# Optionals IAS Mains Mathematics: Questions 99 - 105 of 283

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## Question 99

Appeared in Year: *2009*

### Describe in Detail

Essay▾Let be a linear transformation defined by

then find the rank and Nullity of L. Also, determine Null space and range space of L. (CS main Paper 1)

### Explanation

Given that is linear transformation such that

Range space of for

The range space consist of all values of the type for all .

Let

Then (linear span of s)

Where

Since

.

Now we construct a matrix whose rows are vectors of the subset of and convert into Echelon form by using E-row transformation.

Clearly which is in echelon form and the number of non

… (44 more words) …

## Question 100

Appeared in Year: *2015*

### Describe in Detail

Essay▾State Cauchy’s Residue Theorem using it Evaluate the integral.

(CS main Paper 2)

### Explanation

Cauchy Residue Theorem: - If f is analytic in a domain D expect for isolated singularities at , then, for any closed center r in D on which none of the point lie, we have

Now, given

_{Let}

signature of _{are}

_{Now}

are holes of

_{finite}

finite.

_{and Lt}

Z = 0, -1 are pole of order 1.

and z = i is pole of order 2.

_{Res}

And Res

Then by Cauchy Residue Theorem

## Question 101

Appeared in Year: *2011*

### Describe in Detail

Essay▾Let and be a non-singular Matrix of order . Find the eigen value of the matrix where . (CS main Paper 1)

### Explanation

Given

and are Similar Matrix.

and are same.

Characteristics Polynomial of is

Eigen values of are

Eigen values of are i. e. .

Eigen value of are .

## Question 102

Appeared in Year: *2008*

### Describe in Detail

Essay▾Show that the Matrix is invertible is and only if the is invertible. Hence find . (CS main Paper 1)

### Explanation

Let be invertible matrix of order

Clearly

## Question 103

Appeared in Year: *2012*

### Describe in Detail

Essay▾Let be a Hermitian matrix. Find a Non-singular Matrix P such that is diagonal. (CS main Paper 1)

### Explanation

We have

Apply Row operations and then corresponding conjugate column operations

to get.

Apply and then to get.

where

## Question 104

Appeared in Year: *2014*

### Describe in Detail

Essay▾Investigate the values of and so that the equations have

(1) No Solution

(2) A unique solution

(3) An Infinite number of solutions (CS main Paper 1)

### Explanation

The given equations are

Which can be written as

i. e. where

(ii) The given equation have a unique solution is

Operate

Operate

Operate

Le

, the given set of equations had a unique solution, no metter what is the value of .

(i) Take

Operate

Operate

Now

Rank of is is

Rank of are not equal is

If and the given set of equations does not have any solut

… (23 more words) …

## Question 105

Appeared in Year: *2013*

### Describe in Detail

Essay▾Prove that if where positive and real, then the function are has Zeros in the unit circle. (CS main Paper 1)

### Explanation

Rouche’s Theorem: - If and are analytic inside and on a simple closed curve and if on , then and both have the same number of zero inside C.

Now, Taking

we have the given equation as Here and both are analytic within and upon .

Now,

Hence by Rouche’s Theorem and have the same number of zero inside

Here has roots inside

has roots inside

… (2 more words) …