# Optionals IAS Mains Mathematics: Questions 218 - 225 of 283

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## Question 218

Appeared in Year: *2011*

### Describe in Detail

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 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

Appeared in Year: *2007*

### Describe in Detail

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

Appeared in Year: *2008*

### Describe in Detail

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

Appeared in Year: *2011*

### Describe in Detail

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

Appeared in Year: *2011*

### Describe in Detail

Essay▾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

Appeared in Year: *2012*

### Describe in Detail

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

- 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

Appeared in Year: *2011*

### Describe in Detail

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

Appeared in Year: *2011*

### Describe in Detail

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) …