# 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

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 .