Optionals IAS Mains Mathematics: Questions 255 - 261 of 283

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Question 255

Appeared in Year: 2013

Describe in Detail

Essay▾

Let be the ring of Gaussian integers (subring of ) . Which of the following is J: Euclidean domain, principal ideal domain, unique factorization domain? Justify your answer. (Paper II)

Explanation

  • Define by

  • Then

    i.e.. .

  • Let

  • Now for , we can find such that

    i.e.. .

  • Let

  • Where

    Also

  • Now, either

  • Hence we know that
  • Every Euclidean Domain is P. I. D. and every P. I. D. is U. F. D.

    is P. I. D. and also J is U. F. D.

    So J is Euclidean domain, principal ideal domain and unique factorization domain.

Question 256

Appeared in Year: 2007

Describe in Detail

Essay▾

Let where . Show that R is a ring under matrix addition and multiplication

Then show that A is a left ideal of R but not a right ideal of R. (Paper II)

Explanation

  • First, we show that R is a ring under matrix addition and multiplication

    I. Is an abelian group

  • (a) Closure property:

    Let

    Addition is closed.

  • (b) Associative law:

    Let , then

  • (c) Existence of Additive identity:

    For

    Where

  • Such that

    And

    i.e.. .

    is the additive identity of R

  • (d) Existence of Additive inverse:

    For

    There is

  • Such that

    Also

    i.e.. .

    is additive in…

… (99 more words) …

Question 257

Appeared in Year: 2007

Describe in Detail

Essay▾

Evaluate (by using residue theorem)

(Paper 2)

Explanation

  • Let

  • Let c be the circle

    Where

  • Poles of are given by

  • Since so lies out side one circle

    is simple poles of

Question 258

Appeared in Year: 2010

Describe in Detail

Essay▾

Consider the polynomial Ring . Show is irreducible over . Let be the ideal in generated by . Then show that is a field and that each element of it is of the form with in . (Paper 2)

Explanation

  • Given

    Let prime

    Now

    satisfies all the conditions of Eisenstein՚s criterion

    is irreducible over

  • Lemma: - Let principal ideal domain, which is not a field. Then any proper ideal of R is maximal it is generated by an irreducible element of R.
  • Proof of lemma: -
  • Since R is not a field, therefore there exist such that is not a unit

    is not a maximal ide…

… (179 more words) …

Question 259

Appeared in Year: 2006

Describe in Detail

Essay▾

Using elementary row operations, find the rank of the matrix.

(Paper 1)

Explanation

Let

Which is row echelon form

Since there are three non-zero rows in row echelon form

Question 260

Appeared in Year: 2007

Describe in Detail

Essay▾

Find the equation of the sphere inscribed in the tetrahedron whose faces are (Paper 1)

Explanation

  • Let be the center and r be one radius of the sphere
  • Since one sphere is inscribed in the tetrahedron

    The length of from one centre on each face = radius

    … eq. (1)

  • From first three members

    … eq. (2)

  • From first and last member of (1)

    , also from (1)

    Centre of sphere is and radius

    The equation of the sphere is

  • Which is one required equation of the sp…

… (1 more words) …

Question 261

Appeared in Year: 2007

Describe in Detail

Essay▾

For any constant vector show that the vector represented by curl is always parallel to the vector being the position vector of a point measured from the origin. (Paper 1)

Explanation

Let

Also given is the position vector of a point measured from the origin.

Now

is parallel to