Optionals IAS Mains Mathematics: Questions 99 - 105 of 283

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Question 99

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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

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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

Eigenvalues and Eigenvectors
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Appeared in Year: 2011

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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

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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

Hermitian, Skew-Hermitian
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Appeared in Year: 2012

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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

Solution of System of Linear Equations
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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

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Question 105

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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

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