Optionals IAS Mains Mathematics: Questions 248 - 254 of 283

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Question 248

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Appeared in Year: 2010

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Short Answer▾

Prove that

Where f is a scalar function. (Paper I)

Question 249

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Appeared in Year: 2007

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Essay▾

Determine by using stoke’s theorem, where c is the curve defined by

That starts from the point and goes at first below the z-plane (Paper I)

Explanation

  • The curve c is given by
  • Which is the curve of intersection of the sphere ……………eq. (1)
  • And the plane ………. eq. (2)
  • Now centre of sphere (1)
  • Lie on plane is the great circle of the sphere passing through
C is a circle lying on the plane

C is a Circle Lying on the Plane

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  • Thus, C is a circle lying on the plane with center at D and radius
  • Let S be the region bounded by th

… (31 more words) …

Question 250

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Appeared in Year: 2014

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Essay▾

Verify Gauss divergence theorem for the vector

Taken over the cube (Paper I)

Explanation

  • Let be the surface of the cube bounded by the planes
  • Let v be the volume enclosed by s then we have to prove
Graph of the six faces of cube

Graph of the Six Faces of Cube

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  • Now
  • Now, we shall calculate over the six faces of cube.
  • Over the face ANPL:
  • Over the face OBMC:
  • Over the face BMPN: =1
  • Over the face OCLA:
  • Over the face CLPM:
  • Over the face OANB

… (17 more words) …

Question 251

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Appeared in Year: 2012

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Essay▾

Is the ideal generated by 2 and x is polynomial ring of polynomials in a single variable x with coefficients in the ring of integers, a principal ideal? Justify your answer. (Paper II)

Explanation

  • Let
  • If possible let I be principal ideal of and for some
  • The for some ………………. eq. (1)
  • Similarly for some ………………eq. (2) Each coefficient of is even integer i. e. for some
  • i. e. from equation (2), we have,
  • Which is not true as the polynomial in I have even constant terms is not principal ideal.

… (12 more words) …

Question 252

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Appeared in Year: 2014

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Essay▾

Prove that the set is a commutative ring with identity. (Paper II)

Explanation

  • Given
  • I. is Abelian group (i) Let
  • So closure property is satisfied
  • (ii) Let
  • So associative property hold.
  • (iii) such that So is identity element
  • (iv) such that So inverse of each element exist
  • (v) Commutativity Let Now
  • So Commutativity exist So is Abelian group
  • II. is semi-group.
  • (i) Let Then So closure property is satisfied
  • (ii) Let

… (39 more words) …

Question 253

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Appeared in Year: 2011

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Short Answer▾

Give an example of a group G in which proper subgroup is cyclic but the group itself is not cyclic. (Paper II)

Question 254

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Appeared in Year: 2007

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Essay▾

Consider the vector space

, over the real field R, Define the map by

Where

Is D is a linear transformation on X? If it is then construct the matrix representation for D with respect to the order basis for X (Paper I)

Explanation

D is Linear Transformation

Let

Then

Then

is a linear transformation

Now, given is X is polynomial of degree less than or equal to 3.

is a basis of x

Now

Then matrix representation of D write basis is