Mathematics Class XI

Question 1 of 25

1. Question
In a class 35 students, 24 like to play Cricket and 16 like to play Football. Also, each student likes to play a least one of two games. How many students like to play both games?
- 5
- 8
- 12
- 10
Correct

Incorrect

Hint

$\therefore n(x) = 24,n(y) = 16,n(x \cup y) = 25}$
$\begin{array}{l}\therefore n(x \cup y) = n(x) + n(y) - n(x \cap y)\\ \Rightarrow 35 = 24 + 16 - (x \cap y)\\ \Rightarrow n(x \cap y) = 5\end{array}$
Question 2 of 25

2. Question
If $z$ denotes the set of all integers and $A = \{ (a,b):{a^2} + 3{b^2} = 28,a,b \in z\} $ and $B = \{ (a,b):a > b,a,b \in z\} $, then the number of elements in $A \cap B$ is
- 2
- 4
- 5
- 6
Correct

Incorrect

Hint

$\begin{array}{c}A = \{ (a,b):{a^2} + 3{b^2} = 28,a,b \in z\} \\ = \{ ( \pm 5, \pm 1),( \pm 1, \pm 3),( \pm 4, \pm 2)\} \end{array}$
And $B = \{ (a,b):a > b,a,b \in z\}$
$\therefore A \cap B = \{ (5,1),(5, - 1),( - 1, - 3),(4,2),(4, - 2),(1, - 3)\}$
$n(A \cap B) = 6$
Question 3 of 25

3. Question
Let Universal set, $U = \{ x:{x^5} – 6{x^4} + 11{x^3} – 6{x^2} = 0\} ,$ $A = \{ x:{x^2} – 5x + 6 = 0\} \& B = \{ x:{x^2} – 3x + 2 = 0\} $ then ${(A \cap B)^ \subset }$ is
- $\{ 1,3\} $
- $\{ 1,2,3\} $
- $\{ 0,1,3\} $
- $\{ 0,1\} $
Correct

Incorrect

Hint

$U = \{ x:{x^5} - 5{x^4} + 11{x^3} - 6{x^2} = 0\} = \{ 0,1,2,3\}$
$A = \{ x:{x^2} - 5x + 6 = 0\} = \{ 2,3\}$
And $B = \{ x:{x^2} - 3x + 2\} = \{ 1,2\}$
$\therefore {(A \cap B)^ \subset } = U - (A \cap B) = \{ 0,1,2,3\} - \{ 2\} = \{ 0,1,3\}$ .
Question 4 of 25

4. Question
$A = \{ x:{x^2} – 1 = 0,x \in z\} $ and $B = \{ x:2{x^2} – 5x + 3 = 0,x \in z\} ,$ then $A \cap B$ is equal to
- 1
- 3
- 4
- 2
Correct

Incorrect

Hint

$\begin{array}{c}A = \{ x:{x^2} - 1 = 0,x \in z\} \\ = \{ - 1,1\} \end{array}$
$\begin{array}{c}B = \{ x:2{x^2} - 5x + 3 = 0,x \in z\} \\ = \{ 1,\frac{3}{2}\} \end{array}$
$\therefore A \cap B = \{ 1\}$ .
Question 5 of 25

5. Question
If $A = \{ x:{x^3} – 1 = 0,x \in z\} $ and $B = \{ x:{x^2} – 2x + 1 = 0,x \in z\} ,$ then what is the relation between $A\& B$?
- $A \subset B$
- $B \subset A$
- $A = B$
- All of the above
Correct

Incorrect

Hint

\[A = \{ x:{x^3} – 1 = 0,x \in z\} \]
$B = \{ x:{x^2} - 2x + 1 = 0,x \in z\}$
$\therefore A = \{ 1\} ,B \subset A$
$\therefore A = B$
$\therefore$ All the above conditions are satisfied for relation between $A$ and $B$ .
Question 6 of 25

6. Question
Given $n(A – B) = 3,n(B – A) = 4$ and $n(A \cap B) = 2,$ then find $n(A \cup B)$?
- 7
- 9
- 24
- 31
Correct

Incorrect

Hint

$\begin{array}{c}n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B)\\ = 3 + 4 + 2\\ = 9\end{array}$
Question 7 of 25

7. Question
If $A$ and $B$ are disjoint sets, $n(A) = 3,n(B) = 5,$ then $n(A \cap B)$ is equal to
- 8
- 15
- 0
- 2
Correct

