Properties of Set Operation

Q. The sum of all the elements in the set $\{n\in \{1,2,...,100\}|\text{H.C.F. of}n\text{and 2040 is 1}\}$ is equal to

Q. In a class of $140$ students numbered $1$ to $140$, all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is :

1. $1$
2. $102$
3. $38$
4. $42$

Q.

If $A$ and $B$ are two sets, then $A\cup B=A\cap B$ if

1. $A\subseteq B$

2. $B\subseteq A$

3. $A=B$

4. None of these

Q.

How to calculate percentage of marks of $10\mathrm{th}$ class state board?

Q. In a class of $120$ students numbered from $1$ to $120$, all even numbered students opt for Physics, those whose numbers are divisible by $5$, opt for Chemistry and those whose numbers are divisible by $7$, opt for Maths. How many opt for none of the three subjects?

1. $19$
2. $41$
3. $21$
4. $26$

Q. Let $A,B$ and $C$ be sets such that $\varphi \ne A\cap B\subseteq C$. Then which of the following statements is not true ?

1. $B\cap C\ne \varphi$
2. $(C\cup A)\cap (C\cup B)=C$
3. If $(A-B)\subseteq C,$ then $A\subseteq C$
4. If $(A-C)\subseteq B,$ then $A\subseteq B$

Q.

If $\alpha$and $\beta$are the roots of the equation $a{x}^2+bx+c=0$then find $\alpha /[ɑ\beta +b]+\beta /[aɑ+b]$

1. $2/a$

2. $2/b$

3. $2/c$

4. $-2/a$

Q. If $n\left(U\right)=60,n\left(A\right)=21,n\left(B\right)=43,$ then minimum and maximum value of $n(A\cup B)$ is

1. $43,60$
2. $50,64$
3. $44,60$
4. $38,60$

Q. In a class of $40$ students, $12$ enrolled for both English and German. $22$ enrolled for German. If the students of the class enrolled for at least one of the two subjects, then the number of students who enrolled for English but not German is

Q. Let $X=\{n\in N:1\le n\le 50\}$. If $A=\{n\in X:n\text{is a multiple of 2}\}$ and $B=\{n\in X:n\text{is a multiple of 7}\}$, then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is

Q.

Prove$\sin 3A=3\sin A-4{\sin }^{3}A$

Q.

In a class of $55$ students, the number of students studying different subjects is $23$ in Mathematics, $24$ in physics, $19$ in chemistry, $12$ in Mathematics and Physics, $9$ in Mathematics and chemistry, $7$ in physics and chemistry and $4$ in all the three subjects.

The number of students who have taken exactly one the subjects is

1. $6$

2. $9$

3. $7$

4. None of these

Q. A group of $123$ workers went to a canteen for cold drinks, ice-cream and tea. $42$ workers took ice-cream, $36$ took tea and $30$ took cold drinks. $15$ workers purchased ice-cream and tea, $10$ purchased ice-cream and cold drinks, and $4$ purchased cold drinks and tea but not ice-cream. Then how many workers did not purchase anything ?

1. $44$
2. $46$
3. $48$
4. $42$

Q.

If $N_a=\left[an:n\in N\right]$, then $N_5\cap N_7$ is equal to

1. $N_7$

2. $N$

3. $N_{35}$

4. $N_5$

Q. Let $A,B$ and $C$ be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$. Show that $B=C.$

Q.

If $\alpha$ and $\beta$are the roots of $x^2-2x+2=0$ then the minimum value of $n$ such that ${(\alpha /\beta )}^n=1$

1. $4$

2. $3$

3. $2$

4. $5$

Q. In a survey of $200$ students of a higher secondary school, it was found that $120$ studied mathematics; $90$ studied physics and $70$ studied chemistry; $40$ studied mathematics and physics; $30$ studied physics and chemistry; $50$ studied chemistry and mathematics, and $20$ studied none of these subjects. Then the number of students who studied all the three subjects, is

1. $20$
2. $12$
3. $15$
4. $30$

Q.

