Optionals IAS Mains Mathematics: Questions 270 - 277 of 283

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Question 270

Edit

Appeared in Year: 2006

Describe in Detail

Essay▾

Solve: (Paper 2)

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

Solution of System of Linear Equations
Edit

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

Edit

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

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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

Edit

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

Edit

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

Edit

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

Edit

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