Incorrect

Hint

Since, $A$ and $B$ are disjoint sets, they have no common elements. Hence, $(A \cap B) = \{ \}$ .
$\therefore n(A \cap B) = 0$ .
Question 8 of 25

8. Question
If $n(A) = 3,n(B) = 5$ and $n(A \cup B) = 8$ then $n[P(A \cap B)]$ is equal to
- 4
- 1
- 0
- 7
Correct

Incorrect

Hint

$n(A) = 3,n(B) = 5,n(A \cup B) = 8$
$\Rightarrow n(A) + n(B) = n(A \cup B)$
Hence, $A$ and $B$ are disjoint
$\therefore A \cap B = \phi$
$\therefore P(A \cap B) = \{ \{ \} \}$ or $\{ \phi \}$
$\therefore n[P(A \cap B)] = 1$ .
Question 9 of 25

9. Question
If $A = \{ x:{x^2} – x – 2 = 9,x \in z\} $ and $B = \{ x:{x^3} – 1 = 0,x \in z\} $, then
- $A \subset B$
- $B \subset A$
- $A = B$
- $A \cap B = \phi $
Correct

Incorrect

Hint

$A = \{ x:{x^2} - x - 2 = 0,x \in z\}$
$B = \{ x:{x^3} - 1 = 0,x \in z\}$
$\begin{array}{c}\therefore A = {x^2} - x - 2 = 0\\ \Rightarrow {x^2} - 2x + x - 2 = 0\\ \Rightarrow (x - 2)(x + 1) = 0\end{array}$
$\Rightarrow x = 2$ or $x = - 1$
$\therefore A = \{ - 1,2\}$
$\begin{array}{c}\therefore B = {x^3} - 1 = 0\\ \Rightarrow {x^3} = 1\\ \Rightarrow x = 1\\\therefore B = \{ 1\} \end{array}$
$\therefore A \cap B = \phi$ .
Question 10 of 25

10. Question
If $A = \{ 1,2,3,4\} ,B = \{ 2,3,4\} ,$ and $C = \{ \{ \} ,\{ 2\} ,\{ 3\} ,\{ 4\} ,\{ 2,3\} ,\{ 2,4\} ,\{ 3,4\} ,\{ 2,3,4\} \} $. What is the relation between $A,B$ and $C$?
- $A \subset B$ and $B \subset C$
- $B \subset A$ and $C = P(B)$
- $A \subset B$ and $B \in C$
- None of these
Correct

Incorrect

Hint

$A = \{ 1,2,3,4\} ,B\{ 2,3,4\}$
$\therefore B \subset A$
$C = P(B) = \{ \phi ,\{ 2\} ,\{ 3\} ,\{ 4\} ,\{ 2,3\} ,\{ 2,4\} ,\{ 3,4\} ,\{ 2,3,4\} \}$ .
Question 11 of 25

11. Question
If $A = \{ 1,2,3,4\} $ and $B = \{ \{ 1\} ,\{ 2\} ,\{ 3\} ,\{ 4\} \} $, then $A \cap B$ is equal to
- $A$
- $B$
- $\phi $
- None of these
Correct

Incorrect

Hint

$A = \{ 1,2,3,4\} ,B = \{ \{ 1\} ,\{ 2\} ,\{ 3\} ,\{ 4\} \}$
No elements of $A$ contains in $B$ or no elements of $B$ contains in $A$ .
$\therefore A \cap B = \phi$ .
Question 12 of 25

12. Question
If \[A = \phi \] and $B = \{ \phi \} $, Determine the relation between $A$ and $B$?
- $A \subset B$
- $A = B$
- $A \in B$
- None of these
Correct

Incorrect

Hint

$A = \phi ,B = \{ \phi \}$
$\phi$ is an element of $B$ .
$\therefore A \in B$ .
Question 13 of 25

13. Question
$A = \{ a,b\} ,B = \{ a,b,c\} ,C = \{ c,a,b\} $. Find the relation between $A.B$ and $C$?
- $A = C,A = B$
- $A \subset B \subseteq C$
- $A = B \subset C$
- None of these
Correct

Incorrect

Hint

$A = \{ a,b\} ,B = \{ a,b,c\} ,C = \{ c,a,b\}$
$A \subset B,B = C$
$\Rightarrow B \subseteq C$
$\therefore A \subset B \subseteq C$ .
Question 14 of 25

