Optionals IAS Mains Mathematics: Questions 270 - 277 of 283

Given differential equation is

Standard Form is

Auxiliary Equation is

Complementary Function

being arbitrary constant

Now, Partial Integration

The required solution is

The given equation are

Which can be written as

(ii) The given equation will have unique solution

If

Applying we get

we get

One given set of equation has a unique solution, no matter what is one value of

(i) Take

Applying

Applying

Now

Now given equations has no solution is

Now is 3 is

is

, then given set of equation does not have any solution

(iii) if

• (I). Given ……………. eq. (1) ………………eq. (2) ……………eq. (3) ……………. eq. (4) ……………. . eq. (5) Satisfies Laplace’s Equation is harmonic function
• (II) [By C. R equation] ………………. . eq. (1) Where Now
• Hence equation (1) is exact differential equation. Hence its solution is is required analytic function.

(i) Here

At

At

Let be a unit vector in the direction of z-axis

Required directional derivative

(ii) Here

,

At

Let be a unit vector in the direction of

Required directional derivative

• Given s is one set of all numbers except-1 and
• That Point is a group
• (i) Closure property Let Now Let
• Which is not true
• Do closure property satisfied
• (ii) Associative Property Let Now And Associative Property is satisfied.
• (iii) Existence of Identity Let be any element Let is identity element of
• (iv) Existence of inverse Let be

Given is a vector space of all matrices over field f

Let be any element

Now

Generates V.

Therefore, B is linearly independent

Let

is linearly independent set

is basis of V and

Here

(i)

Now is solenoidal is

i. e.

(ii) Now curl

is irrational for every value of n.

Let be the position vector of any point on the given curve.

And

Also

And

And