Optionals IAS Mains Mathematics: Questions 124 - 129 of 283

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Question 124

Appeared in Year: 2013

Describe in Detail

Essay▾

is a function of , such that the differential coefficient is equal to . Find out a relation between , which is free from any derivative/differential. (CS Paper 1)

Explanation

Given differential Equation

Let

Diff. (1) w. r. t

Using (1) and (2)

But,

(3) reduce to,

Integrating,

Where c is the arbitrary constant

Question 125

Appeared in Year: 2011

Describe in Detail

Essay▾

Use Laplace Transform method to solve the following initial value problem:

(CS Paper 1)

Explanation

Given,

S. F is

Taking Laplace transform of this equation, we get

Putting and we get

Taking inverse Laplace Transform. We get,

Now,

And

Question 126

Edit

Appeared in Year: 2016

Describe in Detail

Essay▾

Prove that every power series represents an analytic function inside its circle of convergence (CS Paper 2)

Explanation

Suppose >

is given power series and

Let R be the radius of convergence of the power series then R is also radius of convergence of the power series which is the derived series of

Let z be any point within the circle

Also there exist a real number such that

Then the series is convergent

For

is bounded

So, there exist a finite real number such tha…

… (40 more words) …

Question 127

Appeared in Year: 2013

Describe in Detail

Essay▾

The velocity of a train which starts from rest is given in the following table. The time is in minutes and velocity is in km/hour.

The Time is in Minutes and Velocity is in Km/Hour

Estimate approximately the total distance run in 20 minutes by using composite Simpson՚s rule. (CS Paper 2)

Explanation

If is the distance covered in t (min) , then

At, the velocity of train

The Time is in Minutes and Velocity is in Km/Hour

Question 128

Appeared in Year: 2015

Describe in Detail

Essay▾

Find the solution of the System

Using Gauss – Seidel Method (make Four Iteration) (CS Paper 2)

Explanation

The system of Equation is,

Given, Equation can be written as

According to the Gauss – Seidel Method, the approximation of the unknown is given by

Let us take the initial approximation to be zero for each unknown

Iteration (1)

Iteration (2)

Iteration (3)

Iteration (4)

Hence, after 4 iteration the solution of given system of equation is

… (1 more words) …

Question 129

Appeared in Year: 2011

Describe in Detail

Essay▾

Obtain the solution of the ordinary differential equation

(CS Paper 1)

Explanation

Given, ordinary differential equation is

Let

Differentiating (1) with respect to we get

From (1)

Integrating, we get

Where is the arbitrary constant

Where is the

Where

Using (1) ,

Putting in (4) , we get

So, required solution is