# Optionals IAS Mains Mathematics: Questions 53 - 60 of 283

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## Question 53

Appeared in Year: *2015*

### Describe in Detail

Essay▾Let for all , be defined by

. What is the Matrix relative to the basis ? [CS (Main) Paper 1]

### Explanation

We know that

is standard basis of .

Now

Then matrix of write Basis

is

## Question 54

Appeared in Year: *2016*

### Describe in Detail

Essay▾Show that every algebraically closed field is infinite. [CS (Main) Paper 2]

### Explanation

- Lemma: - For any filed the following statements are equivalent:
- is algebraically closed.
- Every irreducible polynomial in is of degree
- Every polynomial of positive degree in factors completely into factors in .
- Every polynomial of position degree in has at least one root in .

- Let be an irreducible polynomial of degree in .
Then we can find a field…

… (238 more words) …

## Question 55

Appeared in Year: *2016*

### Describe in Detail

Essay▾Find a particular integral of [CS (Main) ]

### Explanation

- Given differential Equation is
S. F. is

P. I.

- So, particular Integral of given differential equation is

## Question 56

Appeared in Year: *2016*

### Describe in Detail

Essay▾If is such that , then choosing and as bases of and respectively, find the matrix of [CS (Main) Paper 1]

### Explanation

Given linear Transformation is

Let

and

Now

Then Matrix is

## Question 57

Appeared in Year: *2014*

### Describe in Detail

Essay▾Using elementary row or column operations, find the rank of the Matrix.

(CS main Paper 1)

### Explanation

Let

Operate and

Operate

Operate

Here Minor

So,

So, Rank of given Matrix is .

## Question 58

Appeared in Year: *2012*

### Describe in Detail

Essay▾Use Cauchy integral formula to evaluate , where is the circle . (CS main Paper 1)

### Explanation

Cauchy integral formula for Higher Derivative: - If a function is analytic within and on a closed contour and in any point within then derivate of all orders are analytic and are given by.

Let

Clearly lies inside circle

Take

Then

Then by above result

Hence

## Question 59

### Describe in Detail

Essay▾Solve the following linear programing problem by the simplex method. Write its dual. Also, write the optimal table of the dual from the optimal table of the given problem:

Subject to:

### Explanation

Presenting the above in the simplex table, we have

**Simplex table 1**

Min. Ratio |

**Simplex table 2**

Min. Ratio |

Since , so we are with optimum solution

Now, the Above Problem in standard from is

Subject to

∵ No. of Constraint = 2

So, No. of Variables in dual = 2

Let are variables of dual problem is

…

… (77 more words) …

## Question 60

Appeared in Year: *2010*

### Describe in Detail

Essay▾Let . find the unique linear transformation so that is the Matrix of with respect to the basis of and

of . Also find . (CS main Paper 1)

### Explanation

We have

Let,

is defined by