Optionals IAS Mains Mathematics: Questions 278 - 283 of 283

• Given differential equation is

• Hence auxiliary Equation is

  … eq. (1)

• From last two members, we have

• Integrating, we get

  … eq. (2)

• From first and last members if (1) , we have

• Putting

  We get

• Which is ordinary differential equation of order 1

• Hence general solution of given differential equation is

Grad

Now

i.e..

Similarly

And

Adding vertically, we get

The equation of cone is

… eq. (1)

The equation of plane is

… eq. (2)

Comparing (1) with

We get

Here

Necessary and sufficient condition that the cone may have three mutually perpendicular generators

The cone has three mutually perpendicular generators

The plane (2) cuts the cone (1) in perpendicular lines if the normal to the plane (2) through the v…

• Given

  Now is always continuous

• Clearly, if given by

• I. is Abelian group

  (i) let

  are real valued continuous function

  Now

• We know that sum of two continuous function is continuous

  is also continuous

• And sum of real valued function is real valued

  is also real valued

  is closed under addition.

• (ii) let,

  Then

  is associative under addition

• (iii) Existence o…

The answer to c is not correct, I guess.

H is a subspace of V if

• The zero vector is an element of H
• H is closed under addition: if a, b in H, then in H
• H is closed under scalar multiplication: if in H and c a scalar then cx in H

is a subspace, because

• 0, the zero matrix is in H, because
• H is closed under addition: Let A, C in H, thus and , then, thu…

Let X and Y be two given subspace.

Case 1- one subspace is contained in other

X is contained in Y. then union X and Y will be Y. and since Y is already subspace, union will be subspace too.

Case 2- union is subspace.

• Now assume X is not contained in Y. that is x does not lie in Y.
• Let x belongs to X and y belongs to Y.
• Subspace satisfies closure property…