Download GTU BE/B.Tech 2019 Summer 1st Sem And 2nd Sem Old 110014 Calculus Question Paper

Seat No.:

Enrolment No.

GUJARAT TECHNOLOGICAL UNIVERSITY

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

Subject Code: 110014

Date: 06/06/2019

Subject Name: Calculus

Time: 10:30 AM TO 01:30 PM 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

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1. State Euler’s theorem on homogeneous function. If u = tan^-1 [ (x³ + y³) / (x - y) ] , then prove that x² (?²u / ?x²) + 2xy (?²u / ?x?y) + y² (?²u / ?y²) = 2sinucos³u [03]
2. If u=xy² +y³ +x³ +z³ show that x(?u/?x) + y(?u/?y) + z(?u/?z) = 3u [03]
3. If u = f(x³ +y³ +z³ - 3xyz), prove that (?u/?x) + (?u/?y) + (?u/?z) = 0 [05]

Q.2

1. Determine whether, lim _(x,y)->(0,0) (x² - y²) / (x² + y²) exist or not? If they exist find the value of the limit. [05]

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2. Find maxima and minima of the function f(x,y)=x³+y³ -3x-12y+20 [05]
3. Expand e^x tan^-1 y about (1,1) up to second degree in (x—1) and (y —1). [04]

Q.3

1. If x=rcos?, y=rsin?, find ?(x,y) / ?(r, ?) and ?(r, ?) / ?(x,y) [05]
2. Expand cosx in Maclaurin’s series. [02]

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3. Evaluate lim _x->0 (tanx —sin x) / x³ [03]

Q.4

1. Find the values of a and b such that, lim _x->0 (x(1+acosx)—bsinx) / x³ = 1 [04]
2. Using Taylor’s series find ³v27.12 correct to four decimal places. [05]
3. Trace the curve y²(a +x)=x²(3a - x) [05]

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Q.5

1. Trace the curve r = a(1 +cos?) [04]
2. Using reduction formula evaluate (i) ?₀^p/2 cos⁵ x dx and (ii) ?₀^p/2 sin⁶ x dx [02]
3. Test the convergence of S_n=1⁸ n / (n³ +1) [03]

Q.6

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1. Test the convergence of S_n=1⁸ 3ⁿ / n² [04]
2. Test the convergence of S_n=1⁸ ne^-n2 [05]
3. Obtain the reduction formula for ?₀^p/2 cosⁿ xdx [05]

Q.7

1. Evaluate ?_R sin ? dA , where R the region is in the 1^st quadrant. i.e. outside the circle r =2 and inside the cardioids r = 2(1 +cos ?). [05]

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2. Evaluate by changing the order of integration ?₀² ?_x^v12-x2 xy dydx [05]
3. Find the volume bounded by cylinder x² + y² =4 and the planes y+z=4, z=0. [04]

Q.8

1. Prove that ?₁⁸ (1/x^p) dx , converges when p >1 and diverges when p <1 [05]
2. Use triple integral in cylindrical co-ordinate to find the volume of solid, bounded above the hemisphere z=v(25— x² — y²), below by xy — plane and laterally by the cylinder x² +y² =9 [05]

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3. Find the volume of a cone with height 4cm and radius of base 4cm . Use the method of slicing. [04]

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