\begin{aligned} 3x + y - 3z &= 13 \\ 2x - 3y + 7z &= 5 \\ 2x + 19y - 47z &= 32 \end{aligned}

4 Marks
(c)
Find the Fourier Series of the function f(x)=x2f(x) = x^2 in the interval (−π,π)(-\pi, \pi).
7 Marks

Question 2

14 MarksMedium
(a)
Define the improper integrals of the first kind and the second kind. State the relation between Beta and Gamma function.
3 Marks
(b)
Find the area of the surface of revolution generated by revolving the curve x=y3x = y^3 from y=0y = 0 to y=2y = 2.
4 Marks
(c)
Find eigen values and corresponding eigen vectors of the matrix: (011101110)\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}
7 Marks
OR OPTION
(c)
Find the inverse of the matrix using Gauss-Jordan elimination method: A=(123230012)A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{pmatrix}
7 Marks

Question 3

14 MarksMedium
(a)
Determine whether lim⁡(x,y)→(0,0)x2−y2x2+y2\lim_{(x, y) \to (0, 0)} \frac{x^2 - y^2}{x^2 + y^2} exists and find it if exists.
3 Marks
(b)
Find the equation of tangent plane and normal line to the surface 2xz2−3xy−4x=72xz^2 - 3xy - 4x = 7 at the point (1,−1,2)(1, -1, 2).
4 Marks
(c)
Find the extreme values of the function x3+y3−63(x+y)+12xyx^3 + y^3 - 63(x + y) + 12xy.
7 Marks
OR OPTION
(a)
State chain rule for ∂u∂x\frac{\partial u}{\partial x} for u=f(v,w)u = f(v, w), where v=g(x,y),w=h(x,y)v = g(x, y), w = h(x, y). If u=f(x−2y,2y−3z,3z−x)u = f(x - 2y, 2y - 3z, 3z - x) then show that ∂u∂x+12∂u∂y+13∂u∂z=0\frac{\partial u}{\partial x} + \frac{1}{2}\frac{\partial u}{\partial y} + \frac{1}{3}\frac{\partial u}{\partial z} = 0
3 Marks
(b)
Find the directional derivative of f(x,y,z)=xy2+yz2f(x, y, z) = xy^2 + yz^2 at the point (2,−1,1)(2, -1, 1) in the direction of the vector i^+2j^+2k^\hat{i} + 2\hat{j} + 2\hat{k}.
4 Marks
(c)
The temperature T(x,y,z)T(x, y, z) at any point in space is T=400xyz2T = 400xyz^2. Find the highest temperature on surface of the sphere x2+y2+z2=1x^2 + y^2 + z^2 = 1.
7 Marks

Question 4

14 MarksMedium
(a)
Evaluate ∫−11∫02(1−6x2y) dx dy\int_{-1}^1 \int_0^2 (1 - 6x^2 y) \, dx \, dy
3 Marks
(b)
Find the value of ∬R(2x−y2) dA\iint_R (2x - y^2) \, dA over the triangular region RR enclosed between the lines y=−x+1y = -x + 1, y=x+1y = x + 1 and y=3y = 3.
4 Marks
(c)
Change the order of integral and evaluate ∫04a∫x2/4a2axdy dx,(a>0)\int_0^{4a} \int_{x^2/4a}^{2\sqrt{ax}} dy \, dx, \quad (a > 0)
7 Marks
OR OPTION
(a)
Calculate ∫02∫1z∫0yzxyz dx dy dz\int_0^2 \int_1^z \int_0^{yz} xyz \, dx \, dy \, dz
3 Marks
(b)
Evaluate ∫01∫x1sin⁡y2 dy dx\int_0^1 \int_x^1 \sin y^2 \, dy \, dx
4 Marks
(c)
Compute ∫02∫02x−x2xx2+y2 dy dx\int_0^2 \int_0^{\sqrt{2x-x^2}} \frac{x}{x^2+y^2} \, dy \, dx by transforming into polar coordinates.
7 Marks

Question 5

14 MarksMedium
(a)
Define Monotonic sequence. Test the convergence of the sequence {2−(−1)n}\{2 - (-1)^n\}.
3 Marks
(b)
Express the function f(x)=log⁡(1+x)f(x) = \log(1 + x) in power series using the formula of Maclaurin’s series.
4 Marks
(c)
iTest the convergence of ∑n=1∞n!(2n+1)!\sum_{n=1}^\infty \frac{n!}{(2n+1)!}
iiState Cauchy’s root test and discuss the convergence of ∑n=1∞1(log⁡n)n\sum_{n=1}^\infty \frac{1}{(\log n)^n}
7 Marks
OR OPTION
(a)
State sandwich theorem for sequence. Show that the sequence un=sin⁡nnu_n = \frac{\sin n}{n} converges to zero.
3 Marks
(b)
Using Taylor’s theorem find the approximate value of 10\sqrt{10} up to three decimal places.
4 Marks
(c)
iExamine the convergence of the series 11⋅2−12⋅3+13⋅4−15⋅6+17⋅8−…\frac{1}{1 \cdot 2} - \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} - \frac{1}{5 \cdot 6} + \frac{1}{7 \cdot 8} - \dots
iiState Cauchy’s Integral test for convergence of series and test the convergence of the series ∑n=1∞n2e−n3\sum_{n=1}^\infty n^2 e^{-n^3}
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 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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