# F Is One-One: Let

**Chapter 1: Relations and functions.**

Question numbers 1 to 5 are of each 1 mark and question numbers 6 to 10 are of 4 marks each.

Let A = {1, 2, 3}, B = {4, 5, 6} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.

Answer / Since f(1) = 4, f(2) = 5, f(3) = 6 that is different elements of doain have different f-images in the range.

Therefore f is one-one.

If be defined by , then find

Answer /

If binary operation * is defined as , then evaluate

Answer /

If * is a binary operation on positive rational numbers and defined by , find the identity element.

Answer / Let e be the binary element such that

Thus

Find the domain of definition of the function f(x) = log|x|.

Answer / F(x) = log|x|, for real domain, ,

If and , then find the value of

Answer /

Therefore

Show that the function defined by is one-one and onto function. Also find inverse of the function f.

Answer / The function defined by

- F is one-one: Let

Therefore f is One-One.

- F is Onto: Let

Examine which of the following is a binary operation (i) . For binary operation check the commutative and associative property.

Answer /

And for all

Therefore it is binary operation on Rational numbers.

Commutativity:

Associativity:

Therefore,

Or binary operation is not Associative.

Let X be non-empty set. P(X) be its power set. Let ‘*’ be an operation defined on elements of P(x) by , then (i) Prove that * is a binary operation in P(X). (ii) Is * commutative? (iii) Is * Associative? (iv) Find the identity element in P(X) w.r.t *.

Answer / Given:

(i)Let A and B are any two non-empty subsets of X, then we know that

Now for any two non empty elements of P(X), we have

Therefore, ‘*’ is binary operation in P(X).

(ii)For any two non-empty subsets A and B of X, then we know that

Therefore ‘*’ is commutative.

(iii)For any three non-empty subsets A, B, C

We have

Therefore * is associative.

(iv)Let I be the identity element in P(X).

A*I = I*A = A

Therefore

Thus I is the identity element.

Given a non-empty set X, let be define as . Show that the empty set is the identity fr the operation * and all the elements A of P(X) are invertible with

Answer / Given let be define as

I Part: Let

II Part: Let

**Chapter 2: Inverse Trigonometric Functions**

Question numbers 11 to 15 are of each 1 mark and question numbers 16 to 20 are of 4 marks each.

What is the principle value of ?

Answer /

Find the value of

Answer / Let

If , then find the principle value of .

Answer /

Express

Answer / Let

For what value of x,

Answer / Given that but

Add both relations

Find the value of

Answer / Sol:

Solve for x:

Answer / Solution:

Prove that

Answer /

If

Answer / We know that

If

Answer /

Square both the sides

**Chapter 3: Matrices**

Question numbers 21 to 25 are of each 1 mark and question numbers 26 to 30 are of 4 marks each and question numbers 31 to 35 are of 6 marks each.

If

Answer / Since

Construct a 2 x 2 matrix whose element

Answer /

If

Answer /

If

Answer /

If

Answer / Since

If

Answer / Since

Hence proved.

Pre-multiplying

Represent the given matrix as the sum of symmetric and skew symmetric matrix

Answer / Since

Let

Let

Hence the result.

If , where n is any positive integer.

Answer / We shall prove it by using principle of Mathematical Induction.

For n = 1, , which is true.

Therefore given statement is true for n = 1.

Let us suppose that given statement is true for n = k.

Let us take n = k + 1

which is true for n = k +1

Therefore by PMI, it is true for every positive integer k.

Find the values of x, y and z if the given matrix A satisfy the equation A.A’ =I where

Answer /

Given that AA’=I

Prove that inverse of every square matrix, if it exists, is unique.

Answer / Let ‘A’ be any invertible square matrix of order n.

Let B and C are two inverse of A.

Therefore; AB = BA = I - - - - - (1)

And AC = CA = I ------(2)

From (1), since AB = I

Pre-multiply it by C, we get,

C(AB) = CI

Or (CA)B = C [Since C(AB) = (CA)B and CI = C]

Or IB = C [from (2), CA = I]

Or B = C

Hence inverse of a matrix is unique.

Using elementary operations find the inverse of

Answer / Using row operations; therefore, A = IA

Therefore

Using elementary operations find the inverse of

Answer / Let us apply row operations, that is A = IA

Therefore

Therefore

Using elementary operations find the inverse of

Answer /

Using elementary operations find the inverse of

Answer /

Using elementary operations find the inverse of

Answer /

(Page 1 of 12)