14. Question
If $A,B$ and $C$ are disjoint sets, then $n(A \cup B \cup C)$ is equal to
- $n(A) + n(B) + n(C)$
- $n(A) + n(B) + n(C) – n(A \cap B \cap C)$
- $n(A) + n(B) – n(C) – n(B \cap C)$
- None of these
Correct

Incorrect

Hint

$\begin{array}{c}n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C)\\ - n(B \cap C) + n(A \cap B \cap C)\end{array}$
Since, $A,B$ and $C$ are disjoints
$A \cap B = A \cap C = B \cap C = A \cap B \cap C = \phi$
$\therefore n(A \cup B \cup C) = n(A) + n(B) + n(C)$ .
Question 15 of 25

15. Question
In a class of 75 students, if 30 eat sweets, 35 eats chocolates and 15 eat both the items, then the number of students who don’t eat these items is
- 25
- 35
- 20
- 30
Correct

Incorrect

Hint

Let $x$ be total number of students.
Let $A$ number of students eat sweets & $B$ numbers of students eat chocolate.
$n(A) = 30,n(B) = 35,n(A \cap B) = 15$
$\begin{array}{c}n(A \cup B) = n(A) + n(B) - n(A \cap B)\\ = 30 + 35 - 15\\ = 50\end{array}$
So, number of students who do not eat both items is
$\begin{array}{c}n{(A \cup B)^ \subset } = n(x) - n(A \cup B)\\ = 75 - 50\\ = 25\end{array}$
Question 16 of 25

16. Question
In a survey of 400 students in a school, 200 were taking apple juice, 175 were taking orange juice and 125 were taking both the juice. How many students were taking neither apple juice nor orange juice?
- 70
- 140
- 150
- 210
Correct

Incorrect

Hint

$n(x) = 400,n(A) = 200,n(B) = 175$
$n(A \cap B) = 125$
$\begin{array}{c}n(A \cup B) = n(A) + n(B) - n(A \cap B)\\ = 200 + 175 - 125\\ = 250\end{array}$
So, number of students were taking neither juice
$\begin{array}{c}n{(A \cup B)^ \subset } = n(x) - n(A \cup B)\\ = 400 - 250\\ = 150\end{array}$
Question 17 of 25

17. Question
In a survey 21 people like product $A$, 26 like product $B$ and 29 liked product $C$, If 14 people like $A$ and $B$ , 12 like $B$ and $C$, 10 like $C$ and $A$ and 8 like all the three product. Find the total number of people?
- 38
- 48
- 50
- 52
Correct

Incorrect

Hint

$\begin{array}{l}n(A) = 21,n(B) = 26,n(C) = 29,n(A \cap B) = 14,n(B \cap C) = 12,\\n(C \cap A) = 10,n(A \cap B \cap C) = 8\end{array}$
$\begin{array}{c}n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C)\\ - n(C \cap A) + n(A \cap B \cap C)\\ = 21 + 26 + 29 - 14 - 12 - 10 + 8\\ = 48\end{array}$
$\therefore A \cup B \cup C = 48$ .
Question 18 of 25

18. Question
In a group, 100 can speak ODIA, 80 can speak HINDI, 75 can speak ENGLISH, 40 can speak ODIA & HINDI, 50 can speak ODIA & ENGLISH, 45 can speak HINDI & ENGLISH. If the total number of people in this group is 150. Then, find how many people can speak three languages?
- 20
- 35
- 30
- 15
Correct

Incorrect

Hint

$n(O) = 100,n(H) = 80,n(E) = 75$
$n(O \cap H) = 40,n(O \cap E) = 50,n(H \cap E) = 45$
$\begin{array}{c}n(O \cup H \cup E) = n(O) + n(H) + n(E) - n(O \cap H) - n(O \cap E) - n(H \cap E)\\ + n(O \cap H \cap E)\end{array}$
$\Rightarrow 150 = 100 + 80 + 75 - 40 - 50 - 45 + n(O \cap H \cap E)$
$\begin{array}{c} \Rightarrow n(O \cap H \cap E) = 150 + 135 - 255\\ \Rightarrow n(O \cap H \cap E) = 30\end{array}$
Question 19 of 25

