Download GTU BE/B.Tech 2019 Summer 1st Sem And 2nd Sem Old 110008 Maths I Question Paper

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Subject Code: 110008

GUJARAT TECHNOLOGICAL UNIVERSITY

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BE - SEMESTER-I & II (OLD) EXAMINATION - SUMMER-2019

Subject Name: Maths - I

Time: 10:30 AM TO 01:30 PM

Date: 06/06/2019

Total Marks: 70

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Instructions:

1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 (a) (i) Evaluate: lim (1 — x) tan (px/2) 02

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x?1
(ii) Evaluate : lim (x² -1) / (1 - cosx) 02
x?0
(iii) Find Jacobian for functions u = xsiny, v = ysinx. 03

(b) (i) Sketch the region and find the area bounded by ellipse x²/a² + y²/b² = 1 03

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(ii) Using Lagrange’s mean value theorem, prove that 2b²/(1+b²) < tan^-1b — tan^-1a < 2a²/(1+a²) 04

Q.2 (a) (i) Verify Rolle’s theorem for f(x) = x(x + 3)e^-x/2 in -3 < x < 0. 04
(ii) Find two non-negative numbers whose sum is 9 such that the product of one number and the square of the other is maximum. 03

(b) (i) Prove that tan^-1(1/3) + tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/8) = p/4 04
(ii) Find the absolute maximum and minimum values of f(x) = x³ - x² in interval [-1,1]. 03

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Q.3 (a) (i) Test the convergence of the series: S (1/n^x), x > 0. 04
(ii) Test the convergence of the series: S (1/n!) 03

(b) (i) Test the convergence of the series: 1 - 1/2 + 1/3 - 1/4 + ... 04
(ii) Test the convergence of the series: S (n/(n+1)!) 03

Q.4 (a) (i) Find extreme values of f(x,y) = x³ + 3xy² —3x² — 3y² +7 04

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(ii) Find all first and second order partial derivatives for f(x,y) = x²siny + y² cos x. Hence, verify mixed derivative Theorem. 03

(b) (i) If u = tan^-1(x³+y³)/(x+y), show that x(?u/?x) + y(?u/?y) = 2sinucos3u. 04
(ii) If u=f(x—y,y —z,z—x) then prove that (?u/?x) + (?u/?y) + (?u/?z) = 0. 03

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Q.5 (a) Sketch the region of integration and evaluate by reversing the order of Integration for integral ?₀^4a ?_x²/4a^2vax dy dx 07

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(b) (i) Evaluate the integral ?₀⁸ ?₀⁸ e^-(x2+y2)dxdy by changing into polar coordinates. 04
(ii) Evaluate ?₀¹ ?₀^1-x ?₀^x+y e^zdxdydz. 03

Q.6 (a) (i) Find directional derivative of f = xy² + yz² at point (2,—1,1) in the direction of vector i + 2j + 2k. 04
(ii) If F= 3xi + (2xz - y)j + (z - 5)k, then show that curl F = 0. 03

(b) Evaluate ?_C F · dr, where, F = (x² + y²)i — 2xyj . Where, C is the rectangle in XY-plane bounded by y =0,x =a,y = b,x = 0. 07

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Q.7 (a) Verify Green’s theorem for ?_C [(x² — 2xy)dx + (x²y + 3)dy]. Where, C is the boundary of the region bounded by y = x² and the line y = x. 07

(b) (i) Show that F = (y²-z²+ 6xy - 2x)i + (3xz + 2xy)j + (3xy — 2xz + 2z)k is both solenoidal and irrotational. 04
(ii) Evaluate ?_C[e^x dx + 2ydy — dz] by Stoke’s theorem. where, C is the curve x²+y²=4,z=2. 03

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