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Gujarat Technological UniversitySummer 2026 Examination

GTU 3110014 Mathematics - 1 (Maths 1) Summer 2026 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→0x−sin⁡xx3\lim_{x \to 0} \frac{x - \sin x}{x^3}
3 Marks
(b)
Discuss the convergence of the series 1+12+14+18+116+…1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots. If convergent then find it's sum.
4 Marks
(c)
iCheck the convergence of ∫0214−x2 dx\int_0^2 \frac{1}{\sqrt{4-x^2}} \, dx.
iiFind the volume of the solid of revolution of the area bounded by the curve y=xexy = x e^x and lines x=0x = 0 to x=1x = 1.
7 Marks

Question 2

14 MarksMedium
(a)
State Sandwich theorem for the sequences and show that lim⁡n→∞sin⁡2nn=0\lim_{n \to \infty} \frac{\sin^2 n}{n} = 0
3 Marks
(b)
Check the convergence of the series ∑n=1∞5n+6n7n\sum_{n=1}^\infty \frac{5^n + 6^n}{7^n}
4 Marks
(c)
Find Fourier series of f(x)=(π−x)2f(x) = \frac{(\pi - x)}{2} 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
OR OPTION
(c)
Find the Fourier series of f(x)=x2f(x) = x^2 in the interval −π≤x≤π-\pi \le x \le \pi. Hence deduce that 112−122+132−⋯=π212\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \dots = \frac{\pi^2}{12}
7 Marks

Question 3

14 MarksMedium
(a)
If u=log⁡(x2+y2)u = \log(x^2 + y^2) then show that ∂2u∂x∂y=∂2u∂y∂x\frac{\partial^2 u}{\partial x \partial y} = \frac{\partial^2 u}{\partial y \partial x}
3 Marks
(b)
Find the rate of change of xyzxyz in the direction normal to the surface x2y+xy2+yz2=3x^2 y + xy^2 + yz^2 = 3 at point P(1,1,1)P(1, 1, 1).
4 Marks
(c)
If 4x+9y+16z=25\frac{4}{x} + \frac{9}{y} + \frac{16}{z} = 25 then by Lagrange’s method find the values of x,yx, y and zz which makes x+y+zx + y + z maximum.
7 Marks
OR OPTION
(a)
If z=exy,x=tcos⁡t,y=tsin⁡tz = e^{xy}, x = t \cos t, y = t \sin t then find dzdt\frac{dz}{dt} at t=π2t = \frac{\pi}{2}.
3 Marks
(b)
Discuss the maxima and minima of the function 3x2−y2+x33x^2 - y^2 + x^3.
4 Marks
(c)
iIf u=f(ey−z,ez−x,ex−y)u = f(e^{y-z}, e^{z-x}, e^{x-y}) then show that ux+uy+uz=0u_x + u_y + u_z = 0.
iiFind the equations of the tangent plane and the normal line to the surface x2y2+xz−2y3=10x^2 y^2 + xz - 2y^3 = 10 at (2,1,4)(2, 1, 4).
7 Marks

Question 4

14 MarksMedium
(a)
Evaluate ∫01∫01+x2dy dx1+x2+y2\int_0^1 \int_0^{\sqrt{1+x^2}} \frac{dy \, dx}{1 + x^2 + y^2}
3 Marks
(b)
Evaluate the integral ∫01∫02xex2 dy dx\int_0^1 \int_0^{2x} e^{x^2} \, dy \, dx by changing the order of integration.
4 Marks
(c)
Evaluate ∬Rxy dA\iint_R x y \, dA where RR is the region bounded by xx-axis, x=2ax = 2a and the curve x2=4ayx^2 = 4ay.
7 Marks
OR OPTION
(a)
Evaluate ∬Rr3sin⁡2θ dr dθ\iint_R r^3 \sin 2\theta \, dr \, d\theta over the area bounded in the first quadrant between two circles r=2r = 2 and r=4r = 4.
3 Marks
(b)
Evaluate ∫0a∫yaxx2+y2 dx dy\int_0^a \int_y^a \frac{x}{x^2 + y^2} \, dx \, dy by transforming into Polar coordinates.
4 Marks
(c)
Evaluate ∫01∫01−x2∫01−x2−y2xyz dz dy dx\int_0^1 \int_0^{\sqrt{1-x^2}} \int_0^{\sqrt{1-x^2-y^2}} xyz \, dz \, dy \, dx
7 Marks

Question 5

14 MarksMedium
(a)
Find the rank of the matrix AA by reducing it to row echelon form: A=[11−111−12−13101]A = \begin{bmatrix} 1 & 1 & -1 & 1 \\ 1 & -1 & 2 & -1 \\ 3 & 1 & 0 & 1 \end{bmatrix}
3 Marks
(b)
State Cauchy’s Integral Test and discuss the convergence of ∑n=1∞1n(1+(log⁡n)2)\sum_{n=1}^\infty \frac{1}{n(1 + (\log n)^2)}
4 Marks
(c)
Solve the following system of equations using Gauss Jordan Method:

x+2y−z=1x+y+2z=92x+y−z=2\begin{aligned} x + 2y - z &= 1 \\ x + y + 2z &= 9 \\ 2x + y - z &= 2 \end{aligned}

7 Marks
OR OPTION
(a)
Verify Cayley-Hamilton theorem for the matrix A=[1423]A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}
3 Marks
(b)
Expand exsinh⁡xe^x \sinh x in powers of xx upto x4x^4.
4 Marks
(c)
Find the eigen values and eigen vectors for the matrix A=[3−11−15−11−13]A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 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 2026, 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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