Optionals IAS Mains Mathematics: Questions 61 - 65 of 283
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Question 61
Appeared in Year: 2014
Describe in Detail
Essay▾Let and be the following subspace of :
And .
Find a basis and the dimension of
(CS main Paper 1)
Explanation
(i) Let
spans
Now, let for
Hence, is a basis of and hence
let
spans W
Now, let
is basis of and hence
Let
and
and
spans and , being a non-zero singleton set is L. I.
Hence is basis of
Question 62
Appeared in Year: 2015
Describe in Detail
Essay▾Consider the following liner programing problem:
Subject to
(i) Using the definition, find it՚s all basic solutions. Which of these are degenerate basic feasible solutions and which are non-degenerate basic feasible solution (s) is/are optimal?
(ii) Without solving the problem, show that it has optimal solution. Which of the basic feasible solution/ (s) is/are optimal? (CS main Paper 2)
Explanation
Given,
Subject to The given system of Equation can be written in the matrix from as where
Since, Rank of A is 2
Then Maximum No. of linearly independent columns of A is 2. Thus we can take any of the following sub-matrices as basis matrix B.
Let us take first . A basic solution to the given system is now obtained by setting and solving the system.
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Question 63
Appeared in Year: 2015
Describe in Detail
Essay▾Give an example of a ring having identity but a subring of this having a different identity. (CS main Paper 2)
Explanation
Let and S =
First, we prove R is Ring
Let
(1)
(2)
(3)
(4)
(5)
So R is Abelian group of addition.
(6)
(7)
And
So, R is semi group under multiplication.
R is a Ring
Also
So is identity of the Ring R.
Now
Also S satisfies all the property of Ring
So S itself is a Ring.
Now, we find the identity of Ring S
Let is identity of Rings and
So is identity of Ring…
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Question 64
Appeared in Year: 2014
Describe in Detail
Essay▾Prove that the eigen value of a unitary Matrix have absolute value 1. (CS main Paper 1)
Explanation
Let be the unitary Matrix.
Let be given value of
Non- Zero Vector X such that
From and , we get
Hence the Result.
Question 65
Appeared in Year: 2011
Describe in Detail
Essay▾Show that the vectors and are linearly independent in . Let be a linear transformation defined by.
Show that the image of above vectors under are linearly dependent. Give the reason for the same. (CS main Paper 1)
Explanation
Given vectors are and
Let
and are linearly independent.
Now, is defined as
The matrix corresponding to vector and is.
So, the vector and are linearly independent.