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Gujarat Technological UniversityWinter 2025 Examination

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

Question 1

14 MarksMedium
(a)
Evaluate lim⁡x→0(cos⁡x)cot⁡x\lim_{x \to 0} (\cos x)^{\cot x}
3 Marks
(b)
Discuss the convergence of the series 11⋅2+12⋅3+13⋅4+14⋅5+…\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \dots. If convergent find its sum.
4 Marks
(c)
1Investigate the convergence of the integral ∫0111−x dx\int_0^1 \frac{1}{1-x} \, dx.
2Find the length of the curve y(x)=x312+1x;1≤x≤4y(x) = \frac{x^3}{12} + \frac{1}{x}; \quad 1 \le x \le 4.
7 Marks

Question 2

14 MarksMedium
(a)
Discuss the convergence of the series ∑n=1∞2n+1(n+1)2\sum_{n=1}^\infty \frac{2n+1}{(n+1)^2}
3 Marks
(b)
Discuss the convergence of the following:
1∑n=1∞4n+5n6n\sum_{n=1}^\infty \frac{4^n + 5^n}{6^n}
2∑n=1∞2nn3+1\sum_{n=1}^\infty \frac{2^n}{n^3 + 1}
4 Marks
(c)
Obtain the Fourier series of the function f(x)={0if −π<x≤0xif 0≤x<πf(x) = \begin{cases} 0 & \text{if } -\pi < x \le 0 \\ x & \text{if } 0 \le x < \pi \end{cases}
7 Marks
OR OPTION
(c)
Obtain the Fourier series of the function f(x)=e∣x∣;−2≤x≤2f(x) = e^{|x|}; \quad -2 \le x \le 2.
7 Marks

Question 3

14 MarksMedium
(a)
If u=f(x−y,y−z,z−x)u = f(x - y, y - z, z - x), show that ∂u∂x+∂u∂y+∂u∂z=0\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0
3 Marks
(b)
Find the tangent plane and normal line of the surface x3+y3+z3=3x^3 + y^3 + z^3 = 3 at the point (1,1,1)(1, 1, 1).
4 Marks
(c)
Find the extreme values that the function f(x,y)=xyf(x, y) = xy takes on the ellipse x28+y22=1\frac{x^2}{8} + \frac{y^2}{2} = 1.
7 Marks
OR OPTION
(a)
If u=2x3y+y3z2u = 2x^3 y + y^3 z^2, where x=rsetx = r s e^t, y=rs2e−ty = r s^2 e^{-t} and z=r2ssin⁡tz = r^2 s \sin t, find ∂u∂s\frac{\partial u}{\partial s} at r=1,s=1,t=0r = 1, s = 1, t = 0.
3 Marks
(b)
If u=tan⁡−1(x2+y2x−y)u = \tan^{-1}\left(\frac{x^2 + y^2}{x - y}\right), show that x∂u∂x+y∂u∂y=12sin⁡2ux \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = \frac{1}{2} \sin 2u
4 Marks
(c)
Examine the function f(x,y)=2x4+y2−x2−2yf(x, y) = 2x^4 + y^2 - x^2 - 2y for maxima and minima.
7 Marks

Question 4

14 MarksMedium
(a)
Evaluate ∫01∫0xeyx dy dx\int_0^1 \int_0^x e^{\frac{y}{x}} \, dy \, dx
3 Marks
(b)
Evaluate ∬Rf(x,y) dA\iint_R f(x, y) \, dA, where f(x,y)=6x2+2yf(x, y) = 6x^2 + 2y and RR is the region bounded by y=x2y = x^2 and y=4y = 4.
4 Marks
(c)
By using changing the order of integration, evaluate ∫01∫−1−y21−yy dx dy\int_0^1 \int_{-\sqrt{1-y^2}}^{1-y} y \, dx \, dy
7 Marks
OR OPTION
(a)
Evaluate ∫0π/2∫0acos⁡θrsin⁡θ dr dθ\int_0^{\pi/2} \int_0^{a \cos \theta} r \sin \theta \, dr \, d\theta
3 Marks
(b)
Evaluate ∫01∫01−y∫01−y−zz dx dz dy\int_0^1 \int_0^{1-y} \int_0^{1-y-z} z \, dx \, dz \, dy
4 Marks
(c)
Evaluate ∬Rx dx dy\iint_R x \, dx \, dy, where RR is the region bounded by triangle with vertices (0,0),(0,1)(0, 0), (0, 1) and (1,1)(1, 1), using the transformations x=u,y=uvx = u, y = uv.
7 Marks

Question 5

14 MarksMedium
(a)
Verify Cayley-Hamilton theorem for the matrix A=[1423]A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}
3 Marks
(b)
Expand ln⁡(1+sin⁡x)\ln(1 + \sin x) in powers of xx up to x3x^3.
4 Marks
(c)
Solve the following system using Gauss-Jordan method:

x−2y−z+3w=12x−4y+z=5x−2y+2z−3w=4\begin{aligned} x - 2y - z + 3w &= 1 \\ 2x - 4y + z &= 5 \\ x - 2y + 2z - 3w &= 4 \end{aligned}

7 Marks
OR OPTION
(a)
Using row echelon form find the rank of the matrix A=[1111−1−1311]A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -1 & -1 \\ 3 & 1 & 1 \end{bmatrix}
3 Marks
(b)
Discuss the convergence of the series 11+3+21+32+31+33+41+34+…\frac{1}{1+3} + \frac{2}{1+3^2} + \frac{3}{1+3^3} + \frac{4}{1+3^4} + \dots
4 Marks
(c)
Find the modal matrix PP and diagonal matrix DD for the matrix A=[1−6−40420−6−3]A = \begin{bmatrix} 1 & -6 & -4 \\ 0 & 4 & 2 \\ 0 & -6 & -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) (Winter 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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