# Optionals IAS Mains Mathematics: Questions 255 - 261 of 283

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## Question 255

Appeared in Year: *2013*

### Describe in Detail

Essay▾Let be the ring of Gaussian integers (subring of ) . Which of the following is J: Euclidean domain, principal ideal domain, unique factorization domain? Justify your answer. (Paper II)

### Explanation

- Define by
- Then
i.e.. .

- Let
- Now for , we can find such that
i.e.. .

- Let
- Where
Also

- Now, either
- Hence we know that
- Every Euclidean Domain is P. I. D. and every P. I. D. is U. F. D.
is P. I. D. and also J is U. F. D.

So J is Euclidean domain, principal ideal domain and unique factorization domain.

## Question 256

Appeared in Year: *2007*

### Describe in Detail

Essay▾Let where . Show that R is a ring under matrix addition and multiplication

Then show that A is a left ideal of R but not a right ideal of R. (Paper II)

### Explanation

- First, we show that R is a ring under matrix addition and multiplication
I. Is an abelian group

- (a) Closure property:
Let

Addition is closed.

- (b) Associative law:
Let , then

- (c) Existence of Additive identity:
For

Where

- Such that
And

i.e.. .

is the additive identity of R

- (d) Existence of Additive inverse:
For

There is

- Such that
Also

i.e.. .

is additive in…

… (99 more words) …

## Question 257

Appeared in Year: *2007*

### Describe in Detail

Essay▾Evaluate (by using residue theorem)

(Paper 2)

### Explanation

- Let
- Let c be the circle
Where

- Poles of are given by
- Since so lies out side one circle
is simple poles of

## Question 258

Appeared in Year: *2010*

### Describe in Detail

Essay▾Consider the polynomial Ring . Show is irreducible over . Let be the ideal in generated by . Then show that is a field and that each element of it is of the form with in . (Paper 2)

### Explanation

- Given
Let prime

Now

satisfies all the conditions of Eisenstein՚s criterion

is irreducible over

- Lemma: - Let principal ideal domain, which is not a field. Then any proper ideal of R is maximal it is generated by an irreducible element of R.
- Proof of lemma: -
- Since R is not a field, therefore there exist such that is not a unit
is not a maximal ide…

… (179 more words) …

## Question 259

Appeared in Year: *2006*

### Describe in Detail

Essay▾Using elementary row operations, find the rank of the matrix.

(Paper 1)

### Explanation

Let

Which is row echelon form

Since there are three non-zero rows in row echelon form

## Question 260

Appeared in Year: *2007*

### Describe in Detail

Essay▾Find the equation of the sphere inscribed in the tetrahedron whose faces are (Paper 1)

### Explanation

- Let be the center and r be one radius of the sphere
- Since one sphere is inscribed in the tetrahedron
The length of from one centre on each face = radius

… eq. (1)

- From first three members
… eq. (2)

- From first and last member of (1)
, also from (1)

Centre of sphere is and radius

The equation of the sphere is

- Which is one required equation of the sp…

… (1 more words) …

## Question 261

Appeared in Year: *2007*

### Describe in Detail

Essay▾For any constant vector show that the vector represented by curl is always parallel to the vector being the position vector of a point measured from the origin. (Paper 1)

### Explanation

Let

Also given is the position vector of a point measured from the origin.

Now

is parallel to