Let N be the set of natural numbers and R be the relation on N × N defined by ( a , b ) R ( c , d ) iff a d = b c for all a , b , c , d ∈ N . Show that R is an equivalence relation.

Given that, N is the set of natural numbers and R be the relation on N × N defined by ( a , b ) R ( c , d ) iff a d = b c for all a , b , c , d ∈ N We know that, a relation R that is reflexive, symmetric and transitive is called an equivalence relation. Reflexive: For any ( a , b ) ∈ N × N a · b = b · a ∴ ( a , b ) R ( a , b ) Thus, R is reflexive Symmetric: For any ( a , b ) , ( c , d ) ∈ N × N ( a , b ) R ( c , d ) ⇒ a · d = b · c ⇒ c · b = d · a ⇒ ( c , d ) R ( a , b ) Thus, R is symmetric Transitive: For any ( a , b ) , ( c , d ) , ( e , f ) , ∈ N × N ( a , b ) R ( c , d ) and ( c , d ) R ( e , f ) ⇒ a · d = b · c and c · f = d · e ⇒ a · d · c · f = b · c · d · e ⇒ a · f = b · e ∴ ( a , b ) R ( e , f ) Thus, R is transitive Therefore, R is an equivalence Relation.

Check whether the relation R defined in the set 1 , 2 , 3 , 4 , 5 , 6 as R = a , b : b = a + 1 is reflexive, symmetric or transitive.

Suppose A = 1 , 2 , 3 , 4 , 5 , 6 . A relation R is defined on set A as: R = { ( a , b ) : b = a + 1 } ∴ R = { ( 1, 2 ) , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 } Reflexive Property : A relation R in a set A is said to be reflexive if x , x ∈ R , for every x ∈ A Now, 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 ∉ R ∴ R is not reflexive. Symmetric Property : A relation R in a set A is said to be symmetric if x , y ∈ R implies that y , x ∈ R for all x , y ∈ A . Now it can be observed that 1 , 2 ∈ R , but 2 , 1 ∉ R . Hence , R is not symmetric. Transitive Property : A relation R in a set A is said to be transitive if x , y ∈ R and y , z ∈ R implies that x , z ∈ R for all x , y , z ∈ A . Now, 1 , 2 , 2 , 3 ∈ R but, ( 1 , 3 ) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive.

Show that the relation R on R is defined as R = { ( a , b ) : a ≤ b } , is reflexive and transitive but not symmetric.

Given, R = a , b : a ≤ b , a , b ∈ R For the relation R to be reflexive: If the relation R is reflexive, then a , a ∈ R i.e. a ≤ a , which is always true for all the values of a , a ∈ R So, the given relation R is reflexive. For the relation R to be symmetric: If a , b ∈ R then b , a ∈ R i.e. if a ≤ b then b ≤ a , which is not possible for all the values of a , b ∈ R . For example: if a = 2 and b = 5 Here, a < b but b > a So, the given relation R is not symmetric. For the relation R to be transitive: If a , b ∈ R and b , c ∈ R then a , c ∈ R i.e.,if a ≤ b and b ≤ c then a ≤ c , which is true for all the values of a , b , c ∈ R . For example: if a = 2 , b = 3 and c = 4 Here, a < b < c So, the given relation is transitive. Hence, the given relation R is reflexive and transitive but not symmetric.

Find a matrix A such that 2 A - 3 B + 5 C = 0 , where B = - 2 2 0 3 1 4 and C = 2 0 - 2 7 1 6

We have, 2 A - 3 B + 5 C = O ⇒ 2 A = 3 B - 5 C ⇒ 2 A = 3 - 2 2 0 3 1 4 - 5 2 0 - 2 7 1 6 2 A = - 6 6 0 9 3 12 - 10 0 - 10 35 5 30 2 A = - 16 6 10 - 26 - 2 - 18 ⇒ A = - 8 3 5 - 13 - 1 - 9

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Which principle of organising emphasizes that a subordinate should receive orders from one superior only?

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Important Questions on Continuity and Differentiability

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

The identical point charges,

q

each, are kept

2 m

apart in air. A third point charge

Q

of unknown magnitude and sign is placed on the line joining the charges such that the system remains in equilibrium. Find the position and nature of

Q

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

For

x ϵ (0, \frac{3}{2}),

let

f (x) = \sqrt{x}, g (x) = t a n x

and

h (x) = \frac{1 - x^{2}}{1 + x^{2}}

. If

ϕ (x) = (h o f) o g) (x),

then

ϕ (\frac{π}{3})

is equal to:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Four-point charges $Q, q, Q$ and $q$ are placed at the corners of a square of side $' a'$ as shown in the figure.

Question Image

Find the resultant electric force on a charge $Q$ .

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

A = [\begin{matrix} 0 \\ 1 \end{matrix} \begin{matrix} - 1 \\ 0 \end{matrix}]

, then which one of the following statements is not correct?

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

A relation

R

in a set

A

is called _____, if

(a_{1}, a_{2}) \in R

implies

(a_{2}, a_{1}) \in R

, for all

a_{1}, a_{2} \in A

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{matrix}]$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

f (x) = x + 1

, find

\frac{d}{d x} (fof) (x)

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

A

is a square matrix such that

A^{2} = I

, then

(A - I)^{3} + (A + I)^{3} - 7 A

is equal to

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

N

be the set of natural numbers and

R

be the relation on

N \times N

defined by

(a, b) R (c, d) iff a d = b c

for all

a, b, c, d \in N

. Show that

R

is an equivalence relation.

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Check whether the relation

R

defined in the set

\{1, 2, 3, 4, 5, 6\}

R = \{(a, b) : b = a + 1\}

is reflexive, symmetric or transitive.

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $B$ is a $3 \times 3$ matrix such that $B^{2} = 0$ , then $d e t . [{(I + B)}^{50} - 50 B]$ is equal to :

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Show that the relation

R

R

is defined as

R = {(a, b) : a \leq b}

, is reflexive and transitive but not symmetric.

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

A = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}], (a \in R)

such that

A^{32} = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] .

Then, a value of

α

is:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Three-point charges $q, - 4 q$ and $2 q$ are placed at the vertices of an equilateral triangle ABC of side $l$ as shown in the figure. Obtain the expression for the magnitude of the resultant electric force acting on the charge $q$ .

Question Image

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

A

is a symmetric matrix and

B

is skew- symmetric matrix such that

A + B = [\begin{matrix} 2 & 3 \\ 5 & - 1 \end{matrix}]

, then

A B

is equal to:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Find a matrix

A

such that

2 A - 3 B + 5 C = 0

, where

B = [\begin{array}{r} - 2 & 2 & 0 \\ 3 & 1 & 4 \end{array}]

and

C = [\begin{array}{r} 2 & 0 & - 2 \\ 7 & 1 & 6 \end{array}]

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

A

and

B

are two matrices such that

A B = B, B A = A

then

A^{2} + B^{2} =

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

R

be a relation defined on

ℕ

as a

R

b is

2 a + 3 b

is a multiple of

5, a, b \in ℕ

. Then

R

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

A = {x \in Z : 0 \leq x \leq 12}

. Show that

R = {(a, b) : a, b \in A, | a - b | is divisible by 4}

is an equivalence relation. Find the set of all elements related to

1

. Also write the equivalence class

[2]

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Consider the non-empty set consisting of children in a family and a relation

R

defined as

a R b

a

is brother of

b

. Then

R

Which principle of organising emphasizes that a subordinate should receive orders from one superior only?

Important Questions on Continuity and Differentiability

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