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

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Enrolment No.

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER- I & II (OLD) EXAMINATION — WINTER 2019

Subject Code: 110008 Date: 17/01/2020

Subject Name: Maths - I

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Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

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

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1. (1) State Sandwich theorem, using it find lim g(x) if 3—x² < g(x)<3secx for all x.
   x?0
2. (11) Can Rolle’s theorem for f(x)= |x|, x? [-1,1] applied?

If u=f(x—y,y—z,z-x), Prove that ?u/?x + ?u/?y + ?u/?z =0

1. (1) Use Taylor’s series to find the expansion of log_e x in powers of (x—1).

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1. (1) Use L’Hospital rule, Evaluate lim (log x / cot x)
   x?0

Trace the curve r² =a² cos 2? .

1. (i) Test the convergence of S (2ⁿ -1) / n³
2. (11) Does the sequence {n / (n+3)} monotone?

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If u=tan^-1[(x³+y³)/(x-y)], Prove that x² ?²u/?x² +2xy ?²u/?x?y +y² ?²u/?y² =2cos3usinu.

1. (1) Test the convergence of S (3ⁿ n²) / n! , by Ratio Test.
2. (11) Discuss the convergence if the series 4—9+5+4—9+5+4—9+5................

Find the extremum values for f(x, y) = x³ +y³ —3xy.

1. (i) Expand x³y+3y—2 in the neighbourhood of the point (1,-2).

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2. (11) Find the equation for tangent plane and normal line at the point (1,1,1) on the surface x²+ y² +z²=3.

Find the Volume of sphere x² +y² +z² =a².

1. (1) Evaluate ?₀⁸ ?_x⁸ e^-y2 dy dx , by changing the order of integration.
2. (ii) Find the value of m if F = (x+2y)i+(my+4z)j+(5z+6x)k is solenoidal.

Evaluate ?_R (x+ y)dydx , where R is the region bounded by x=0,x=2,y=x,y=x+2.

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1. (1) Evaluate ?₀¹ ?₀^2p ?₀² (r² cos² ? + z²)rd?drdz

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1. (11) Using Green’s theorem to evaluate the integral §(yzdx +x²dy), where C:The triangle bounded by x=0,x+y=1,y=0.

Use divergence theorem to evaluate ?_s (x³dydz +x² ydzdx + xzzdxdz) where S is the closed surface consisting of the cylinder x² +y² = a² and the circular discs z =0 and z=b.

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