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Gujarat Technological UniversitySummer 2025 Examination

GTU 3110019 Remedial Mathematics (Remedial Maths) Summer 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 Hours10:30 AM – 1:00 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Find the slope of the line passing through the points (3,−2)(3, -2) and (3,4)(3, 4).
3 Marks
(b)
Find the equation of line passing through (−4,3)(-4, 3) with slope -5.
4 Marks
(c)
1Find the distance between the parallel lines 3x−4y+7=03x - 4y + 7 = 0 and 3x−4y+5=03x - 4y + 5 = 0. (3 marks)
2Find the angles between the lines 3x+y=1\sqrt{3}x + y = 1 and x+3y=1x + \sqrt{3}y = 1. (4 marks)
7 Marks

Question 2

14 MarksMedium
(a)
State domain and range of the function f(x)=cos⁡xf(x) = \cos x.
3 Marks
(b)
Deduce trigonometric identity cos⁡2x=1−tan⁡2x1+tan⁡2x\cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x}.
4 Marks
(c)
1Find d2ydx2\frac{d^2 y}{dx^2} if y=cos⁡2xy = \cos 2x. (3 marks)
2Find derivative of cos⁡x1+sin⁡x\frac{\cos x}{1 + \sin x}. (4 marks)
7 Marks
OR OPTION
(c)
1Find derivative of y=sin⁡xcos⁡xy = \sin x \cos x. (3 marks)
2Find dydx\frac{dy}{dx} if y=x1+tan⁡xy = \frac{x}{1 + \tan x}. (4 marks)
7 Marks

Question 3

14 MarksMedium
(a)
Evaluate lim⁡x→0sin⁡4xsin⁡2x\lim_{x \to 0} \frac{\sin 4x}{\sin 2x}.
3 Marks
(b)
If y=sin⁡−1xy = \sin^{-1} x, show that (1−x2)d2ydx2−xdydx=0(1 - x^2)\frac{d^2 y}{dx^2} - x\frac{dy}{dx} = 0.
4 Marks
(c)
Find the solution of the following:
1dydx=x+12−y\frac{dy}{dx} = \frac{x+1}{2-y}; (y≠2)(y \neq 2) (3 marks)
2dydx+3y=e−3x\frac{dy}{dx} + 3y = e^{-3x} (4 marks)
7 Marks
OR OPTION
(a)
Evaluate lim⁡x→2x3−2x2x2−5x+6\lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6}.
3 Marks
(b)
Find dydx\frac{dy}{dx} if x=cos⁡tx = \cos t and y=cos⁡2ty = \cos 2t.
4 Marks
(c)
Solve the following:
1xydydx=(x+2)(y+2)xy \frac{dy}{dx} = (x + 2)(y + 2) (3 marks)
2dydx=x+xy\frac{dy}{dx} = x + xy (4 marks)
7 Marks

Question 4

14 MarksMedium
(a)
Find ∫(1+log⁡x)2x dx\int \frac{(1 + \log x)^2}{x} \, dx.
3 Marks
(b)
Find ∫dxx2+2x+2\int \frac{dx}{x^2 + 2x + 2}.
4 Marks
(c)
Evaluate the following:
1∫04(x+e2x) dx\int_0^4 (x + e^{2x}) \, dx (3 marks)
2∫0π/4(2sec⁡2x+x3+2) dx\int_0^{\pi/4} (2\sec^2 x + x^3 + 2) \, dx (4 marks)
7 Marks
OR OPTION
(a)
Find ∫sin⁡−1x1−x2 dx\int \frac{\sin^{-1} x}{\sqrt{1 - x^2}} \, dx.
3 Marks
(b)
Find ∫x(x+1)(x+2) dx\int \frac{x}{(x+1)(x+2)} \, dx.
4 Marks
(c)
Evaluate the following:
1∫01xex2 dx\int_0^1 x e^{x^2} \, dx (3 marks)
2∫0πsin⁡3x dx\int_0^\pi \sin^3 x \, dx (4 marks)
7 Marks

Question 5

14 MarksMedium
(a)
Evaluate the determinant ∣3−1−200−13−50∣\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}.
3 Marks
(b)
If A=[31−12]A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, Show that A2−5A+7I=0A^2 - 5A + 7I = 0. Hence find A−1A^{-1}.
4 Marks
(c)
Solve the system of linear equations:

3x−2y+3z=83x - 2y + 3z = 8 2x+y−z=12x + y - z = 1 4x−3y+2z=44x - 3y + 2z = 4

7 Marks
OR OPTION
(a)
Find the value of xx, if ∣x218x∣=∣62186∣\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}.
3 Marks
(b)
By using the properties of determinants prove ∣1aa21bb21cc2∣=(a−b)(b−c)(c−a)\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = (a-b)(b-c)(c-a).
4 Marks
(c)
Solve the system of linear equations:

x−y+z=4x - y + z = 4 2x+y−3z=02x + y - 3z = 0 x+y+z=2x + y + z = 2

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) (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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