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Gujarat Technological UniversityWinter 2024 Examination

GTU 3110019 Remedial Mathematics (Remedial Maths) Winter 2024 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 lim⁡x→2(x2−6x+8x2−4)\lim_{x \to 2} \left(\frac{x^2 - 6x + 8}{x^2 - 4}\right).
3 Marks
(b)

Find dydx\frac{dy}{dx} for the following:

1y=ecot⁡xy = e^{\cot x}
2y=log⁡(sin⁡x2)y = \log(\sin x^2)
4 Marks
(c)
Show that the function y=e3x+e2xy = e^{3x} + e^{2x} is a solution of differential equation d2ydx2−5dydx+6y=0\frac{d^2 y}{dx^2} - 5\frac{dy}{dx} + 6y = 0.
7 Marks

Question 2

14 MarksMedium
(a)
Find the equation of a line passing through the points (1,−3)(1, -3) & (5,6)(5, 6).
3 Marks
(b)
Find the equation of a line passing through (−3,5)(-3, 5) and perpendicular to the line through the points (2,5)(2, 5) and (−3,6)(-3, 6).
4 Marks
(c)
1If a line passing through the points (−2,5)(-2, 5) and (4,9)(4, 9) is perpendicular to the line passing through the points (6,8)(6, 8) and (x,20)(x, 20) then find the value of xx. (3 marks)
2Find the angle between two lines 3x−y+5=03x - y + 5 = 0 and 2x−4y+11=02x - 4y + 11 = 0. (4 marks)
7 Marks
OR OPTION
(c)
1Represent cos⁡(x+y)\cos(x+y) in terms of sine and cosine and hence prove that cos⁡(2x)=cos⁡2x−sin⁡2x\cos(2x) = \cos^2 x - \sin^2 x. (3 marks)
2Find the distance of the point (5,2)(5, 2) from the line 3x−4y+13=03x - 4y + 13 = 0. (4 marks)
7 Marks

Question 3

14 MarksMedium
(a)
Find the value of xx, such that ∣x32x∣=∣14435∣\begin{vmatrix} x & 3 \\ 2 & x \end{vmatrix} = \begin{vmatrix} 14 & 4 \\ 3 & 5 \end{vmatrix}.
3 Marks
(b)
If A=[3124]A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} & B=[−1205]B = \begin{bmatrix} -1 & 2 \\ 0 & 5 \end{bmatrix} then verify that AB≠BAAB \neq BA.
4 Marks
(c)
Solve the following system of linear equations:

x+y+2z=8x + y + 2z = 8 −x−2y+3z=1-x - 2y + 3z = 1 3x−7y+4z=103x - 7y + 4z = 10

7 Marks
OR OPTION
(a)
Define (a) Identity matrix (b) Scalar matrix and (c) Diagonal matrix.
3 Marks
(b)
If A=[3124]A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} & B=[−123054]B = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 5 & 4 \end{bmatrix} then find A⋅BA \cdot B.
4 Marks
(c)
Solve the following system of linear equations:

x+2y−z=1x + 2y - z = 1 x+y+2z=9x + y + 2z = 9 2x+y−z=22x + y - z = 2

7 Marks

Question 4

14 MarksMedium
(a)
Determine the order and degree of following differential equations:
1d2ydx2−3(dydx)3+4y=0\frac{d^2 y}{dx^2} - 3\left(\frac{dy}{dx}\right)^3 + 4y = 0
2(d2ydx2)3−(d3ydx3)13−y=0\left(\frac{d^2 y}{dx^2}\right)^3 - \left(\frac{d^3 y}{dx^3}\right)^{\frac{1}{3}} - y = 0
3 Marks
(b)
Solve: dydx+yx=x2\frac{dy}{dx} + \frac{y}{x} = x^2.
4 Marks
(c)
1Solve: x2dy=y(x2+x+1)dxx^2 dy = y(x^2 + x + 1)dx. (3 marks)
2If y=xe3xy = x e^{3x} find d2ydx2\frac{d^2 y}{dx^2}. (4 marks)
7 Marks
OR OPTION
(a)
Evaluate: lim⁡x→1(x100−1x−1)\lim_{x \to 1} \left(\frac{x^{100} - 1}{x - 1}\right).
3 Marks
(b)
Solve: dydx+3y=e−x\frac{dy}{dx} + 3y = e^{-x}.
4 Marks
(c)
1If y=exx2+1y = \frac{e^x}{x^2 + 1} then find dydx\frac{dy}{dx}. (3 marks)
2Solve: dydx=1+y21+x2\frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. (4 marks)
7 Marks

Question 5

14 MarksMedium
(a)
Evaluate: ∫(x2−7x+10x) dx\int \left(\frac{x^2 - 7x + 10}{x}\right) \, dx.
3 Marks
(b)
Evaluate: ∫(e2x1+e2x) dx\int \left(\frac{e^{2x}}{1 + e^{2x}}\right) \, dx.
4 Marks
(c)
Evaluate the following integrations:
1∫(2e3x+sin⁡2x+3) dx\int (2e^{3x} + \sin 2x + 3) \, dx (3 marks)
2∫021(x+1)(x+2) dx\int_0^2 \frac{1}{(x+1)(x+2)} \, dx (4 marks)
7 Marks
OR OPTION
(a)
Evaluate: ∫(1x3+1x2) dx\int \left(\frac{1}{x^3} + \frac{1}{x^2}\right) \, dx.
3 Marks
(b)
Evaluate: ∫(etan⁡−1x1+x2) dx\int \left(\frac{e^{\tan^{-1} x}}{1 + x^2}\right) \, dx.
4 Marks
(c)
Evaluate the integration:
1∫sin⁡3xcos⁡2x dx\int \sin 3x \cos 2x \, dx (3 marks)
2∫23(1x2−1) dx\int_2^3 \left(\frac{1}{x^2 - 1}\right) \, dx (4 marks)
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 2024, 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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