Optionals IAS Mains Mathematics: Questions 218 - 225 of 283

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Question 218

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Appeared in Year: 2011

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Essay▾

If , calculate the double integral over the hemisphere given by (Paper I)

Explanation

  • Here is the surface of the sphere lying above the plane. Let c be one foundry ABED of the surface S. Then the curve C is a circle in the plane and equation of C are …………………eq. (1)
The curve C is a circle in the xy

The Curve C is a Circle in the Xy

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  • The parametric equation of C are …………. eq. (2)
  • We want to find i. e.
  • Now According to the stoke’s theorem …………. . eq.

… (22 more words) …

Question 219

Lagrange's Method of Multipliers, Jacobian
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Appeared in Year: 2007

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Essay▾

Find a rectangular parallelepiped of greatest volume for a given total surface areas, using Lagrange’s method of multipliers. (Paper I)

Explanation

  • Let v be the volume of the parallelepiped whose edges are
  • We have to find the maximum value of …………. . eq. (1) …………. eq. (2)
  • Where s is the total surface area consider the function
  • Where is Lagrange’s multiplier to be determined.
  • For extreme points, ………………eq. (3) ………………eq. (4) ………………eq. (5)
  • Multiplying (3) by x, (4) by y and subtractive, we get

… (55 more words) …

Question 220

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Appeared in Year: 2008

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Essay▾

Let be two group and let be a homomorphism. For any element

(i) Prove that.

(ii) is a normal subgroup of G. (Paper II)

Explanation

  • (i)
  • Let
  • Now – [m times. ] [m times. ] Where is identity of Where is identity of
  • (ii) is a normal subgroup of G.
  • We know
  • Where are identity of respectively. is non-empty Let
  • Now
  • Thus is subgroup of G. Farther, let
  • Then
  • Hence is normal subgroup of G.

… (1 more words) …

Question 221

Lagrange's Method of Multipliers, Jacobian
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Appeared in Year: 2011

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Essay▾

Find the shortest distance from the origin to the hyperbola. (Paper II)

Explanation

  • Let be the distance of the origin from any point ………………. . eq. (1)
  • Where ……………………eq. (2) Let
  • Where is Lagrange’s multiplier.
  • For extreme points, …………. . eq. (3) …………. . eq. (4)
  • Since x and y are both non-zero
  • Take
  • Which is not possible.
  • Take
  • Now
  • Which is positive definite is minimum and minimum value of is Required dist

… (23 more words) …

Question 222

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Appeared in Year: 2011

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How many generators are there of the cyclic group of order 8? (Paper II)

Explanation

  • Lemma: - The number of generators of a finite cyclic group of order n is is the number of integers less than n and coprimes to
  • Proof: - Let G be a cyclic group of order n generated by a
  • We know that all the element of G is of the form, for some integer
  • Therefore, we have ……………. . eq. (1)
  • We know that order of generator of cyclic group is equal to t

… (187 more words) …

Question 223

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Appeared in Year: 2012

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Essay▾

Verify Green’s theorem in the plane for

Where c is the closed curve of the region bounded by (Paper I)

Explanation

By Green’s theorem in a plane

c is the closed curve of the region

C is the Closed Curve of the Region

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  • We verify this result Now
  • The curve intersect at
  • And the positive direction in traversing C is shown in
  • Now we evaluate the line integral along C
  • Along the curve , line integral
  • Along the line Line integral Required line integral
  • Hence the theorem is veri

… (4 more words) …

Question 224

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Appeared in Year: 2011

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Essay▾

Prove that a group of prime order is abelian. (Paper II)

Explanation

  • Lemma: - Every cyclic group is Abelian.
  • Proof: - consider a cyclic group G generated by a i. e. Let be arbitrary element for some integer The is Abelian group
  • Main Proof: - Let G be a group of order p, a positive prime has atleast two elements. Consider be the cyclic subgroup of G. Therefor
  • By Lagrange’s theorem i. e. Also must be cyc

… (13 more words) …

Question 225

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Appeared in Year: 2011

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Essay▾

Let F be the set of all real valued continuous functions defined on the closed interval prove that is a commutative ring with unity with respect to addition and multiplication of function defined point wise as below:

Where (Paper II)

Explanation

  • Given T. P. is a ring
  • (I) is Abelian group
  • a) Let are real valued continuous function on Now We know that sum of two continuous function is continuous is continuous function is continuous real valued function is closed under addition
  • b) Let
  • c) Let be defined as for all Also O is continuous function and real valued Also is the a

… (84 more words) …