19. Question
For any sets $A$ and $B$, $(A \cap B) – A$ is equal to
- $B$
- $A$
- $\phi $
- $A – B$
Correct

Incorrect

Hint

$\begin{array}{c}{(A \cup B)^ \subset } - {B^ \subset } = ({A^ \subset } \cap {B^ \subset }) - {B^ \subset }\\ = ({A^ \subset } \cap {B^ \subset }) \cap {(B)^ \subset }\\ = ({A^ \subset } \cap {B^ \subset }) \cap B\end{array}$ (De-Morgan’s law)
$\begin{array}{c}{(A \cup B)^ \subset } - {B^ \subset } = {A^ \subset } \cap ({B^ \subset } \cap B)\\ = {A^ \subset } \cap \phi \\ = \phi \end{array}$ (Associative law)
Question 20 of 25

20. Question
${(A \cup B)^ \subset } – {B^ \subset }$ is equal to _______, where $A$ and $B$ are any sets.
- $\phi $
- ${A^ \subset }$
- $A – B$
- None of these
Correct

Incorrect

Hint

$\begin{array}{c}{(A \cup B)^ \subset } - {B^ \subset } = ({A^ \subset } \cap {B^ \subset }) - {B^ \subset }\\ = ({A^ \subset } \cap {B^ \subset }) \cap {(B)^ \subset }\\ = ({A^ \subset } \cap {B^ \subset }) \cap B\end{array}$ (De-Morgan’s law)
$\begin{array}{c}{(A \cup B)^ \subset } - {B^ \subset } = {A^ \subset } \cap ({B^ \subset } \cap B)\\ = {A^ \subset } \cap \phi \\ = \phi \end{array}$ (Associative law)
Question 21 of 25

21. Question
Let $A = \{ {x^y},{y^x}\} $, the $P(A)$ is equal to
- $\{ \{ {x^y}\} ,\{ {y^x}\} \} $
- $\{ \phi ,\{ {x^y}\} ,\{ {y^x}\} ,\{ {x^y},{y^x}\} \} $
- $\{ \phi ,{x^y},{y^x},{x^y}{y^x}\} $
- None of these
Correct

Incorrect

Hint

$A = \{ {x^y},{x^y}\}$
$P(A) = \{ \phi ,\{ {x^y}\} ,\{ {y^x}\} ,\{ {x^y},{y^x}\} \}$ .
Question 22 of 25

22. Question
$A = \{ x;x$ is an integer which is both even and odd$\} $, $B = \{ x:x$ is an integer and $x \ne x\} $, then $A \cup B$ is equal to
- $\phi $
- $A$
- $B$
- None of these
Correct

Incorrect

Hint

$A = \{ x:x$ is an integer which is both even and odd $\}$
$A = \phi$ (Because any integer can’t be both even and odd)
$B = \{ x:x$ is an integer and $x \ne x\}$
$B = \phi ( - 5 \ne - 5)$
$\therefore A \cup B = \phi \cup \phi = \phi$ .
Question 23 of 25

23. Question
If $A$ and $B$ are sets, and $A = \{ 1,2,3,4\} ,B = \{ x:x$ is a divisor of $60,x \in N\} $, $A$ and $B$is related to
- $A \subset B$
- $B \subset A$
- $A = B$
- None of these
Correct

Incorrect

Hint

$A = \{ 1,2,3,4\}$
$B = \{ x:x$ is a divisor of $60,x \in N\}$
$B = \{ 1,2,3,4,5,6,10,12,15,20,30,60\}$
$\therefore A \subset B$ .
Question 24 of 25

24. Question
If $A = \phi $A, then $n[P(P(A))]$ is equal to
- 1
- 0
- 2
- 4
Correct

Incorrect

Hint

$\begin{array}{c}A = \phi \\P(A) = P(\phi )\\ = \{ \phi \} \end{array}$
$\begin{array}{c}P(P(A)) = P(\{ \phi \} )\\ = \{ \phi ,\{ \phi \} \} \end{array}$
$\therefore n[P(P(A))] = 2$ .
Question 25 of 25

25. Question
If $A$ and $B$ are any sets, $n(A) = 3,n(B) = 4,n(A \cap B)2,$ then $n[P(A \cup B)]$ is equal to
- 12
- 64
- 8
- 32
Correct

Incorrect

Hint

$\n(A) = 3,n(B) = 4,n(A \cap B) = 2$ .
$\begin{array}{c}n(A \cup B) = n(A) + n(B) - n(A \cap B)\\ = 3 + 4 - 2\\ = 5\end{array}$
$\therefore A \cup B$ have 5 elements.
$\begin{array}{c}n[P(A \cup B)] = {2^5}\\ = 32\end{array}$
$\therefore n[P(A \cup B)] = 32$ .

Mathematics Class XI

Practice Questions 2 – Sets – Class XI

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