Ordered Pair

Q. There are total $18$ balls in a bag. Out of them, $6$ are red in colour, $4$ are green in colour and $8$ are blue in colour. If Vishal picks three balls randomly from the bag, then what is the probability that all the three balls are not of the same colour?

1. $\frac{46}{51}$
2. $\frac{49}{51}$
3. $\frac{43}{51}$
4. $\frac{41}{51}$

Q. There are $15$ boys and $10$ girls in a class. If three students are selected at random, then what is the probability that $1$ girl and $2$ boys are selected?

1. $\frac{21}{46}$
2. $\frac{25}{46}$
3. $\frac{1}{2}$
4. $\frac{19}{46}$

Q. If $\{R=(x,y):x+2y=8\}$ is a relation on N, write the Range of R.

Q.

What is the relationship between a function and its inverse?

Q.

Which of the following is a true statement?

1. $\left\{a\right\}\in \left\{a,b,c\right\}$

2. $\left\{a\right\}\subseteq \{a,b,c\}$

3. $\varphi \in \{a,b,c\}$

4. None of these

Q. Let $A=\{y:y={\log }_{2}x,x<16,x,y\in N\},B=\{x:x^2-7x+12=0\}$ then $(A\cup B)\times (A\cap B)$ is

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

Q.

Write the following relations as sets of ordered pairs and find which of them are functions :
(i) $\left\{\right(x,y):y=3x,xϵ(1,2,3),yϵ(3,6,9,12\left)\right\}$
(ii) $\left\{\right(x,y):y>x+1,x=1,2\}$ and y = 2, 4, 6
(iii) $\left\{\right(x,y):x+y=3,x,yϵ\{0,1,2,3\left\}\right\}$

Q.

If the ordered pairs (x, -1) and (5, y) belong to the set (a, b) ; b = 2a - 3}, find the values of x and y.

Q. Let $A=\{x_1,x_2,x_3,x_4,x_5,x_6\}$ and $f:A\to A$. The number of bijective functions such that $f\left(x_i\right)\ne x_i$ for exactly $3$ elements $(i=1\text{to}6)$ is

Q.

If $aϵ\{-1,2,3,4,5\}$ and $bϵ\{0,3,6\},$ write the set of all ordered pairs (a, b) such that a + b = 5.

Q.

If $A$ is subset of $B$, then $BUA$ is equal to

1. $B\cap A$

2. $A$

3. $B$

4. None of these

Q. If two ordered pairs are related as $(4a-3,a+2b)=(3a,2a-b)$, then $a-b=$

Q. For two finite disjoint sets $A$ and $B$, if the number of elements in power set of $A$ is $224$ more than the number of elements in power set of $B$, then the number of elements present in either of the sets is

1. $12$
2. $0$
3. $15$
4. $13$

Q. If $R=\left\{\right(x,y):x,y\in W,x^2+y^2=25\}$,
then the number of elements in $R$ is

1. $3$
2. $2$
3. $5$
4. $4$

Q. Let A be the set of first ten natural numbers and let R be a relation in A defined by (x, y) $ϵ$ R is and only if x + 2y = 10. Which of the following is false?

1. R = {(2, 4), (4, 3), (6, 2), (8, 1)}
2. Range of R = {1, 2, 3, 4}
3. Domain of R = {2, 4, 6, 8}
4. At least one is false

Q. Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
$\left(i\right)\left\{\right(2,1),(5,1),(8,1),(11,1\left)\right(14,1),(17,1\left)\right\}$
$\left(ii\right)\left\{\right(2,1),(4,2),(6,3),(8,4\left)\right(10,5),(12,6),(14,7\left)\right\}$
$\left(iii\right)\left\{\right(1,3),(1,5),(2,5\left)\right\}$

Q. Define a function as a set of ordered pairs.

Q. Let $A=[a_{ij}{]}_{2\times 2},$ where $a_{ij}=(i^2-j^2),$ then, which of the following is correct

1. $A=\left[\begin{array}{cc}0& -3\\ 3& 0\end{array}\right]$
2. Trace of $A$ is a negative number
3. Trace of $A$ is a positive number
4. $A=\left[\begin{array}{cc}0& -1\\ 3& -3\end{array}\right]$

Q.

Transposing $-5$ from LHS to RHS of an equality means adding ________to RHS

Q.

If \(a \epsilon {2, 4, 6, 9} and \(b \epsilon \{4, 6, 18, 27\}, then form the set of all ordered pairs (a, b) such that a divides b and a < b.

Q. If $X=\{1,3,5,7\}$, then which of the following is a function from $X$ to itself?

1. $f_1=\left\{\right(x,y):y=x+2\}$
2. $f_2=\left\{\right(x,y):x+y<8\}$
3. $f_3=\left\{\right(x,y):x+y=8\}$
4. $f_4=\left\{\right(x,y):x<y\}$

Q. The number lock of a suitcase has $4$ wheels each labelled with ten digits i.e. from $0$ to $9.$ What is the probability of a person getting the right sequence to open the suitcase, if
$A:$ Repetition is not allowed.
$B:$ Repetition is allowed.

1. $P\left(A\right)=\frac{1}{5040}$
2. $P\left(A\right)=\frac{1}{1080}$
3. $P\left(B\right)=\frac{1}{{10}^5}$
4. $P\left(B\right)=\frac{1}{{10}^4}$

Q. A bag contains $4$ white, $3$ black and $2$ red balls. Balls are drawn from the bag one by one without replacement.
The probability that the $4^{\text{th}}$ ball is red is

1. $\frac{1}{16}$
2. $\frac{1}{9}$
3. $\frac{2}{9}$
4. $\frac{3}{11}$

Q.

If $A=\{x\in R:|x|<2\}$ and $B=\{x\in R:|x-2|\ge 3\}$ then :

1. $A-B=[-1,2]$
2. $B-A=R-(-2,5)$
3. $A\cup B=R-(2,5)$
4. $A\cap B=(-2,-1)$

Q. The number of ways one can choose a set of distinct positive integers, each smaller than or equal to $20,$ such that there sum is odd

1. $2^{18}$
2. $2^{19}$
3. $2^{20}$
4. $2^{21}$

Q. Let $A=\left\{\right(a,b,c):a,b,c\in N,a^{\left(bc\right)}=64\},$ then the total number of ordered triplets is

Q. Let $A=\{1,2,4\},B=\{2,4,5\},C=\{2,5\}$, then $(A-B)\times (B-C)$ is

1. $\left\{\right(1,2),(1,5),(2,5\left)\right\}$
2. $\left\{\right(1,4\left)\right\}$
3. $\left\{\right(4,1\left)\right\}$
4. $\varphi$

Q. Let $A=\{1,2,4\},B=\{2,4,5\},C=\{2,5\}$, then $(A-B)\times (B-C)$ is

1. $\left\{\right(4,1\left)\right\}$
2. $\left\{\right(1,2),(1,5),(2,5\left)\right\}$
3. $\varphi$
4. $\left\{\right(1,4\left)\right\}$

Q. If $A=\left\{\right(a,b,c):a,b,c\text{are prime numbers and}a+b+c=9\}$, then the number of ordered triplets is

Q. If $A=\{\alpha ,\beta ,\gamma \},B=\{1,2,3,4\}$, then the number of elements in set $A\times B\times B$ is

1. $48$
2. $36$
3. $10$
4. $2^{3\times 4\times 4}$