Optionals IAS Mains Mathematics: Questions 87 - 92 of 283

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Question 87

Appeared in Year: 2015

Describe in Detail

Essay▾

Show that the function is harmonic. Find its conjugate harmonic function . Also, find the corresponding analytic function in the terms of Z. (CS main Paper 2)

Explanation

Given

and

And

is harmonic function.

Now

[By C. R. Equation ]

Its solution is

is harmonic conjugate of

is analytic function.

Now Let

Question 88

Appeared in Year: 2012

Describe in Detail

Essay▾

Let be the vector space of all matrices over the field of real numbers.

Let be the set consisting of all matrices with zero determinant. Is a subspace of Justify your Answer. (CS main Paper 1)

Explanation

lemma: Let be a vector space over a field Then is a subspace of its and

Proof: - Let itself is a vector space over .

Then itself is a vector space over

for all and .

Conversely.

Let and .

we get

is an additive abelian group.

Also taking we have

is closed under scalar multiplication. The remaining axioms for scalar multiplication also hold for…

… (52 more words) …

Question 89

Appeared in Year: 2013

Describe in Detail

Essay▾

Let be the linear transformation defined by

Find a basis and the dimension of the image of and the Kernel of . (CS Paper I)

Explanation

We know that Basis of is

First we shall find Basis for Image of

is Basis of

generates Image of .

Now is defined as

Here

Generate Image of

To find basis of Range , we have to find the L. I Vectors from for this consider the Matrix whose rows are generator of and reduce it to Echelon form.

i.e..

Operate

Operate

from L. I set of vectors which genera…

… (66 more words) …

Question 90

Appeared in Year: 2015

Describe in Detail

Essay▾

Find all possible Talon՚s and Laurent՚s series expansion of the function.

about the point (CS main Paper 2)

Explanation

Clearly the function f (z) is not analytic at z = 1 and z = 2.

When

When

(Since Binomial expansion is valid when )

(iii) when|Z|> 2

Binomial expansion is valid when

Question 91

Appeared in Year: 2012

Describe in Detail

Essay▾

Show that the function defined by

Is not analytic at the origin through it satisfies Cauchy-Riemann Equations at the origin. (CS main Paper 1)

Explanation

Given,

Now,

and

Again

Let along we have

And if let along we have

Hence does not exist.

So, is not analytic at origin.

Question 92

Solution of System of Linear Equations

Appeared in Year: 2011

Describe in Detail

Essay▾

Let

Solve the System of Equations given by using the Above, also solve the system of equations where denotes the transpose of matrix . (CS main Paper 1)

Explanation

Now,

Operate

Operate

Operate

So are solutions of

Now,

Operate

Operate

So, are solution of .