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DEPARTMENT OF MATHEMATICS �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 –A (2Ă— đ?&#x;?đ?&#x;Ž = đ?&#x;?đ?&#x;Ž). Attempt any two questions from part B (đ?&#x;?đ?&#x;“đ?‘´đ?’‚đ?’“đ?’Œđ?’” đ?’†đ?’‚đ?’„đ?’‰). Attempt any two questions from part C (đ?&#x;?đ?&#x;“đ?‘´đ?’‚đ?’“đ?’Œđ?’” đ?’†đ?’‚đ?’„đ?’‰).

PART A 1(a) A particle moves along a curve whose parametric equation in terms of time t, is đ?‘Ľ = 3đ?‘Ą 2 , đ?‘Ś = đ?‘Ą 2 − 2đ?‘Ą, đ?‘§ = đ?‘Ą 3 . Find its velocity and acceleration at time đ?‘Ą = 2 (b) Write the statement of Stoke’s theorem. (c) Write the Physical interpretation of Curl. (d) If đ?‘&#x;⃗ = đ?‘Ľđ?‘–Ě‚ +đ?‘Śđ?‘—Ě‚ +đ?‘§đ?‘˜Ě‚ , prove that div (đ?‘&#x; đ?‘› đ?‘&#x;⃗) = (đ?‘› + 3) đ?‘&#x; đ?‘› . (e) Find a unit vector normal to the surface đ?‘Ľ 3 + đ?‘Ś 3 + 3đ?‘Ľđ?‘Śđ?‘§ = 3 at point (1, 2, −1). (f) Find the angle between tangent. If đ?‘&#x;⃗ = đ?‘Ą 2 đ?‘– âˆ’Ě‚ 2đ?‘Ąđ?‘—Ě‚ + đ?‘Ą 3 đ?‘˜Ě‚ at the points đ?‘Ą = 1 đ?‘Žđ?‘›đ?‘‘ đ?‘Ą = 2. (g) Find the Directional Derivatives of the function x đ?‘Ś 2 + đ?‘Śđ?‘§ 3 at points (2, −1, 1). (h) Write sine and cosine half range series formulae. (i) Write Fourier coefficients formulae. (j) Write period đ?‘ đ?‘–đ?‘›đ?‘Ľ đ?‘Žđ?‘›đ?‘‘ đ?‘?đ?‘œđ?‘ đ?‘Ľ.


PART B

2. .Obtain the Fourier series to represent đ?‘“(đ?‘Ľ) = Hence obtain the following relation: (i) (ii)

1 12

1

1

1

1

12

(đ?œ‹ − đ?‘Ľ)2 in the interval 0 ≤ đ?‘Ľ ≤ 2đ?œ‹.

4 1

1

+ 22 + 32 + 42 +‌‌‌‌‌‌. =

đ?œ‹2

1

1

− 22 + 32 − 42 +‌‌‌‌‌‌. = 12

(iii) 1 − 4

đ?‘Ľ

3(a) Find the helf range sine series for đ?‘“(đ?‘Ľ) = { 3 đ?‘Ľâˆ’4

1 12

1

1

đ?œ‹2 6

.

1

+ 32 + 52 + 72 +‌‌‌‌‌‌. = đ?‘“đ?‘œđ?‘&#x;

0 ≤� ≤ 1 2

for

đ?œ‹2

15

8

1 2

≤x ≤1

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

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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;.

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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 .

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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).

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

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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

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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) ?

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(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.

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