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

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 1st And 2nd Sem Old 110014 Calculus Previous Question Paper

This post was last modified on 20 February 2020

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GTU BE/B.Tech 2019 Winter Question Papers || Gujarat Technological University

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GUJARAT TECHNOLOGICAL UNIVERSITY

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

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Subject Code: 110014 Date: 17/01/2020

Subject Name: Calculus

Time: 10:30 AM TO 01:30 PM

Instructions:

Attempt any five questions.

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Make suitable assumptions wherever necessary.
Figures to the right indicate full marks.

Q.1	(a)	(i) Test the convergence of the sequence $(\frac{n^2 + n}{n^2 - n})$ .	03
	(ii) Expand log x in powers of (x—1).	04
(b)	(i) Evaluate $lim_{x \to 0} (\frac{cot x}{cos x})$ .	03
	(ii) Test the convergence of the series $\sum_{n=1}^{\infty} \frac{2^n}{5n^2 + 11n + 7}$ .	04
Q.2	(a)	(i) Evaluate $lim_{x \to \infty} x - 2log(x-1)$ .	03
	(ii) Determine the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{x^n}{n}, x > 0$ .	04
(b)	Expand logcos (x+ $\frac{\pi}{3}$ ) using Taylor's theorem in ascending powers of x and hence find the value of log(cos48°) correct up to three decimal places.	07
Q.3	(a)	(i) Evaluate $\int \frac{1}{\sqrt{x^2 - 2}} dx$ .	03
	(ii) Test the convergence of the improper integral $\int_1^{\infty} \frac{3x+5}{x^4+7} dx$ .	04
(b)	(i) Show that $f(x,y) = \begin{cases} \frac{x^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ is continuous at origin.	04
	(ii) If u = $e^{xyz}$ , show that $\frac{\partial^3 u}{\partial x \partial y \partial z} = (1 + 3xyz + x^2y^2z^2)e^{xyz}$ .	03
Q.4	(a)	If u = f(r) and $r^2 = x^2 + y^2 + z^2$ , prove that $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = f''(r) + \frac{2}{r}f'(r)$ .	07
(b)	(i) If u = f( $\frac{x}{y}, \frac{y}{z}, \frac{z}{x}$ ), prove that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z} = 0$ .	04
	(ii) Find the equations of the tangent plane and normal line to the surface z=2x²+y² at the point (1,1,3).	03
Q.5	(a)	If u = cos^-1( $\frac{x+y}{\sqrt{x}+\sqrt{y}}$ ), show that (i) $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = -\frac{1}{2}cot u$ (ii) $x^2\frac{\partial^2 u}{\partial x^2} + 2xy\frac{\partial^2 u}{\partial x \partial y} + y^2\frac{\partial^2 u}{\partial y^2} = \frac{sin^2 u(sin u - 4cos u)}{sin^3 u}$	07
(b)	(i) Trace the curve x³ + y³ = 3axy, a > 0	04
	(ii) Discuss the maxima and minima of the function x² + y² + 6x + 12	03
Q.6	(a)	(i) Evaluate $\int_0^1 \int_0^{\sqrt{1-x^2}} \frac{1}{1+x^2+y^2} dy dx$	03
	(ii) Evaluate $\int_0^1 \int_x^1 e^{y^2} dy dx$ by changing the order of integration.	04
(b)	The temperature at any point (x, y, z) in space is T = 400xyz². Find the highest temperature on the surface of unit sphere x² + y² + z² = 1 by the method of Lagrange's multipliers.	07
Q.7	(a)	Evaluate $\iint_R (x^2 + y^2) dA$ , by changing the variables, where R is the region lying in the first quadrant and bounded by the hyperbolas x² - y² = 1, x² - y² = 9, xy = 2 and xy = 4	07
(b)	Evaluate $\int_{-1}^1 \int_0^1 \int_0^{\sqrt{1-y^2}} e^{-(x^2+y^2+z^2)^{3/2}} dx dy dz$	07

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This post was last modified on 20 February 2020

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