# Optionals IAS Mains Mathematics: Questions 171 - 178 of 283

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## Question 171

Appeared in Year: 2014

### Describe in Detail

Essay▾

Solve by method of variation Parameters: (CS Paper -1)

### Explanation

Given Differential Equation is: … eq. (1)

Compare (1) with , we get

Now with , we consider the following equation

Integrating, log y =

Where A is arbitrary constant

Let

Then the required general solution is given by

Where

Is required solution

… (1 more words) …

## Question 172

Appeared in Year: 2007

### Describe in Detail

Essay▾

Form a Partial Differential Equation by eliminating the function f from:

(CS paper -2)

### Explanation

Given differential equation is

… eq. (1)

Differentiate partially (1) w. r. t. ‘x’ and ‘y’ , we get

… eq. (2)

And

… eq. (3)

Eliminating from (2) and (3) , we get

And

Which is required partial differential equation.

… (7 more words) …

## Question 173

Appeared in Year: 2010

### Describe in Detail

Essay▾

Construct the dual of the primal problem:

Maximize

Subject to the constraints

And (CS paper -2)

### Explanation

Given

Maximize

Constraints

And

To make the dual in case of Maximisation problem all the constrains should be of type. Hence we multiply I by (-1) and divide the constraint no. II and III into 2 parts and rewrite the above problem as:

Maximize

Subject to

Where

Dual of the above problem is as below:

Min

Subject to

Where

Put in primal there are 3 constr…

… (25 more words) …

## Question 174

Appeared in Year: 2008

### Describe in Detail

Essay▾

Using Laplace transform, solve the initial value problem

With (CS paper -1)

### Explanation

Given … eq. (1)

Taking, Laplace transform, we get

Now

Comparing the coefficient of both side, we get

… eq. (2)

… eq. (3)

… eq. (4)

… eq. (5)

(3)

Putting the value of A and B in (2) & (3) , we get

… eq. (6)

And … eq. (7)

Now subtract equation (6) from (7) , we get

(6)

Applying inverse Laplace transform, we get

… (7 more words) …

## Question 175

Appeared in Year: 2011

### Describe in Detail

Essay▾

Obtain Clairaut՚s form the differential equation

Also find its general solution (CS paper -1)

### Explanation

Given Differential Equation is

It can also be written as

When

… eq. (1)

Putting … eq. (2)

… eq. (3)

Using (2) and (3) , (1) becomes

It can also be written as

Which is clairaut՚s form.

So, Replacing P by arbitrary constant c, the required solution is

Where c is arbitrary constant

… (5 more words) …

## Question 176

Appeared in Year: 2013

### Describe in Detail

Essay▾

A Sphere s has points at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere s with the plane as a great circle. (CS Paper -1)

### Explanation

The equation of the sphere s has the points (0,1, 0) , (3, -5,2) at opposite ends of a diameter is

… eq. (1)

Also, equation of given plane is … eq. (2)

The equation of any sphere through the circle

is

… eq. (3)

Its centre is

If the circle of intersection of (1) and (2) is great circle of sphere (3) then the centre

Of sphere (3) must lie on plane (2)

P…

… (15 more words) …

## Question 177

Appeared in Year: 2008

### Describe in Detail

Essay▾

Solve the Equation (CS paper -1)

### Explanation

Given equation is

… eq. (1)

Differentiating (1) w. r. t. y, we get

Integrating, we get

… eq. (2)

To eliminate p between (1) and (2) , first solve (2) for p

Putting this value of p in (1) , we get

… (2 more words) …

## Question 178

Appeared in Year: 2012

### Describe in Detail

Essay▾

By the method of Vogel, determine an initial basic feasible solution for the following transportation problem:

Products have to be sent to destinations . The cost of sending product to destinations where the matrix

The total requirements of destinations are given by respectively and the availability of the products are respectively . (CS paper -2)

### Explanation

Give Transportation Problem is Transportation Problem in Table

By using Vogel Approximation Method Vogel Approximation Method

Noted that there are allocations which are necessary to proceed further.

The minimum transportation cost is