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

Edit

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:
    1. is algebraically closed.
    2. Every irreducible polynomial in is of degree
    3. Every polynomial of positive degree in factors completely into factors in .
    4. 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

Table Show the Simplex
Min. Ratio

Simplex table 2

Table Show the Simplex
Min. Ratio
Table Show the Simplex

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