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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

Tags:GTUBEB.Tech2019WinterQuestion PaperCalculus1100141st Semester2nd SemesterOldGujarat Technological UniversityPDFDownload
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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:

  1. Attempt any five questions.

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  2. Make suitable assumptions wherever necessary.
  3. 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