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Gujarat Technological UniversitySummer 2023 Examination

GTU 3110019 Remedial Mathematics (Remedial Maths) Summer 2023 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)
Evaluate the determinant A=∣124−130410∣A = \begin{vmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{vmatrix}.
3 Marks
(b)
Find A⋅BA \cdot B if A=[6923]A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix} and B=[260798]B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}.
4 Marks
(c)
1If [x+3z+42y−7−6a−10b−3−210]=[063y−2−6−32c+22b+4−210]\begin{bmatrix} x+3 & z+4 & 2y-7 \\ -6 & a-1 & 0 \\ b-3 & -21 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 6 & 3y-2 \\ -6 & -3 & 2c+2 \\ 2b+4 & -21 & 0 \end{bmatrix}, find a,b,c,x,ya, b, c, x, y and zz.
2Solve the following system of equations by matrix method: 2x+5y=12x + 5y = 1 3x+2y=73x + 2y = 7
7 Marks

Question 2

14 MarksMedium
(a)
Find distance of the point (3,−5)(3, -5) from the line 3x−4y−26=03x - 4y - 26 = 0.
3 Marks
(b)
If the angle between two lines is π4\frac{\pi}{4} and slope of one of the lines is 12\frac{1}{2}, find the slope of the other line.
4 Marks
(c)
1Define modulus function and greatest integer function.
2Deduce trigonometric identity sin⁡3x=3sin⁡x−4sin⁡3x\sin 3x = 3\sin x - 4\sin^3 x.
7 Marks
OR OPTION
(c)
1Define exponential function and logarithmic function.
2Deduce trigonometric identity sin⁡2x=2tan⁡x1+tan⁡2x\sin 2x = \frac{2\tan x}{1+\tan^2 x}, x≠nπ+π2,n∈Zx \neq n\pi + \frac{\pi}{2}, n \in \mathbb{Z}.
7 Marks

Question 3

14 MarksMedium
(a)
Evaluate: lim⁡x→2(x3−4x2+4xx2−4)\lim_{x \to 2} \left(\frac{x^3 - 4x^2 + 4x}{x^2 - 4}\right).
3 Marks
(b)
Find dydx\frac{dy}{dx} if y=x5−cos⁡xsin⁡xy = \frac{x^5 - \cos x}{\sin x}.
4 Marks
(c)
1Find derivative of y=sin⁡(x2)y = \sin(x^2).
2If y=3e2x+2e3xy = 3e^{2x} + 2e^{3x}, prove that d2ydx2−5dydx+6y=0\frac{d^2 y}{dx^2} - 5\frac{dy}{dx} + 6y = 0.
7 Marks
OR OPTION
(a)
Evaluate: lim⁡x→0(1−cos⁡xx)\lim_{x \to 0} \left(\frac{1 - \cos x}{x}\right).
3 Marks
(b)
Find dydx\frac{dy}{dx} if x=acos⁡θ,y=asin⁡θx = a\cos\theta, y = a\sin\theta.
4 Marks
(c)
1Find derivative of y=cos⁡−1(ex)y = \cos^{-1}(e^x).
2Find d2ydx2\frac{d^2 y}{dx^2} if y=x3+tan⁡xy = x^3 + \tan x.
7 Marks

Question 4

14 MarksMedium
(a)
Find all points of discontinuity for the function given by f(x)={x+2x<10x=1x−2x>1f(x) = \begin{cases} x+2 & x < 1 \\ 0 & x = 1 \\ x-2 & x > 1 \end{cases}.
3 Marks
(b)
Find order and degree of following differential equations:
1xyd2ydx2+x(dydx)2−ydydx=0xy \frac{d^2 y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0
2(d3ydx3)2+(d2ydx2)3+(dydx)4+y5=0\left(\frac{d^3 y}{dx^3}\right)^2 + \left(\frac{d^2 y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^4 + y^5 = 0
4 Marks
(c)
Solve the following:
1dydx=1+y21+x2\frac{dy}{dx} = \frac{1+y^2}{1+x^2}
2xcos⁡(yx)dydx=ycos⁡(yx)+xx\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x.
7 Marks
OR OPTION
(a)
Discuss the continuity of f(x)={x3+1x≠01x=0f(x) = \begin{cases} x^3 + 1 & x \neq 0 \\ 1 & x = 0 \end{cases} at x=0x = 0.
3 Marks
(b)

Find dydx\frac{dy}{dx} if:

1y+sin⁡y=cos⁡xy + \sin y = \cos x
2x−y=πx - y = \pi
4 Marks
(c)
Solve the following:
1x dy=(2x2+1) dxx \, dy = (2x^2 + 1) \, dx
2xdydx+2y=x2x \frac{dy}{dx} + 2y = x^2 (x≠0)(x \neq 0).
7 Marks

Question 5

14 MarksMedium
(a)
State integration by parts and find ∫xex dx\int x e^x \, dx.
3 Marks
(b)
Find the following integrals:
1∫x3−1x2 dx\int \frac{x^3 - 1}{x^2} \, dx
2∫(x2/3+1) dx\int (x^{2/3} + 1) \, dx
4 Marks
(c)
Evaluate followings:
1∫121(x+1)(x+2) dx\int_1^2 \frac{1}{(x+1)(x+2)} \, dx
2∫12tan⁡−1x1+x2 dx\int_1^2 \frac{\tan^{-1} x}{1+x^2} \, dx.
7 Marks
OR OPTION
(a)
Find ∫1−sin⁡xcos⁡2x dx\int \frac{1 - \sin x}{\cos^2 x} \, dx.
3 Marks
(b)
Find ∫1x2−6x+13 dx\int \frac{1}{x^2 - 6x + 13} \, dx.
4 Marks
(c)
Find ∫x2+x+1(x+1)(x2+1) dx\int \frac{x^2 + x + 1}{(x+1)(x^2 + 1)} \, dx.
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 2023, 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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