Optionals IAS Mains Mathematics: Questions 61 - 65 of 283

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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

Edit

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

Eigenvalues and Eigenvectors
Edit

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.