# 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

By using Vogel Approximation Method

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

The minimum transportation cost is