Optionals IAS Mains Mathematics: Questions 218 - 225 of 283

• Here is the surface of the sphere lying above the plane. Let c be one foundry ABED of the surface S. Then the curve C is a circle in the plane and equation of C are …………………eq. (1)

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• The parametric equation of C are …………. eq. (2)
• We want to find i. e.
• Now According to the stoke’s theorem …………. . eq.

• Let v be the volume of the parallelepiped whose edges are
• We have to find the maximum value of …………. . eq. (1) …………. eq. (2)
• Where s is the total surface area consider the function
• Where is Lagrange’s multiplier to be determined.
• For extreme points, ………………eq. (3) ………………eq. (4) ………………eq. (5)
• Multiplying (3) by x, (4) by y and subtractive, we get

• (i)
• Let
• Now – [m times. ] [m times. ] Where is identity of Where is identity of
• (ii) is a normal subgroup of G.
• We know
• Where are identity of respectively. is non-empty Let
• Now
• Thus is subgroup of G. Farther, let
• Then
• Hence is normal subgroup of G.

• Let be the distance of the origin from any point ………………. . eq. (1)
• Where ……………………eq. (2) Let
• Where is Lagrange’s multiplier.
• For extreme points, …………. . eq. (3) …………. . eq. (4)
• Since x and y are both non-zero
• Take
• Which is not possible.
• Take
• Now
• Which is positive definite is minimum and minimum value of is Required dist

• Lemma: - The number of generators of a finite cyclic group of order n is is the number of integers less than n and coprimes to
• Proof: - Let G be a cyclic group of order n generated by a
• We know that all the element of G is of the form, for some integer
• Therefore, we have ……………. . eq. (1)
• We know that order of generator of cyclic group is equal to t

By Green’s theorem in a plane

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• We verify this result Now
• The curve intersect at
• And the positive direction in traversing C is shown in
• Now we evaluate the line integral along C
• Along the curve , line integral
• Along the line Line integral Required line integral
• Hence the theorem is veri

• Lemma: - Every cyclic group is Abelian.
• Proof: - consider a cyclic group G generated by a i. e. Let be arbitrary element for some integer The is Abelian group
• Main Proof: - Let G be a group of order p, a positive prime has atleast two elements. Consider be the cyclic subgroup of G. Therefor
• By Lagrange’s theorem i. e. Also must be cyc

• Given T. P. is a ring
• (I) is Abelian group
• a) Let are real valued continuous function on Now We know that sum of two continuous function is continuous is continuous function is continuous real valued function is closed under addition
• b) Let
• c) Let be defined as for all Also O is continuous function and real valued Also is the a