Tag: mathematics and statistics

Questions Related to mathematics and statistics

If $(3p+q,p-q)=(p-q,3p+q)$, then:

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $p=q=0$

  2. $p=q$

  3. $p=2q$

  4. $p+q=0$

Show answer
Correct Option: D
Explanation:

As the ordered pairs are equal,
$ 3p + q = p - q $
$  => 2p = -2q $
$ => 2p + 2q = 0 $
$ => p + q = 0 $


If $\displaystyle n\left ( P\times Q \right )=0$  then n(P) can possibly be

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. 0

  2. 10

  3. 20

  4. Any value

Show answer
Correct Option: D
Explanation:

$ n(P) $ can be of any value as we are not sure of $ n(Q) $
Hence, $ n(P) $ can take any of the given values.

If $A=\left {1, 2,3\right }$ and $B=\left {3,8\right }$, then $(A\cup B)\times (A\cap B)$ is equal to

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $\left {(8,3), (8,2), (8,1), (8,8)\right }$

  2. $\left {(1,2), (2,2), (3,3), (8,8)\right }$

  3. $\left {(3,1), (3,2), (3,3), (3,8)\right }$

  4. $\left {(1,3), (2,3), (3,3), (8,3)\right }$

Show answer
Correct Option: D
Explanation:

Given, $A=\left {1, 2,3\right }$ and $B=\left {3,8\right }$,
Therefore, $A\cup B=\left {1,2,3\right }\cup \left {3,8\right }=\left {1,2,3,8\right }$
and $A\cap B=\left {1,2,3\right }\cap \left {3,8\right }=\left {3\right }$
$\therefore (A\cup B)\times (A\cap B)=\left {1,2,3,8\right }\times \left {3\right }$
$=\left {(1,3), (2,3), (3,3), (8,3)\right }$

What is the Cartesian product of $A = \left {1, 2\right }$ and $B = \left {a, b\right }$?

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $\left {(1, a), (1, b), (2, a), (b, b)\right }$

  2. $\left {(1, 1), (2, 2), (a, a), (b, b)\right }$

  3. $\left {(1, a), (2, a), (1, b), (2, b)\right }$

  4. $\left {(1, 1), (a, a), (2, a), (1, b)\right }$

Show answer
Correct Option: C
Explanation:

If $A $ and $B$ are two non empty sets, then the Cartesian product $A \times B$ is set of all ordered pairs $(a,b)$ such that $a\in A$ and $b\in B$.


Given $A ={1,2}$ and $B = {a,b}$

Hence $A\times B = {(1,a),(1,b),(2,a),(2,b)}$ 

Let a relation $R$ be defined by $R=\left {(4,5), (1,4), (4,6), (7,6), (3,7)\right }$. The relation $R^{-1}\circ R$ is given by

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $\left {(1,1), (4,4), (7,4), (4,7), (7,7)\right }$

  2. $\left {(1,1), (4,4), (4,7), (7,4), (7,7),(3,3)\right }$

  3. $\left {(1,5), (1,6), (3,6)\right }$

  4. None of these

Show answer
Correct Option: B
Explanation:
We have $R=\left \{(4,5), (1,4), (4,6), (7,6), (3,7)\right \}$.
$\therefore R^{-1}=\left \{(5,4), (4,1), (6,4), (6,7), (7,3)\right \}$
$(4,4)\in R^{-1}\circ R$ because $(4,5)\in R$ and $(5,4)\in R^{-1}$
$(1,1)\in R^{-1}\circ R$ because $(1,4)\in R$ and $(4,1)\in R^{-1}$
$(4,4)\in R^{-1}\circ R$ because $(4,6)\in R$ and $(6,4)\in R^{-1}$
$(4,7)\in R^{-1}\circ R$ because $(4,6)\in R$ and $(6,7)\in R^{-1}$
$(7,4)\in R^{-1}\circ R$ because $(7,6)\in R$ and $(6,4)\in R^{-1}$
$(7,7)\in R^{-1}\circ R$ because $(7,6)\in R$ and $(6,7)\in R^{-1}$
$(3,3)\in R^{-1}\circ R$ because $(3,7)\in R$ and $(7,3)\in R^{-1}$
$\therefore R^{-1}\circ R=\left \{(4,4), (1,1), (4,7), (7,4), (7,7), (3,3)\right \}$.
$\therefore$ The correct answer is $B$.

Given $(a - 2, b + 3) = (6, 8)$, are equal ordered pair. Find the value of $a$ and $b$.

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $a = 8$ and $b = 5$

  2. $a = 8$ and $b = 3$

  3. $a = 5$ and $b = 5$

  4. $a = 8$ and $b = 6$

Show answer
Correct Option: A
Explanation:

By equality of ordered pairs, we have
$(a - 2, b + 3) = (6, 8)$
On equating we get
$a - 2 = 6$
$a = 8$
$b + 3 = 8$
$b = 5$
So, the value of$ a = 8 $ and $ b = 5.$

What is the second component of an ordered pair $(3, -0.2)$?

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $3$

  2. $0.2$

  3. $1$

  4. $-0.2$

Show answer
Correct Option: D
Explanation:

In an ordered pair $(x,y)$, the first component is $x$ and the second component is $y$.

Therefore, in an ordered pair $(3,-0.2)$, the second component is $-0.2$.

What is the first component of an ordered pair $(1, -1)$?

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $1$

  2. $-1$

  3. $2$

  4. $0$

Show answer
Correct Option: A
Explanation:

In an ordered pair $(x,y)$, the first component is $x$ and the second component is $y$.

Therefore, in an ordered pair $(1,-1)$, the first component is $1$.

Ordered pairs $(x, y)$ and $(-1, -1)$ are equal if $y = -1$ and $x =$ _____

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $1$

  2. $-1$

  3. $0$

  4. $2$

Show answer
Correct Option: B
Explanation:

Given , $(x-y)=(-1,1)$

$x=-1, y=-1$
The value of $x=-1$.

Ordered pairs $(x, y)$ and $(3, 6)$ are equal if $x = 3$ and $y = ?$

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $3$

  2. $6$

  3. $-6$

  4. $-3$

Show answer
Correct Option: B
Explanation:

Given 

$(x,y)= (3,6)$
$x=3$
$y=6$
The value of $x=3$ and $y=6$.