Optionals IAS Mains Mathematics: Questions 146 - 153 of 283

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Question 146

Maxima and Minima

Appeared in Year: 2016

Describe in Detail

Essay▾

Find the relative Maxima and Minima value of the function

(CS Paper 2)

Explanation

Given

Step I

Step II

and

Adding (1) and (2) , we get

From (1) ,

Critical points are

Step III

Step IV

At

At further investigation is needed

Consider

Which does not keep the same sign for all small value of

There is neither Maxima nor minima at

At

Also,

has a minima value at

And Minimum value

At

Also

has a Minima value at

And Minimum value

Question 147

Appeared in Year: 2013

Describe in Detail

Essay▾

Use Euler՚s method with step size to compute the approximate value of Correct up to five decimal place from the initial value Problem.

(CS Paper 2)

Explanation

Here,

Step (1)

Step (2)

Now,

Step (3)

Step (4)

Hence, for

Question 148

Appeared in Year: 2011

Describe in Detail

Essay▾

Calculate (upto 3 places of decimal) by dividing the range into 8 equal parts by Simpson՚s rule. (CS Paper 2)

Explanation

We have,

The values of and are tabulated as

Table Shows the Value of X and F (X)

By Simpson՚s rule, we have

… (2 more words) …

Question 149

Appeared in Year: 2012

Describe in Detail

Essay▾

Find at from the following data:

Table Shows the Value of X and Y
x:0.10.20.30.4
y:0.99750.99000.97760.9604

(CS Paper 2)

Explanation

The forward difference table is:

Shows the Forward Difference Table

Taking we have

By newton՚s forward difference formula at

We have

at

… (2 more words) …

Question 150

Appeared in Year: 2014

Describe in Detail

Essay▾

Use Runge – Kutta formula of forth order to find the value of y at where , Take the Step length (CS Paper 2)

Explanation

We have

Step (1) Here,

Step (2) Here

Question 151

Appeared in Year: 2011

Describe in Detail

Essay▾

Obtain the general solution of the second order ordinary differential equation

Where dashes denotes derivatives w. r. t (CS Paper 1)

Explanation

Given, Ordinary differential equation is

S. F. is

A. E. is

C. F

Now, P. I

Now,

And

(Case fail)

So,

Hence, P. I

The required solution is

Question 152

Appeared in Year: 2013

Describe in Detail

Essay▾

Solve the Differential Equation

(CS Paper 1)

Explanation

Comparing (1) with

and

We have,

Which is function of alone

I. F of (1)

Multiplying (1) by , we have

Which must be exact equation and so its solution is

(Taking y as a constant)

Where c is the arbitrary constant.

Question 153

Appeared in Year: 2013

Describe in Detail

Essay▾

By using Laplace Transform method, solve the differential equation

subject to the initial conditions and at in which are constants. (CS Paper I)

Explanation

Given,

Taking Laplace transform of the given differential equation

Where

Now

Now (1) becomes

Taking inverse Laplace transform (2)

Now

And

So, equation (3) become