The probability of finding a green signal on a busy crossing X is 30 % . What is the probability of finding a green signal on X on two consecutive days out of three?

Let A be the event of finding a green signal on a busy crossing X The probability of finding a green signal on a busy crossing X is 30 % . P A = 30 100 = 3 10 The probability of not finding a green signal on a busy crossing X is P A ¯ = 1 - 3 10 = 7 10 . The probability of finding a green signal on X on two consecutive days out of three = P A P A P A ¯ + P A ¯ P A P A = 3 10 × 3 10 × 7 10 + 7 10 × 3 10 × 3 10 = 63 500 .

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.

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What is the formula used to calculate the sacrificing ratio when a new partner is admitted?

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Important Questions on Determinants

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

The probability of finding a green signal on a busy crossing

X

30 %

. What is the probability of finding a green signal on

X

on two consecutive days out of three?

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

A problem in mathematics is given to three students

A, B, C

and their respective probability of solving the problem is

\frac{1}{2}, \frac{1}{3}

and

\frac{1}{4}

, the probability that the problem will be solved is

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

A die marked

1, 2, 3

in red

4, 5, 6

in green tossed. Let

A

be the event "number is even" and

B

be the event "number is marked red". Find whether the events

A

and

B

are independent or not.

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

Out of

{[{CoF}_{6}]}^{- 3}

and

[Co {(en)}_{3}]^{3 +}

, which complex compound is high spin complex.

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

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

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

Out of

{[{CoF}_{6}]}^{- 3}

and

[Co {(en)}_{3}]^{3 +}

which complex compound is paramagnetic.

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

Four persons can hit a target correctly with probabilities

\frac{1}{2}, \frac{1}{3}, \frac{1}{4}

and

\frac{1}{8}

respectively. If all hit at the target independently, then the probability that the target would be hit, is

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

A square loop of side $' a'$ carrying a current $I_{2}$ is kept at distance $x$ from an infinitely long straight wire carrying a current $I_{1}$ as shown in the figure. Obtain the expression for the resultant force acting on the loop.

Question Image

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

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>Algebra>Determinants>Minor and Cofactors

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.

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

Out of

{[{CoF}_{6}]}^{- 3}

and

[Co {(en)}_{3}]^{3 +}

, which complex compound is inner orbital complex.

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

Show that the relation

R

R

is defined as

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

, is reflexive and transitive but not symmetric.

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

On the basis of valence bond theory explain the oxidation state, hybridisation, geometry and magnetic nature of metal in complex

{[{CoF}_{6}]}^{3 -}

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

Explain

[Co {({NH}_{3})}_{6}]^{3 +}

is an inner orbital complex whereas

[Ni {({NH}_{3})}_{6}]^{2 +}

is an outer orbital complex.

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

An arrangement of three parallel straight wires placed perpendicular to plane of paper carrying same current $I$ along the same direction is shown in Figure. Magnitude of force per unit length on the middle wire $B$ is given by

Question Image

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

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>Algebra>Determinants>Minor and Cofactors

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>Algebra>Determinants>Minor and Cofactors

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

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

Test whether the following relation $R_{2}$ is reflexive, symmetric and transitive :

$R_{2}$ on $Z$ defined by $(a, b) \in R_{2} \Leftrightarrow | a - b | \leq 5$

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

The following relation is defined on the set of real numbers.
$a R b$ if

1 + a b > 0

Find whether relation is reflexive, symmetric or transitive.

What is the formula used to calculate the sacrificing ratio when a new partner is admitted?

Important Questions on Determinants

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