# Optionals IAS Mains Mathematics: Questions 278 - 283 of 283

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## Question 278

Appeared in Year: *2006*

### Describe in Detail

Essay▾Solve: (Paper 2)

### Explanation

- Given differential equation is
- Hence auxiliary Equation is
… eq. (1)

- From last two members, we have
- Integrating, we get
… eq. (2)

- From first and last members if (1) , we have
- Putting
We get

- Which is ordinary differential equation of order 1
- Hence general solution of given differential equation is

… (2 more words) …

## Question 279

Appeared in Year: *2009*

### Describe in Detail

Essay▾Show that where (Paper 1)

### Explanation

Grad

Now

i.e..

Similarly

And

Adding vertically, we get

## Question 280

Appeared in Year: *2007*

### Describe in Detail

Essay▾Show that the planes cuts the cone in perpendicular lines. (Paper 1)

### Explanation

The equation of cone is

… eq. (1)

The equation of plane is

… eq. (2)

Comparing (1) with

We get

Here

Necessary and sufficient condition that the cone may have three mutually perpendicular generators

The cone has three mutually perpendicular generators

The plane (2) cuts the cone (1) in perpendicular lines if the normal to the plane (2) through the v…

… (21 more words) …

## Question 281

Appeared in Year: *2010*

### Describe in Detail

Essay▾Let . Show that C is a commutative ring with 1 under point wise addition and multiplication. Determine whether C is an integral domain. Explain. (Paper 2)

### Explanation

- Given
Now is always continuous

- Clearly, if given by
**I**.**is Abelian group**(i) let

are real valued continuous function

Now

- We know that sum of two continuous function is continuous
is also continuous

- And sum of real valued function is real valued
is also real valued

is closed under addition.

- (ii) let,
Then

is associative under addition

- (iii) Existence o…

… (153 more words) …

## Question 282

### Describe in Detail

Essay▾Let C be the field of complex numbers and let n be a positive integer . Let V be the vector space of all matrices over C. Which of the following sets of matrices A in V are subspaces of V?

1. All invertible A;

2. All non-invertible A;

3. All A such that , where B is some fixed matrix in V.

### Explanation

The answer to c is not correct, I guess.

H is a subspace of V if

- The zero vector is an element of H
- H is closed under addition: if a, b in H, then in H
- H is closed under scalar multiplication: if in H and c a scalar then cx in H

is a subspace, because

- 0, the zero matrix is in H, because
- H is closed under addition: Let A, C in H, thus and , then, thu…

… (17 more words) …

## Question 283

### Describe in Detail

Essay▾Prove that the union of two subspaces is a subspace if and only if one is contained in the other.

### Explanation

Let X and Y be two given subspace.

**Case 1- one subspace is contained in other**

X is contained in Y. then union X and Y will be Y. and since Y is already subspace, union will be subspace too.

**Case 2- union is subspace**.

- Now assume X is not contained in Y. that is x does not lie in Y.
- Let x belongs to X and y belongs to Y.
- Subspace satisfies closure property…

… (57 more words) …