In a group of 500 persons, 300 take tea, 150 take coffee, 250 take cold drinks, 90 take tea and coffee, 110 take tea and cold drinks, 80 take coffee and cold drink, 30 none of them. Find the no. of persons who take all the three drinks.

Q. In a class of $345$ students, the students who took English, Math and Physics are equal in number. $30$ students took English and Math, $26$ choose Math and Physics, $28$ choose Physics and English and $14$ choose all three subjects. $43$ students didn’t take any of the subjects. How many students have taken English as subject?

1. $108$
2. $124$
3. $246$
4. $286$

Q.

If $A=\{x:x^2–5x+6=0\},B=\{2,4\},C=\{4,5\}$, then $A\times (B\cap C)$ is

Q. Let $A,B,C$ be finite sets. Suppose that $n\left(A\right)=10,n\left(B\right)=15,n\left(C\right)=20,n(A\cap B)=8$ and $n(B\cap C)=9$. Then the maximum possible value of $n(A\cup B\cup C)$ is

Q. Among a group of students, $50$ played cricket, $50$ played hockey and $40$ played volley ball. $15$ played both cricket and hockey, $20$ played both hockey and volley ball, $15$ played cricket and volley ball and $10$ played all three. If every student played at least one game, then

1. The sum of the number of students who played only cricket, only hockey and only volley ball is $70$
2. The sum of the number of students who played only cricket, only hockey and only volley ball is $75$
3. The number of students who played only cricket is $30$
4. The number of students who played only volley ball is $15$

Q.

Let $f\left(x\right)=x^2$ and $g\left(x\right)=2^x$. Then the solution set of $\left(fog\right(x\left)\right)=\left(gof\right(x\left)\right)$ is:

1. $R$

2. $\left\{0\right\}$

3. $\left\{0,2\right\}$

4. None of these

Q.

Let $z_1,z_2$ be the roots of the equations $z^2+az+12=0and{z}_1,z_2$ form an equilateral triangle with origin. Then, the value of $\left|a\right|$ is

Q.

If $x=1+a+a^2+....\infty \mathrm{and}y=1+b+b^2+....\infty$ where a and b are proper fractions, then $1+ab+a^2{b}^2+...\infty =?$

1. $\frac{xy}{x+y-1}$

2. $\frac{xy}{x-y}$

3. $\frac{\left(x^2+y^2\right)}{x-y}$

4. none of the above

Q.

If$\alpha$ and$\beta$ are the roots of the equation$a{x}^2+bx+c=0$, $\alpha \beta =3$ and$a,bandc$ are in AP, then$\alpha +\beta$ is equal to

1. $-4$

2. $1$

3. $4$

4. $-2$

Q.

The set $S=\{1,2,3,..........,12\}$ is to be partitioned into three sets $A,B,C$ of equal size. Thus, $A\cup B\cup C=S$, $A\cap B=B\cap C=A\cap C=\varnothing$. The number of ways to partition $S$ is

1. $\frac{12!}{3!{\left(4!\right)}^3}$

2. $\frac{12!}{3!{\left(3!\right)}^3}$

3. $\frac{12!}{{\left(4!\right)}^3}$

4. $\frac{12!}{{\left(3!\right)}^3}$

Q.

A survey shows that $76\%$ of the Indians like oranges, whereas $62\%$ like bananas. What percentage of the Insians like botth oranges and bananas ?

Q.

If A = {1, 3, 5, 7, 9, 11, 13, 15, 17}, B ={2, 4, ......, 18} and N, the set of natural numbers is the universal set, then $(A^{\prime }\cup [(A\cup B)\cap B^{\prime }\left]\right)$ is

1. N

2. A

3. $\varphi$

4. B

Q. If $n\left(U\right)=48,n\left(A\right)=28,n\left(B\right)=33$ and $n(B–A)=12$, then $n(A\cap B{)}^C=$