DEPARTMENT OF MATHEMATICS đ?&#x2018;&#x2021;3 EXAMINATION, MAY 2017-18

Semester: 2nd Subject: Advance Mathematics Branch: Chemistry Course Type: Core Time: 3:00 Hrs. Max. Marks: 80

Date of Exam: 15/5/2018 Subject Code: MAH 117-T Session: II Course Nature: Hard Program: B.Sc.

Note: -All questions are compulsory from part â&#x20AC;&#x201C;A (2Ă&#x2014; đ?&#x;?đ?&#x;&#x17D; = đ?&#x;?đ?&#x;&#x17D;). Attempt any two questions from part B (đ?&#x;?đ?&#x;&#x201C;đ?&#x2018;´đ?&#x2019;&#x201A;đ?&#x2019;&#x201C;đ?&#x2019;&#x152;đ?&#x2019;&#x201D; đ?&#x2019;&#x2020;đ?&#x2019;&#x201A;đ?&#x2019;&#x201E;đ?&#x2019;&#x2030;). Attempt any two questions from part C (đ?&#x;?đ?&#x;&#x201C;đ?&#x2018;´đ?&#x2019;&#x201A;đ?&#x2019;&#x201C;đ?&#x2019;&#x152;đ?&#x2019;&#x201D; đ?&#x2019;&#x2020;đ?&#x2019;&#x201A;đ?&#x2019;&#x201E;đ?&#x2019;&#x2030;).

PART A 1(a) A particle moves along a curve whose parametric equation in terms of time t, is đ?&#x2018;Ľ = 3đ?&#x2018;Ą 2 , đ?&#x2018;Ś = đ?&#x2018;Ą 2 â&#x2C6;&#x2019; 2đ?&#x2018;Ą, đ?&#x2018;§ = đ?&#x2018;Ą 3 . Find its velocity and acceleration at time đ?&#x2018;Ą = 2 (b) Write the statement of Stokeâ&#x20AC;&#x2122;s theorem. (c) Write the Physical interpretation of Curl. (d) If đ?&#x2018;&#x;â&#x192;&#x2014; = đ?&#x2018;Ľđ?&#x2018;&#x2013;Ě&#x201A; +đ?&#x2018;Śđ?&#x2018;&#x2014;Ě&#x201A; +đ?&#x2018;§đ?&#x2018;&#x2DC;Ě&#x201A; , prove that div (đ?&#x2018;&#x; đ?&#x2018;&#x203A; đ?&#x2018;&#x;â&#x192;&#x2014;) = (đ?&#x2018;&#x203A; + 3) đ?&#x2018;&#x; đ?&#x2018;&#x203A; . (e) Find a unit vector normal to the surface đ?&#x2018;Ľ 3 + đ?&#x2018;Ś 3 + 3đ?&#x2018;Ľđ?&#x2018;Śđ?&#x2018;§ = 3 at point (1, 2, â&#x2C6;&#x2019;1). (f) Find the angle between tangent. If đ?&#x2018;&#x;â&#x192;&#x2014; = đ?&#x2018;Ą 2 đ?&#x2018;&#x2013; â&#x2C6;&#x2019;Ě&#x201A; 2đ?&#x2018;Ąđ?&#x2018;&#x2014;Ě&#x201A; + đ?&#x2018;Ą 3 đ?&#x2018;&#x2DC;Ě&#x201A; at the points đ?&#x2018;Ą = 1 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x2018;Ą = 2. (g) Find the Directional Derivatives of the function x đ?&#x2018;Ś 2 + đ?&#x2018;Śđ?&#x2018;§ 3 at points (2, â&#x2C6;&#x2019;1, 1). (h) Write sine and cosine half range series formulae. (i) Write Fourier coefficients formulae. (j) Write period đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ľ đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x2018;Ľ.

PART B

2. .Obtain the Fourier series to represent đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) = Hence obtain the following relation: (i) (ii)

1 12

1

1

1

1

12

(đ?&#x153;&#x2039; â&#x2C6;&#x2019; đ?&#x2018;Ľ)2 in the interval 0 â&#x2030;¤ đ?&#x2018;Ľ â&#x2030;¤ 2đ?&#x153;&#x2039;.

4 1

1

+ 22 + 32 + 42 +â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. =

đ?&#x153;&#x2039;2

1

1

â&#x2C6;&#x2019; 22 + 32 â&#x2C6;&#x2019; 42 +â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. = 12

(iii) 1 â&#x2C6;&#x2019; 4

đ?&#x2018;Ľ

3(a) Find the helf range sine series for đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) = { 3 đ?&#x2018;Ľâ&#x2C6;&#x2019;4

1 12

1

1

đ?&#x153;&#x2039;2 6

.

