Optionals IAS Mains Mathematics: Questions 211 - 217 of 283

Let … eq. (1)

Subjected to the constraints

… eq. (2)

And … eq. (3)

Consider the function

Where are Lagrange՚s multiplies to be determined

For extreme values,

… eq. (4)

… eq. (5)

… eq. (6)

Multiplying (4) by , (5) by , (6) by and adding, we get

Putting this value of in (4) , (5) , (6) , we get

Putting value of in (3) , we get

Or

This equation gives max…

Let

Poles of are given by

At

Now

So is simple pole of

At

So is simple pole of

At

Finite

Is pole of order

Cauchy՚s Residue Theorem: - If is analytic within and on a closed contour except at a finite number of poles , then

Where R. H. S. denotes sum of residues of at its poles lying within c.

Using Cauchy Residue Theorem,

• First, of all we find the unit of

  Let be a unit in

• Then there exist such that

  … eq. (1)

• Taking complex conjugate on both sides of (1) , we get

  … eq. (2)

• Multiply (1) and (2) , we get

  Only units in are

  is neither zero nor a unit of

• Let … eq. (3)
• Taking complex conjugate on both sides, we get

  … eq. (4)

• Multiply (3) and (4) , we get

• But has no integr…

Let r be the radius of the base and h be the height of cylinder ABCD which is inscribed in a sphere of radius a.

• It is obvious that for maximum volume the axis of the cylinder must be along the diameter of the sphere. Let O be the center of the sphere such that . By symmetry, O is the midpoint of LM.
• Applying Pythagoras the…

• (i) Let

  Let

  is not closed under multiplication

  is not a Ring

  is not Integral domain

  The set of number of the form is not integral domain.

• (ii) Let

  is Abelian group

• (i) Let

• So is closed under addition.
• (ii) Let

  is associative under addition.

• (iii) such that

  is identity element of under addition

• (iv)
• Such that

  Inverse of every element is exist

• (v) …

• The boundary c of the surface s is the circle
• The parametric equation of c are

• Now by Stoke՚s theorem