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Gujarat Technological UniversitySummer 2024 Examination

GTU 3110019 Remedial Mathematics (Remedial Maths) Summer 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 Hours2:30 PM – 5:00 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Define Signum 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)
iFind dydx\frac{dy}{dx} if y=x5−cos⁡xsin⁡xy = \frac{x^5 - \cos x}{\sin x}. (3 marks)
iiSolve the following system of linear equations by matrix method: x+y+z=6x + y + z = 6 y+3z=11y + 3z = 11 x−2y+z=0x - 2y + z = 0 (4 marks)
7 Marks

Question 2

14 MarksMedium
(a)
Without expanding prove that ∣x+yy+zx+zzxy111∣=0\begin{vmatrix} x+y & y+z & x+z \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} = 0.
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. (5 marks)
iiDerive sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2\sin x \cos x. (2 marks)
7 Marks
OR OPTION
(c)
iFind dydx\frac{dy}{dx}, if x=acos⁡θ,y=asin⁡θx = a\cos\theta, y = a\sin\theta. (5 marks)
iiDerive the trigonometry identity: cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x - \sin^2 x. (2 marks)
7 Marks

Question 3

14 MarksMedium
(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)

Find the distance (i) of the point (3,−1)(3, -1) from the line 3x−4y−26=03x - 4y - 26 = 0.

iibetween two parallel lines: 15x+8y−34=015x + 8y - 34 = 0 and 15x+8y+31=015x + 8y + 31 = 0.
4 Marks
(c)
iDefine the matrices: (a) Identity matrix (b) Scalar matrix (c) Diagonal matrix. (3 marks)
iiFind 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. (4 marks)
7 Marks
OR OPTION
(a)
State the product rule of derivative and find the derivative of f(x)=x+3x2+5x3+7x4+9x5f(x) = x + 3x^2 + 5x^3 + 7x^4 + 9x^5 at x=1x = 1.
3 Marks
(b)
Find the point on the x-axis which is at equidistance from the point (7,6)(7, 6) and (3,4)(3, 4).
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. (3 marks)
iiIf 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)
7 Marks

Question 4

14 MarksMedium
(a)
Determine the order of the following differential equations:
id3ydx3+4(d2ydx2)3+2y=0\frac{d^3 y}{dx^3} + 4\left(\frac{d^2 y}{dx^2}\right)^3 + 2y = 0
ii(d4ydx4)2+(d2ydx2)+sin⁡y=0\left(\frac{d^4 y}{dx^4}\right)^2 + \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)
Solve the differential equation y dx−(x+2y2) dy=0y \, dx - (x + 2y^2) \, dy = 0.
4 Marks
(c)
Solve the following equations:
idydx=1+y21+x2\frac{dy}{dx} = \frac{1 + y^2}{1 + x^2} (3 marks)
iixcos⁡(yx)dydx=ycos⁡(yx)+xx\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x (4 marks)
7 Marks
OR OPTION
(a)
Find all points of discontinuity for the function given by f(x)={x+2x<12x=1x−2x>1f(x) = \begin{cases} x+2 & x < 1 \\ 2 & x = 1 \\ x-2 & x > 1 \end{cases}.
3 Marks
(b)
If y=xcos⁡xy = x\cos x then find d2ydx2\frac{d^2 y}{dx^2}.
4 Marks
(c)
Solve the following equations:
ix dy=(2x2+1) dxx \, dy = (2x^2 + 1) \, dx (3 marks)
iixdydx+2y=x2x\frac{dy}{dx} + 2y = x^2 (x≠0)(x \neq 0) (4 marks)
7 Marks

Question 5

14 MarksMedium
(a)
Find the integrals: (i) ∫x2(1−1x2) dx\int x^2 \left(1 - \frac{1}{x^2}\right) \, dx (ii) ∫(4e3x+sin⁡x+1) dx\int (4e^{3x} + \sin x + 1) \, dx.
3 Marks
(b)
Evaluate the integrals: (i) ∫dxx2−81\int \frac{dx}{x^2 - 81} (ii) ∫x2−1x dx\int \frac{x^2 - 1}{x} \, dx.
4 Marks
(c)
iEvaluate ∫121(x+1)(x+2) dx\int_1^2 \frac{1}{(x+1)(x+2)} \, dx (3 marks)
iiFind ∫1x2−6x+13 dx\int \frac{1}{x^2 - 6x + 13} \, dx (4 marks)
7 Marks
OR OPTION
(a)
State the integration by parts and find ∫xex dx\int x e^x \, dx.
3 Marks
(b)
Find ∫sin⁡x(1+cos⁡x)2 dx\int \frac{\sin x}{(1 + \cos x)^2} \, 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 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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