1

+ 32 + 52 + 72 +â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. = đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x;

0 â&#x2030;¤đ?&#x2018;Ľ â&#x2030;¤ 1 2

for

đ?&#x153;&#x2039;2

15

8

1 2

â&#x2030;¤x â&#x2030;¤1

8

(b) Expand đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) = đ?&#x2018;Ľ 2 as a half range cosine series in 0 < đ?&#x2018;Ľ < 3.

7

4(a) . Find the Fourier series to represent đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x2018;Ľ 2 fromđ?&#x2018;Ľ = â&#x2C6;&#x2019;đ?&#x153;&#x2039; đ?&#x2018;Ąđ?&#x2018;&#x153; đ?&#x2018;Ľ = đ?&#x153;&#x2039;.

5

â&#x2C6;&#x2019;đ?&#x2018;&#x2DC; đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; â&#x2C6;&#x2019; đ?&#x153;&#x2039; < đ?&#x2018;Ľ < 0 (b) Determine the Fourier series of square wave function defined by đ??š(đ?&#x2018;Ľ) = { đ?&#x2018;&#x2DC; đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; 0 < đ?&#x2018;Ľ < đ?&#x153;&#x2039; 1

1

1

đ?&#x153;&#x2039;

Where đ??š(đ?&#x2018;Ľ) = đ??š(đ?&#x2018;Ľ + 2đ?&#x153;&#x2039;) deduce that 1 â&#x2C6;&#x2019; 3 + 5 â&#x2C6;&#x2019; 7 + â&#x2039;Ż â&#x20AC;Ś . . = 4 .

10

PART C 5(a) if đ??šâ&#x192;&#x2014; = (5xy â&#x20AC;&#x201C; 6 đ?&#x2018;Ľ 2 )đ?&#x2018;&#x2013;Ě&#x201A; +(2y â&#x20AC;&#x201C; 4x)đ?&#x2018;&#x2014;Ě&#x201A; , find the value of â&#x2C6;Ťđ?&#x2018;? đ??šâ&#x192;&#x2014; . đ?&#x2018;&#x2018;đ?&#x2018;&#x;â&#x192;&#x2014; along the curve C in the xy plane y = đ?&#x2018;Ľ 3 from the point (1, 1) to (2, 8).

6

(b) Apply Greenâ&#x20AC;&#x2122;s theorem to prove that the area enclosed by a plane curve is

1

â&#x2C6;Ť (x đ?&#x2018;&#x2018;đ?&#x2018;Ś â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;&#x2018;đ?&#x2018;Ľ ).

2 đ?&#x2018;?

Hence find the area of an ellipse whose semi major and minor axes are of lengths a and b.

6(a) Evaluate; â&#x2C6;­ đ?&#x2018;Ľđ?&#x2018;Śđ?&#x2018;§đ?&#x2018;&#x2018;đ?&#x2018;Ľđ?&#x2018;&#x2018;đ?&#x2018;Śđ?&#x2018;&#x2018;đ?&#x2018;§ 1

â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ľ 2

9

over the volume in positive octant of the sphere đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 + đ?&#x2018;§ 2 = đ?&#x2018;&#x17D;2 . 7

â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ľ 2 â&#x2C6;&#x2019;đ?&#x2018;Ś 2

1

đ?&#x2018;&#x2018;đ?&#x2018;§ đ?&#x2018;&#x2018;đ?&#x2018;Ś đ?&#x2018;&#x2018;đ?&#x2018;Ľ, by changing to spherical polar co-ordinates. 8

(b) Evaluate:â&#x2C6;Ť0 â&#x2C6;Ť0

â&#x2C6;Ť0

7(a) Find the constants

a and b so that the surface đ?&#x2018;&#x17D; đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;Śđ?&#x2018;§ = (đ?&#x2018;&#x17D; + 2)đ?&#x2018;Ľ is orthogonal to the surface

â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ľ 2 â&#x2C6;&#x2019;đ?&#x2018;Ś 2 â&#x2C6;&#x2019;đ?&#x2018;§ 2

4 đ?&#x2018;Ľ 2 đ?&#x2018;Ś + đ?&#x2018;§ 3 = 4 at the point (1, -1, 2) ?

3

(b) Verify divergence theorem for đ??šâ&#x192;&#x2014; = (đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;§) đ?&#x2018;&#x2013;Ě&#x201A; + (đ?&#x2018;Ś 2 â&#x2C6;&#x2019; zx )đ?&#x2018;&#x2014;Ě&#x201A; +(đ?&#x2018;§ 2 - xy) đ?&#x2018;&#x2DC;Ě&#x201A; taken over the rectangular parallelepiped 0 â&#x2030;¤ x â&#x2030;¤ a, 0 â&#x2030;¤ y â&#x2030;¤ b, 0 â&#x2030;¤ z â&#x2030;¤ c.

12