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Gujarat Technological UniversitySummer 2026 Examination

GTU 3110019 Remedial Mathematics (Remedial Maths) Summer 2026 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)
Define exponential function and logarithmic function.
3 Marks
(b)
Find the limits: (a) lim⁡x→1x2+1x+99\lim_{x \to 1} \frac{x^2 + 1}{x + 99} (b) lim⁡x→1x15−1x10−1\lim_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}.
4 Marks
(c)
Solve (x4+y4)dx−xy3dy=0(x^4 + y^4)dx - xy^3 dy = 0.
7 Marks

Question 2

14 MarksMedium
(a)
Evaluate the determinant A=∣1−2302401−1∣A = \begin{vmatrix} 1 & -2 & 3 \\ 0 & 2 & 4 \\ 0 & 1 & -1 \end{vmatrix}.
3 Marks
(b)
Find A⋅BA \cdot B if A=[58−23]A = \begin{bmatrix} 5 & 8 \\ -2 & 3 \end{bmatrix} and B=[1−14320]B = \begin{bmatrix} 1 & -1 & 4 \\ 3 & 2 & 0 \end{bmatrix}.
4 Marks
(c)
iSolve dydx+2ytan⁡x=sin⁡x\frac{dy}{dx} + 2y\tan x = \sin x.
iiDerive sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2\sin x \cos x.
7 Marks
OR OPTION
(c)
iFind dydx\frac{dy}{dx}, if x=acos⁡θ,y=asin⁡θx = a\cos\theta, y = a\sin\theta.
iiDerive the trigonometry identity: cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x - \sin^2 x.
7 Marks

Question 3

14 MarksMedium
(a)
Define the matrices: (a) Identity matrix (b) Scalar matrix (c) Diagonal matrix.
3 Marks
(b)
Find the coordinates of the point which divides the line segment joining the points (6,3)(6, 3) and (4,−5)(4, -5) in the ratio 3:23:2 (i) internally and (ii) externally.
4 Marks
(c)
iIf ∣2451∣=∣2x46x∣\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix} then find the value of xx.
iiIf the angle between two lines is π4\frac{\pi}{4} and slope of one of the lines is 14\frac{1}{4}, find the slope of the other line.
7 Marks
OR OPTION
(a)
State the Quotient (division) rule of derivative and compute the derivative of f(x)=sin⁡xcos⁡xf(x) = \frac{\sin x}{\cos x}.
3 Marks
(b)
A quadrilateral has the vertices at the points (−4,2),(2,6),(8,5)(-4, 2), (2, 6), (8, 5) and (9,−7)(9, -7). Show that the mid points of the sides of this quadrilateral are the vertices of a parallelogram.
4 Marks
(c)
iFind the angle between the lines y−3x−5=0y - \sqrt{3}x - 5 = 0 and 3y−x+6=0\sqrt{3}y - x + 6 = 0.
iiSolve the following system of linear equations by matrix method: x+y+2z=8x + y + 2z = 8 −x−2y+3z=1-x - 2y + 3z = 1 3x−7y+4z=103x - 7y + 4z = 10
7 Marks

Question 4

14 MarksMedium
(a)
Determine the order of the following differential equations:
i(d3ydx3)3+5(d2ydx2)4+2y=0\left(\frac{d^3 y}{dx^3}\right)^3 + 5\left(\frac{d^2 y}{dx^2}\right)^4 + 2y = 0
ii(d4ydx4)4+(d2ydx2)+sin⁡y=0\left(\frac{d^4 y}{dx^4}\right)^4 + \left(\frac{d^2 y}{dx^2}\right) + \sin y = 0
iii(y′′)2+3(y′)5−y=0(y'')^2 + 3(y')^5 - y = 0
3 Marks
(b)
If y=(cos⁡x)log⁡xy = (\cos x)^{\log x} then find dydx\frac{dy}{dx}.
4 Marks
(c)
Solve the differential equation by method of separation of variables: ∂u∂x+∂u∂y=2(x+y)u\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 2(x+y)u.
7 Marks
OR OPTION
(a)
Test the function for continuity at x=0x = 0, where f(x)={xsin⁡1xx≠00x=0f(x) = \begin{cases} x \sin\frac{1}{x} & x \neq 0 \\ 0 & x = 0 \end{cases}.
3 Marks
(b)
Find dydx\frac{dy}{dx} of the following function: y=1+cos⁡xsin⁡xy = \frac{1 + \cos x}{\sin x}.
4 Marks
(c)
Solve the differential equation by method of separation of variables: ∂u∂x=4∂u∂y\frac{\partial u}{\partial x} = 4\frac{\partial u}{\partial y}, given that u(0,y)=8e−3yu(0, y) = 8e^{-3y}.
7 Marks

Question 5

14 MarksMedium
(a)
Evaluate ∫x4tan⁡x5 dx\int x^4 \tan x^5 \, dx.
3 Marks
(b)
Evaluate by partial fractions: ∫x+43+2x−x2 dx\int \frac{x+4}{3 + 2x - x^2} \, dx.
4 Marks
(c)
Show that the matrix A=[2−11−12−11−12]A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} satisfies the equation A3−6A2+9A−4I=0A^3 - 6A^2 + 9A - 4I = 0. Hence find A−1A^{-1}.
7 Marks
OR OPTION
(a)
Evaluate ∫sin⁡xsin⁡(x−a) dx\int \frac{\sin x}{\sin(x-a)} \, dx.
3 Marks
(b)
Evaluate by partial fractions: ∫x(2x+1)(x2+1) dx\int \frac{x}{(2x+1)(x^2 + 1)} \, dx.
4 Marks
(c)
Find the inverse of the matrix A=[234431124]A = \begin{bmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \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 Remedial Mathematics (Remedial Maths) (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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