# Optionals IAS Mains Mathematics: Questions 248 - 254 of 283

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## Question 248

Appeared in Year: *2010*

### Write in Short

Short Answer▾Prove that

Where f is a scalar function. (Paper I)

## Question 249

Appeared in Year: *2007*

### Describe in Detail

Essay▾Determine by using stoke’s theorem, where c is the curve defined by

That starts from the point and goes at first below the z-plane (Paper I)

### Explanation

- The curve c is given by
- Which is the curve of intersection of the sphere ……………eq. (1)
- And the plane ………. eq. (2)
- Now centre of sphere (1)
- Lie on plane is the great circle of the sphere passing through

- Thus, C is a circle lying on the plane with center at D and radius
- Let S be the region bounded by th

… (31 more words) …

## Question 250

Appeared in Year: *2014*

### Describe in Detail

Essay▾Verify Gauss divergence theorem for the vector

Taken over the cube (Paper I)

### Explanation

- Let be the surface of the cube bounded by the planes
- Let v be the volume enclosed by s then we have to prove

- Now
- Now, we shall calculate over the six faces of cube.
**Over the face ANPL:****Over the face OBMC:****Over the face BMPN:**=1**Over the face OCLA:****Over the face CLPM:****Over the face OANB**

… (17 more words) …

## Question 251

Appeared in Year: *2012*

### Describe in Detail

Essay▾Is the ideal generated by 2 and x is polynomial ring of polynomials in a single variable x with coefficients in the ring of integers, a principal ideal? Justify your answer. (Paper II)

### Explanation

- Let
- If possible let I be principal ideal of and for some
- The for some ………………. eq. (1)
- Similarly for some ………………eq. (2) Each coefficient of is even integer i. e. for some
- i. e. from equation (2), we have,
- Which is not true as the polynomial in I have even constant terms is not principal ideal.

… (12 more words) …

## Question 252

Appeared in Year: *2014*

### Describe in Detail

Essay▾Prove that the set is a commutative ring with identity. (Paper II)

### Explanation

- Given
**I.****is Abelian group**(i) Let- So closure property is satisfied
- (ii) Let
- So associative property hold.
- (iii) such that So is identity element
- (iv) such that So inverse of each element exist
- (v) Commutativity Let Now
- So Commutativity exist So is Abelian group
**II.****is semi-group.**- (i) Let Then So closure property is satisfied
- (ii) Let

… (39 more words) …

## Question 253

Appeared in Year: *2011*

### Write in Short

Short Answer▾Give an example of a group G in which proper subgroup is cyclic but the group itself is not cyclic. (Paper II)

## Question 254

Appeared in Year: *2007*

### Describe in Detail

Essay▾Consider the vector space

, over the real field R, Define the map by

Where

Is D is a linear transformation on X? If it is then construct the matrix representation for D with respect to the order basis for X (Paper I)

### Explanation

D is Linear Transformation

Let

Then

Then

is a linear transformation

Now, given is X is polynomial of degree less than or equal to 3.

is a basis of x

Now

Then matrix representation of D write basis is