# Optionals IAS Mains Mathematics: Questions 270 - 277 of 283

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## Question 270

### Explanation

Given differential equation is

Standard Form is

Auxiliary Equation is

Complementary Function

being arbitrary constant

Now, Partial Integration

The required solution is

## Question 271

Appeared in Year: *2006*

### Describe in Detail

Essay▾Investigate for what values of the equation

Have –

(i) No solution

(ii) A unique solution

(iii) Infinitely many solution (Paper 1)

### Explanation

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

… (23 more words) …

## Question 272

Appeared in Year: *2010*

### Describe in Detail

Essay▾Show that is harmonic function. Find a harmonic conjugate of. Hence find the analytic function for which is the real part. (Paper 2)

### Explanation

- (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.

… (33 more words) …

## Question 273

Appeared in Year: *2009*

### Describe in Detail

Essay▾Find the directional derivative of –

(i)

(ii) in the direction of (Paper 1)

### Explanation

(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

## Question 274

Appeared in Year: *2006*

### Describe in Detail

Essay▾Let s be the set of all real number except –

Is group?

Find the solution of the equation

in s (Paper 2)

### Explanation

- 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

… (15 more words) …

## Question 275

Appeared in Year: *2006*

### Describe in Detail

Essay▾Let V be vector space of all matrices over the field F. above that V has dimension 4 by exhibition a basis for V. (Paper 1)

### Explanation

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

## Question 276

Appeared in Year: *2007*

### Describe in Detail

Essay▾If , find the value of n in order that may be (i) solenoidal, (ii) irrational (Paper 1)

### Explanation

Here

(i)

Now is solenoidal is

i. e.

(ii) Now curl

is irrational for every value of n.

## Question 277

Appeared in Year: *2007*

### Describe in Detail

Essay▾Find curvature and torsion at any point of the curve (Paper 1)

### Explanation

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

And

Also

And

And