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Gujarat Technological UniversityWinter 2025 Examination

GTU 3110019 Remedial Mathematics (Remedial Maths) Winter 2025 Paper Solution & PDF

B.E. · Common Engineering · Semester 1 · Subject Code: 3110019
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Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours2:30 PM – 5:00 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Define slope of a line and find the slope of the line joining the points (2,3)(2, 3) and (−1,5)(-1, 5).
3 Marks
(b)
Find the equation of a line passing through (2,−3)(2, -3) and perpendicular to the line 3x−4y+7=03x - 4y + 7 = 0.
4 Marks
(c)
Find the distance of the point (4,−3)(4, -3) from the line 3x+4y−5=03x + 4y - 5 = 0.
7 Marks

Question 2

14 MarksMedium
(a)
Define real-valued function with examples.
3 Marks
(b)
Express sin⁡3x\sin 3x and cos⁡3x\cos 3x.
4 Marks
(c)
Prove sin⁡(x+y)sin⁡(x−y)=sin⁡2x−sin⁡2y\sin(x+y)\sin(x-y) = \sin^2 x - \sin^2 y.
7 Marks
OR OPTION
(c)
Find lim⁡x→0sin⁡3xx\lim_{x \to 0} \frac{\sin 3x}{x}. Also Define limits and continuity of function.
7 Marks

Question 3

14 MarksMedium
(a)
Check continuity of f(x)={x2x<13x−2x≥1f(x) = \begin{cases} x^2 & x < 1 \\ 3x - 2 & x \ge 1 \end{cases} at x=1x = 1.
3 Marks
(b)
If y=x2sin⁡xy = x^2\sin x, find dydx\frac{dy}{dx} and d2ydx2\frac{d^2 y}{dx^2}.
4 Marks
(c)
Differentiate the function sin⁡(3x+5)cos⁡x\frac{\sin(3x + 5)}{\cos x}.
7 Marks
OR OPTION
(a)
Find ∫(sin⁡x+cos⁡2x) dx\int (\sin x + \cos 2x) \, dx.
3 Marks
(b)
Find ∫12(4x3−5x2+6x+9) dx\int_1^2 (4x^3 - 5x^2 + 6x + 9) \, dx.
4 Marks
(c)
Integrate the function ∫x(x+1)(x+2) dx\int \frac{x}{(x+1)(x+2)} \, dx.
7 Marks

Question 4

14 MarksMedium
(a)
Find the transpose of the given matrix (1208)\begin{pmatrix} 1 & 2 \\ 0 & 8 \end{pmatrix}.
3 Marks
(b)
Write Minors and Cofactors of the determinant ∣2−403∣\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}.
4 Marks
(c)
Find the inverse of the matrix (123024005)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{pmatrix}.
7 Marks
OR OPTION
(a)
Compute the products [1−223](123231)\begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}.
3 Marks
(b)
If A and B are symmetric matrices, prove that AB−BAAB - BA is a skew symmetric matrix.
4 Marks
(c)
Solve system of linear equation, using matrix method: 4x−3y=34x - 3y = 3 and 3x−5y=73x - 5y = 7.
7 Marks

Question 5

14 MarksMedium
(a)
Find the degree of the differential equations:
1y′′+sin⁡x−5=0y'' + \sin x - 5 = 0
2dydx=sin⁡(2x−8)\frac{dy}{dx} = \sin(2x - 8)
3 Marks
(b)
Find the solution of the differential equation x5dydx=−y5x^5 \frac{dy}{dx} = -y^5.
4 Marks
(c)
Find a particular solution of the differential equation dydx+ycot⁡x=4xcosec⁡x\frac{dy}{dx} + y\cot x = 4x\operatorname{cosec} x given that y=0y = 0 when x=0x = 0.
7 Marks
OR OPTION
(a)
Is y=x2+2x+cy = x^2 + 2x + c a solution of y′−2x−2=0y' - 2x - 2 = 0?
3 Marks
(b)
Find the particular solution of the differential equation: x(x2−1)dydx=1x(x^2 - 1)\frac{dy}{dx} = 1; y=0,x=2y = 0, x = 2.
4 Marks
(c)
Find the solution of the differential equation dydx+2ytan⁡x=sin⁡x\frac{dy}{dx} + 2y\tan x = \sin x when y=0,x=0y = 0, x = 0.
7 Marks
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About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Remedial Mathematics (Remedial Maths) (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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