Optionals IAS Mains Mathematics: Questions 99 - 105 of 283

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

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

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 .

Let be invertible matrix of order

Clearly

We have

Apply Row operations and then corresponding conjugate column operations

to get.

Apply and then to get.

where

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

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