DEPARTMENT OF MATHEMATICS â&#x20AC;&#x153;T3, Examination, May-2018â&#x20AC;? Semester:Second Subject: Calculus-I Branch: Mathematics Course Type: Core Time: 3 Hours Max.Marks: 80

Date of Exam: 24/May/2018 Subject Code: MAH116T Session: I Course Nature: Hard Program: B.Sc Signature: HOD/Associate HOD:

Note: All questions are compulsory from Part A (2*10=20 Marks). Attempt any two questions from part B (15 marks each). Attempt any two questions from part C (15 marks each) Part-A Q1 (a) Show that đ?&#x2018;Ś = đ?&#x2018;&#x2019; đ?&#x2018;Ľ is everywhere concave upwards. (b) Examine the nature of the origin for the curveđ?&#x2018;Ś 2 = đ?&#x2018;&#x17D;đ?&#x2018;Ľ 2 + đ?&#x2018;&#x17D;đ?&#x2018;Ľ 3 according as đ?&#x2018;&#x17D; is positive, zero or negative. (c) Find the equations of tangent to the curve (đ?&#x2018;Ľ â&#x2C6;&#x2019; 2)2 = đ?&#x2018;Ś(đ?&#x2018;Ś â&#x2C6;&#x2019; 1)2 at (2,1) and discuss the nature of the point. (d) Define Double points. Give its types. (e) Give conditions for curve đ?&#x2018;&#x; = đ?&#x2018;&#x201C;(đ?&#x153;&#x192;) to be symmetrical about initial line and the line through the pole perpendicular to the initial line. đ?&#x2018;Ľđ?&#x2018;Ś đ?&#x2018;&#x2013;đ?&#x2018;&#x201C; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2030; (0,0) 2 +đ?&#x2018;Ś 2 2 đ?&#x2018;Ľ (f) Let đ?&#x2018;&#x201C;: đ?&#x2018;&#x2026; â&#x2020;&#x2019; đ?&#x2018;&#x2026; be defined as đ?&#x2018;&#x201C;(đ?&#x2018;Ľ, đ?&#x2018;Ś) = { 0 đ?&#x2018;&#x2013;đ?&#x2018;&#x201C; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ, đ?&#x2018;Ś) = (0,0) Show that đ?&#x2018;&#x201C; is not continuous at (0,0) (g) State and Prove Eulerâ&#x20AC;&#x2122;s theorem on Homogeneous functions. đ?&#x153;&#x2022;(đ?&#x2018;Ľ,đ?&#x2018;Ś) (h) In polar coordinates đ?&#x2018;Ľ = đ?&#x2018;&#x; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x153;&#x192; , đ?&#x2018;Ś = đ?&#x2018;&#x; đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x192;. Evaluate đ?&#x153;&#x2022;(đ?&#x2018;&#x;,đ?&#x153;&#x192;). (i) Define maximum value and minimum value of a function of two variables. đ?&#x2018;&#x2018;đ?&#x2018;˘ (j) If đ?&#x2018;˘ = đ?&#x2018;Ľđ?&#x2018;&#x2122;đ?&#x2018;&#x153;đ?&#x2018;&#x201D;đ?&#x2018;Ľđ?&#x2018;Ś where đ?&#x2018;Ľ 3 + đ?&#x2018;Ś 3 + 3đ?&#x2018;Ľđ?&#x2018;Ś = 1. Find đ?&#x2018;&#x2018;đ?&#x2018;Ľ `

Part-B đ?&#x2018;Ľ 3 â&#x2C6;&#x2019;đ?&#x2018;Ľ

Q2(a) Find the points of inflexion on the curve đ?&#x2018;Ś = 3đ?&#x2018;Ľ 2 +1. (b) Find the position and nature of the double points on the curve đ?&#x2018;Ľ 4 â&#x2C6;&#x2019; 4đ?&#x2018;&#x17D;đ?&#x2018;Ľ 3 + 2đ?&#x2018;&#x17D;đ?&#x2018;Ś 3 + 4đ?&#x2018;&#x17D;2 đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 3đ?&#x2018;&#x17D;2 đ?&#x2018;Ś 2 â&#x2C6;&#x2019; đ?&#x2018;&#x17D;4 = 0

