# Optionals IAS Mains Mathematics: Questions 26 - 35 of 283

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## Question 26

Appeared in Year: *2016*

### Describe in Detail

Essay▾If then find dim and dim [CS (Main) Paper 1]

### Explanation

Given

Now

i.e.. is the solution space of equation

Now, we shall find at basis.

Hence a bases of solution space

and dim

Now given

i.e.. is the solution space of the equation

Now we shall find it basis

Here we have 3 unknown out of which an independent.

Hence a basis of solution space

and dim

Now

Now, we find to find its basis

Here

Here, we have thre…

… (38 more words) …

## Question 27

Appeared in Year: *2015*

### Describe in Detail

Essay▾Obtain Laplace Inverse Transform of

[CS (Main) Paper 1]

### Explanation

- Given is
- Now
- Where
Now

Let

- Now
- But
- Now
- We know
By second shifting theorem

## Question 28

Appeared in Year: *2015*

### Describe in Detail

Essay▾Using Laplace transform, solve [CS (Main) Paper 1]

### Explanation

- Given
- Apply Laplace transform on both sides, we get
- using given condition we get
- Taking inverse Laplace Transform, we get
- Now
- Putting these value in we get

## Question 29

Appeared in Year: *2015*

### Describe in Detail

Essay▾Evaluate , where is the reactangle with vertices [CS (Main) Paper 1]

### Explanation

- By Green՚s Theorem in plane,
- Here

… (4 more words) …

## Question 30

Appeared in Year: *2015*

### Describe in Detail

Essay▾If Matrix then find [CS (Main) Paper 1]

### Explanation

Given

where and

Now

Now

## Question 31

Appeared in Year: *2015*

### Describe in Detail

Essay▾Find the constant so that is the integrating factor of

and hence solve the differential equation [CS (Main) Paper 1]

### Explanation

- Given differentia; equation is
… eq. (1)

If is I. F. of

- Then
is exact

- Comparing with we get
Now

is Exact

- Comparing coefficient both sides, adges
- and
- and
- and
- So solution is
(treating as constant.)

… (1 more words) …

## Question 32

Appeared in Year: *2016*

### Describe in Detail

Essay▾Solve [CS (Main) Paper 1]

### Explanation

- Given Differential equation is
- Now compare it with
Linear differential equation.

We get

- Thus, the solution of this differential equation is
i.e.. .

Now

- Then Solution is
Put

## Question 33

Appeared in Year: *2016*

### Describe in Detail

Essay▾Determine the characteristic of the equation and find the integral surface which passes through the parabola [CS (Main) Paper 2]

### Explanation

- The given differential equation is
- The initial condition for are
- The initial condition of are determined by
and

and

and

and

and

- The characteristic equations are
- Integrating, we have
- Appling Initial condition
we get

- Also
- On integrating, we have
- Appling initial condition
We have

Also

- On integrating, we get
- Apply initial condition
Now

- on integrating both …

… (45 more words) …

## Question 34

Appeared in Year: *2016*

### Describe in Detail

Essay▾Using elementary two operations, find the condition that the linear equations.

Have a solution [CS (Main) ]

### Explanation

- Given linear equation is
Here

Now

Operate:

Operate:

- Now, here
- The linear equation have a solution if
Now iff

(Answer)

- If then given linear equation will not have any solution.

## Question 35

Appeared in Year: *2015*

### Describe in Detail

Essay▾For what positive value of , the plane touches the sphere and hence find the point of contact. [CS (Main) Paper 1]

### Explanation

- The equation of sphere is
- Its centre is
and radius

- The Equation of plane is
- The plane will touch the sphere is perpendicular distance of centre from plane
if

if

if

if

if

if

if

- To find point of contact
- The direction ratio՚s of the normal to the plane are
- The equation of line through the centre of the sphere plane are
Any point on it is if it lies …

… (9 more words) …