# 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

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…

… (11 more words) …

## 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,

… (2 more words) …

## 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…

… (42 more words) …

## Question 215

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.

- 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…

… (36 more words) …

## 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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