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Q3(a) Sketch the graph of the curve đ?&#x2018;Ś = đ?&#x2018;Ľ(đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 3) (b) Trace the curve đ?&#x2018;&#x; = đ?&#x2018;&#x17D; cos 2đ?&#x153;&#x192; , đ?&#x2018;&#x17D; > 0

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Q4(a) Find the intervals in which the curveđ?&#x2018;Ś = (cos đ?&#x2018;Ľ + sin đ?&#x2018;Ľ)đ?&#x2018;&#x2019; đ?&#x2018;Ľ is concave upwards or downwards in 0 < đ?&#x2018;Ľ < 2đ?&#x153;&#x2039;

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(b) Trace the curve đ?&#x2018;Ľ = đ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; 3 đ?&#x153;&#x192;, đ?&#x2018;Ś = đ?&#x2018;?đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;3 đ?&#x153;&#x192;

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Part- C đ?&#x153;&#x2022;2đ?&#x2018;˘

đ?&#x153;&#x2022;2 đ?&#x2018;˘

1

Q5 (a) If đ?&#x2018;˘ = đ?&#x2018;&#x201C;(đ?&#x2018;&#x;) and đ?&#x2018;Ľ = đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x153;&#x192;; đ?&#x2018;Ś = đ?&#x2018;&#x;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x192;. Prove that đ?&#x153;&#x2022;đ?&#x2018;Ľ 2 + đ?&#x153;&#x2022;đ?&#x2018;Ś 2 = đ?&#x2018;&#x201C; â&#x20AC;˛â&#x20AC;˛ (đ?&#x2018;&#x;) + đ?&#x2018;&#x; đ?&#x2018;&#x201C; â&#x20AC;˛ (đ?&#x2018;&#x;) (b) x

đ?&#x2018;Ľ+đ?&#x2018;Ś If đ?&#x2018;Ľ = sinâ&#x2C6;&#x2019;1 đ?&#x2018;Ľ+ đ?&#x2018;Ś â&#x2C6;&#x161; â&#x2C6;&#x161; 2 đ?&#x153;&#x2022; 2đ?&#x2018;˘ 2 đ?&#x153;&#x2022; đ?&#x2018;˘ đ?&#x153;&#x2022;đ?&#x2018;Ľ 2

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Prove that đ?&#x153;&#x2022;2đ?&#x2018;˘

+ 2đ?&#x2018;Ľđ?&#x2018;Ś đ?&#x153;&#x2022;đ?&#x2018;Ľđ?&#x153;&#x2022;đ?&#x2018;Ś + đ?&#x2018;Ś 2 đ?&#x153;&#x2022;đ?&#x2018;Ś 2 =

â&#x2C6;&#x2019; sin đ?&#x2018;˘ cos2 đ?&#x2018;˘

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4đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; 3đ?&#x2018;˘

Q6 (a) Expand đ?&#x2018;Ľ 2 đ?&#x2018;Ś + 3đ?&#x2018;Ś â&#x2C6;&#x2019; 2 in powers of (đ?&#x2018;Ľ â&#x2C6;&#x2019; 1)& (đ?&#x2018;Ś + 2) using Taylorâ&#x20AC;&#x2122;s theorem. (b) Locate the stationary points of đ?&#x2018;Ľ 4 + đ?&#x2018;Ś 4 â&#x2C6;&#x2019; 2đ?&#x2018;Ľ 2 + 4đ?&#x2018;Ľđ?&#x2018;Ś â&#x2C6;&#x2019; 2đ?&#x2018;Ś 2 and determine their nature.

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Q7(a) A rectangular box open at the top is to have volume of 32 cubic feet. Find the dimension of box requiring least material for its construction by Lagrangeâ&#x20AC;&#x2122;s method. (9) 1 đ?&#x2018;Ľ đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

(b) Evaluate the integral â&#x2C6;Ť0

log đ?&#x2018;Ľ

đ?&#x2018;&#x2018;đ?&#x2018;Ľ by applying differentiation under integral sign (đ?&#x2018;&#x17D; â&#x2030;Ľ 0)

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