Download GTU BE/B.Tech 2019 Winter 1st And 2nd Sem New And Spfu 2110014 Calculus Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- I & II (NEW) EXAMINATION — WINTER 2019

Subject Code: 2110014 Date: 17/01/2020
Subject Name: Calculus
Time: 10:30 AM TO 01:30 PM Total Marks: 70

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Instructions:

1. Question No. 1 is compulsory. Attempt any four out of remaining Six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 Objective Question (MCQ) Mark

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(a) 07

1. The sum of the series 1+1/2+1/4+1/8+...
   A) 1 B) 2 C) 3 D) Infinity
2. The series S 1/n is
   n=1
   

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   (A) convergent (B) divergent (C) Oscillating (D) none
3. The series S 1/n² is
   n=1
   (A) convergent (B) divergent (C) Oscillating (D) none
4. The curve y² = x(x-1)² is symmetric about
   

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   (A) x-axis (B) y-axis (C) Line y=x (D) origin
5. A point (a,b) is said to be a saddle point if at (a,b)
   (A) r*t-s²>0 B) r*t-s²<0 C) r*t-s²=0 (D) r*t-s²<0
6. The volume of solid generated by revolving a circle x² +y² =9 about x-axis
   (A) 4p/3 (B) 36 (C) 36p (D) 36

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7. The value of lim (sin x)/x
   X?8
   A) 1 B) 0 C) 2 (D) Infinity

(b) 07

1. Which of the following is homogeneous function of degree one?
   

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   A) x/y B) y/x (C) x+y (D) x/(x²+xy)
2. The value of lim (x-y)/(x²-y²)
   (x,y)->(1,1)
   a) 1/2 B) 0 (C) Infinity (D) 2
3. If x=rcos?,y=rsin? then the value of ?(x,y)/?(r,?) is
   

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   (A) cos? (B) sec? (C) cosec? (D) None
4. The value of ?₀²?_x² e^-y2 dxdy
   (A) 2ln2-2 (B) 2ln2-1 (C) ln2 (D) 1/2
5. The value of lim x^x
   x?0
   

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   (A) 1 (B) e (C) 8 (D) 0
6. The value of ?₀¹?₀¹?₀¹ dxdydz
   (A) 1/2 (B) 1/3 (C) 1/6 (D) 1
7. The value of ?₀^p/2?₀⁴ rdrd?
   (A) 16 (B) 8p (C) 3p/2 (D) 64

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Q.2 (a) Define Jacobian and show that J J' = 1. 03

(b) Find the equations of tangent plane and normal line to x² + y² +z² =81 at the point (—1,—4,8) 04

(c) A rectangular box open at the top is to have a volume of 108 c.c. find the dimension of the box requiring least material for its construction. 07

Q.3 (a) Show that ?²Q/?u?v = ?²Q/?v?u where Q= y³+x² 03

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(b) Discuss the continuity of f(x,y)= xy/(x²+y²) if (x,y)?(0,0) and f(x,y)=0 if (x,y)=(0,0) 04

(c) State and prove Euler’s Theorem for Homogeneous functions. Also, if u=sin^-1(x²+y²)^1/2/(x³+y³)^1/3 then show that
i. Xu_x +yu_y =1/3 tanu
ii. x²u_xx +2xyu_xy +y²u_yy =(2/9)(tan²u—tanu) 07

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Q.4 (a) Evaluate ? rsin ?drd? over the area of the curve r = a(1+cos?) above the initial line. 03

(b) Evaluate the integral by the changing the order of integration, ?₀¹?₀^y (x² + 1)dxdy 04

(c) i. Use triple integral to find the volume of the cylinder x² +y² =1 between the planes z=1and z=2. 03
ii. Evaluate ?₀⁸?₀⁸ e^-(x2+y2) dxdy by changing to polar coordinates 07

Q.5 (a) Test the convergence of the series S 1/(n(ln(n+1))), if convergent then find its value. 03

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(b) Test the convergence of the series 1/1+2+3 + 3/2+3+4 + 5/3+4+5 +... 04

(c) For which value of x does the series x-x²/2+x³/3-x⁴/4+x⁵/5... is absolute or conditionally convergent or divergent? What is the radius of convergent of S xⁿ/n 07

Q.6 (a) Determine the convergent of S tan^-1(1/n)/(1+n) 03

(b) Find the volume of the solid generated by revolving the region bounded by x =y² and the lines x = 0,x=2 about the x-axis. 04

(c) Trace the curve r=a (1 +cos ?);a > 0. 07

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Q.7 (a) Expand sin(x+p/4) in powers of x by using the Taylor’s series. Also, find the value of sin46°. 03

(b) Find lim(x^x-1)/x
x>0 04

(c) Discuss the convergence of the following integrals:
(i) ?₀² 1/(x-1)²dx (ii) ?₀⁸ e^-x2dx 07

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