# Optionals IAS Mains Mathematics: Questions 61 - 65 of 283

Access detailed explanations (illustrated with images and videos) to **283** questions. Access all new questions- tracking exam pattern and syllabus. View the complete topic-wise distribution of questions. *Unlimited Access, Unlimited Time, on Unlimited Devices*!

View Sample Explanation or View Features.

Rs. 750.00 -OR-

How to register? Already Subscribed?

## Question 61

Appeared in Year: *2014*

### Describe in Detail

Essay▾Let and be the following subspace of :

And .

Find a basis and the dimension of

(CS main Paper 1)

### Explanation

(i) Let

spans

Now, let for

Hence, is a basis of and hence

let

spans W

Now, let

is basis of and hence

Let

and

and

spans and , being a non-zero singleton set is L. I.

Hence is basis of

## Question 62

Appeared in Year: *2015*

### Describe in Detail

Essay▾Consider the following liner programing problem:

Subject to

(i) Using the definition, find it՚s all basic solutions. Which of these are degenerate basic feasible solutions and which are non-degenerate basic feasible solution (s) is/are optimal?

(ii) Without solving the problem, show that it has optimal solution. Which of the basic feasible solution/ (s) is/are optimal? (CS main Paper 2)

### Explanation

Given,

Subject to The given system of Equation can be written in the matrix from as where

Since, Rank of A is 2

Then Maximum No. of linearly independent columns of A is 2. Thus we can take any of the following sub-matrices as basis matrix B.

Let us take first . A basic solution to the given system is now obtained by setting and solving the system.

…

… (143 more words) …

## Question 63

Appeared in Year: *2015*

### Describe in Detail

Essay▾Give an example of a ring having identity but a subring of this having a different identity. (CS main Paper 2)

### Explanation

Let and S =

First, we prove R is Ring

Let

(1)

(2)

(3)

(4)

(5)

So R is Abelian group of addition.

(6)

(7)

_{And}

So, R is semi group under multiplication.

R is a Ring

_{Also}

So is identity of the Ring R.

Now

Also S satisfies all the property of Ring

So S itself is a Ring.

Now, we find the identity of Ring S

Let is identity of Rings and

So is identity of Ring…

… (10 more words) …

## Question 64

Appeared in Year: *2014*

### Describe in Detail

Essay▾Prove that the eigen value of a unitary Matrix have absolute value 1. (CS main Paper 1)

### Explanation

Let be the unitary Matrix.

Let be given value of

Non- Zero Vector X such that

From and , we get

Hence the Result.

## Question 65

Appeared in Year: *2011*

### Describe in Detail

Essay▾Show that the vectors and are linearly independent in . Let be a linear transformation defined by.

Show that the image of above vectors under are linearly dependent. Give the reason for the same. (CS main Paper 1)

### Explanation

Given vectors are and

Let

and are linearly independent.

Now, is defined as

The matrix corresponding to vector and is.

So, the vector and are linearly independent.