Optionals IAS Mains Mathematics: Questions 53 - 60 of 283

We know that

is standard basis of .

Now

Then matrix of write Basis

is

• 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…

• Given differential Equation is

  S. F. is

  P. I.

• So, particular Integral of given differential equation is

Given linear Transformation is

Let

and

Now

Then Matrix is

Let

Operate and

Operate

Operate

Here Minor

So,

So, Rank of given Matrix is .

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

Presenting the above in the simplex table, we have

Simplex table 1

Simplex table 2

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

…

We have

Let,

is defined by