# Optionals IAS Mains Mathematics: Questions 262 - 269 of 283

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## Question 262

Appeared in Year: *2007*

### Describe in Detail

Essay▾Show that the function given by is not continuous at but its partial derivatives exist at (Paper 2)

### Explanation

Here

Let along the line

Which is not unique as it assumes different values for different value of m.

Does not exist

is not continuous at

Now

And

both exist

## Question 263

Appeared in Year: *2006*

### Describe in Detail

Essay▾Find quadratic form corresponding to the symmetric matrix

Is this quadratic form positive definite? Justify your answer. (Paper 1)

### Explanation

- Given matrix A can be considered as the matrix of a symmetric bilinear form B with request to standard basis
Corresponding quadratic form

i.e..

- Now
- Applying and then apply Corresponding column operation we get
- Applying and then to get
- All entries are positive in diagonal matrix
This quadratic form is positive definite.

## Question 264

Appeared in Year: *2006*

### Describe in Detail

Essay▾State Cayley – Hamilton theorem and using it, find the invers of (Paper 1)

### Explanation

- Cayley – Hamilton Theorem: - every square matrix satisfies its characteristic equation now, let
- Characteristic Polynomial of A is
is characteristic polynomial of A

- By Cayley Hamilton theorem, every require matrix satisfies its characteristic polynomial.
- Now

… (1 more words) …

## Question 265

Appeared in Year: *2009*

### Describe in Detail

Essay▾Prove that the set of all real symmetric matrices farms a linear subspace of the space all real matrices. What is the dimension of this subspace? Find at least one of the bases for V. (Paper 1)

### Explanation

That Point V is subspace of M

Since O is a symmetric matrix

Therefore

Also, clearly

Now, let

Now

is symmetric matrix

Thus V is subspace of M

Now, let such that be any matrix

Since A is any element

generates V

Now, let

B is linearly independent subset of V

B is basis of V

And

## Question 266

Appeared in Year: *2007*

### Describe in Detail

Essay▾Using Lagrange՚s mean value theorem, show that (Paper 2)

### Explanation

(i) is continuous in

(ii) is derivable in

By Lagrange՚s mean value theorem there exist at least on real number such that

Now

… (2 more words) …

## Question 267

Appeared in Year: *2007*

### Describe in Detail

Essay▾Show that the feel of the normal from the point on the paraboloid lie on the sphere (Paper 1)

### Explanation

- The equation of the paraboloid is
… eq. (1)

- The normal to (1) at is
it passes through

… eq. (2)

- From first and last members of (2)
… eq. (3)

- Similarly … eq. (4)
And … eq. (5)

- Since lies on (1)
- Putting value of (3) , (4) , (5) , we get
… eq. (6)

- Point will lie on the sphere
Which is true

- Hence one Result.

… (6 more words) …

## Question 268

Appeared in Year: *2006*

### Describe in Detail

Essay▾If is defined by

Computer the matrix of T relation to the basis

(Paper 1)

### Explanation

- First, we express ant element
as a linear combination of the element of basis B

- Let for some real
- Now
- Given is defined by
And is a basis of

- Now
- And

## Question 269

Appeared in Year: *2007*

### Describe in Detail

Essay▾Show that the sphere cut orthogonally. Find the radius of their common circle. (Paper 1)

### Explanation

- The equation of the two spheres is
… eq. (1)

And

… eq. (2)

- Result: - (1) The condition that the sphere
And

- Are orthogonal its
Sphere (1) and (2) cut orthogonally

Iff

i.e.. if

i.e.. if which is true

- Hence one sphere (1) and (2) are orthogonal.
- Result: - (2) Two sphere of radii cut orthogonally. Prove that the radius of the common circle is
Proof of re…

… (58 more words) …