Optionals IAS Mains Mathematics: Questions 211 - 217 of 283

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Question 211

Appeared in Year: 2013

Write in Short

Short Answer▾

What are the orders of the following permutations in so?

And (Paper II)

Question 212

Maxima and Minima

Appeared in Year: 2014

Describe in Detail

Essay▾

Find the maximum or minimum values of subject to the conditions . Interpret the result geometrically. (Paper I)

Explanation

Let … eq. (1)

Subjected to the constraints

… eq. (2)

And … eq. (3)

Consider the function

Where are Lagrange՚s multiplies to be determined

For extreme values,

… eq. (4)

… eq. (5)

… eq. (6)

Multiplying (4) by , (5) by , (6) by and adding, we get

Putting this value of in (4) , (5) , (6) , we get

Putting value of in (3) , we get

Or

This equation gives max…

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Question 213

Appeared in Year: 2015

Describe in Detail

Essay▾

State Cauchy՚s residue theorem. Using it, evaluate the integral

(Paper I)

Explanation

Let

Poles of are given by

At

Now

So is simple pole of

At

So is simple pole of

At

Finite

Is pole of order

Cauchy՚s Residue Theorem: - If is analytic within and on a closed contour except at a finite number of poles , then

Where R. H. S. denotes sum of residues of at its poles lying within c.

Using Cauchy Residue Theorem,

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Question 214

Appeared in Year: 2007

Describe in Detail

Essay▾

Is an irreducible element, but not prime, justify your answer. (Paper II)

Explanation

  • First, of all we find the unit of

    Let be a unit in

  • Then there exist such that

    … eq. (1)

  • Taking complex conjugate on both sides of (1) , we get

    … eq. (2)

  • Multiply (1) and (2) , we get

    Only units in are

    is neither zero nor a unit of

  • Let … eq. (3)
  • Taking complex conjugate on both sides, we get

    … eq. (4)

  • Multiply (3) and (4) , we get

  • But has no integr…

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Question 215

Differentiability
Edit

Appeared in Year: 2014

Describe in Detail

Essay▾

Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a. (Paper I)

Explanation

Let r be the radius of the base and h be the height of cylinder ABCD which is inscribed in a sphere of radius a.

The Height of Cylinder ABCD
  • It is obvious that for maximum volume the axis of the cylinder must be along the diameter of the sphere. Let O be the center of the sphere such that . By symmetry, O is the midpoint of LM.
  • Applying Pythagoras the…

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Question 216

Appeared in Year: 2005

Describe in Detail

Essay▾

Do the following sets form integral domains with respect to ordinary addition and multiplication. If so, state if they are fields:

(i) The set of numbers of the form with b rational.

(ii) The set of even integers

(iii) The set of positive integers (Paper II)

Explanation

  • (i) Let

    Let

    is not closed under multiplication

    is not a Ring

    is not Integral domain

    The set of number of the form is not integral domain.

  • (ii) Let

    is Abelian group

  • (i) Let

  • So is closed under addition.
  • (ii) Let

    is associative under addition.

  • (iii) such that

    is identity element of under addition

  • (iv)
  • Such that

    Inverse of every element is exist

  • (v) …

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Question 217

Appeared in Year: 2012

Describe in Detail

Essay▾

If evaluate

Where S is the surface of the sphere above the plane. (Paper I)

Explanation

  • The boundary c of the surface s is the circle
  • The parametric equation of c are

  • Now by Stoke՚s theorem

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