# Optionals IAS Mains Mathematics: Questions 106 - 114 of 283

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## Question 106

Appeared in Year: *2012*

### Describe in Detail

Essay▾Find the dimension and basis for the space all solutions of the following homogeneous

System using matrix rotation:

(CS main Paper 1)

### Explanation

The given equations are:

These equations can be written as

Where

Now,

Operate and

Operate

Operate

The given system reduces to

and

and

The solution set is

Since we have five unknowns and two non-zero equation in echelon form.

The three variables are independent and two variable are dependent

Hence basis of solution space and

## Question 107

Appeared in Year: *2015*

### Describe in Detail

Essay▾Solve for the general solution

Where and (CS Paper 2)

### Explanation

We have

The Lagrange՚s auxiliary equations are

Choosing as multipliers, each fraction of (1)

Choosing as multipliers, each fraction of (1)

From (1) , (2) , and (3)

Taking the first two fraction of (4)

Putting

(5) reduces to

Integrating

Taking the two fraction of (4)

On R. H. S of (7) , Putting

Then (7) Reduces to

On Integrating, we get

From (6) and (8) , t…

… (10 more words) …

## Question 108

Appeared in Year: *2009*

### Describe in Detail

Essay▾Find the inverse Laplace transform of

(CS Paper 1)

### Explanation

>

Then,

## Question 109

Appeared in Year: *2014*

### Describe in Detail

Essay▾Solve the system of equations

Using Gauss – Seidel iteration method (perform Three Iteration) (CS Paper 2)

### Explanation

Given equation can be written as

According to Gauss – Seidel Method, the approximations for the unknowns are given by

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

Iteration (1)

Iteration (2)

Iteration (3)

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

… (1 more words) …

## Question 110

Appeared in Year: *2015*

### Describe in Detail

Essay▾Solve the following Assignment problem to minimize the Sales

(CS Paper 2)

### Explanation

Using Hungarian method by Kuhn

Step (1) : Row Reduction

Subtracting the mimimum element of each row from all the element of that row.

Table 1

Step (2) : Column Reduction

Table 2

Step (3) : Making Assignments

Table 3

The No. of assignment are less than 5 so, we draw minimum number if lines to …

… (176 more words) …

## Question 111

Appeared in Year: *2009*

### Describe in Detail

Essay▾Solve:

(CS Paper 1)

### Explanation

Given

Put

From (2)

Take integration both sides we get,

Put

## Question 112

Appeared in Year: *2011*

### Describe in Detail

Essay▾Solve the PDF

(CS Paper 2)

### Explanation

Given >

Now P. I

P. I

P. I

And

P. I

So, required solution is

## Question 113

Appeared in Year: *2013*

### Describe in Detail

Essay▾Reduce the equation

to its canonical form when

### Explanation

Given

Comparing (1) with

Here,

Equation reduce to

Then, the corresponding equation are given by,

Integrating, these

In order to reduce (1) to its canonical form, we choose

Now,

Also,

Using (5) , (6) and (7) in (1) , we get

It is the required canonical form of (1)

## Question 114

Appeared in Year: *2016*

### Describe in Detail

Essay▾For an integral , show that the two – point Gauss Quadrature rule is given by using this rule, estimate (CS Paper 2)

### Explanation

every Finite interval [a, b] can always be transformed to [-1,1] using the transformation

We consider the integral in the form

We consider the integral in the form

Let the weight function be then (1) reduces to

Where nodes and weight are unknown

For two point formula

Here four unknowns,

Making the method exact for

we get

Eliminating from (3) and (4) …

… (37 more words) …