Intersection

Q.

Prove that : $A\subseteq B,B\subseteq CandC\subseteq A\Rightarrow A=C$.

Q.

A survey shows that $63\%$ of the Americans like cheese whereas $76\%$ like apples. If $x\%$ of the Americans like both cheese and apples, then

1. $x=39$

2. $x=63$

3. $39\le x\le 63$

4. none of these

Q. If $A=\{x\in R:|x-2|>1\},B=\{x\in R:\sqrt{x^2-3}>1\},$ $C=\{x\in R:|x-4|\ge 2\}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C{)}^c\cap Z$ is

Q.

Let A = {1, 2, 4, 5}, B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities :

$\left(i\right)A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

(ii) $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

(iii) $A\cap (B-C)=(A\cap B)-(A\cap C)$

(iv) $A-(B\cup C)=(A-B)\cap (A-C)$

(v) $A-(B\cap C)=(A-B)\cup (A-C)$

(vi) $A\cap \left(B\Delta C\right)=(A\cap B)\Delta (A\cap C)$

Q.

If sets $A$ and $B$ are defined as $A=\left\{(x,y):y=\frac{1}{x},0\equiv x\in R\right\}$ and $B=\left\{\right(x,y):y=-x,x\in R\}$, then

1. $A\cap B=A$

2. $A\cap B=B$

3. $A\cap B=\varnothing$

4. None of these

Q.

For any two sets A and B , prove that :

$A\cap B=\varphi \Rightarrow A\subseteq B^{\prime }$.

Q.

200 logs are stacked in the following manner:

20 logs in the bottom row, 19 in the next row, 18 in the row next to it, and so on.

In how many rows are the 200 logs placed and how many logs are in the top row?

Q.

There are $100$ students in a class. In the examination, $50$ of them failed in Mathematics, $45$ failed in Physics, $40$ failed in Biology and $32$ failed in exactly two of the three subjects. Only one student passed in all the subjects. Then, the number of students failing in all three subjects

1. $12$

2. $4$

3. $2$

4. Cannot be determined

Q.

In a meeting, 60% of the members favour and 40% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var(X).

1. 0.4 and 0.24

2. 0.6 and 0.44

3. 0.6 and 0.24

4. 0.6 and 0

Q.

$\text{If A and B are two events such that P(A) = 0.4, P(A + B) = 0.7 and P(AB) = 0.2, then P(B) =?}$

1. $0.1$

2. $0.3$

3. $0.5$

4. $\text{None of these}$

Q.

The proposition $p\Rightarrow ~(p\wedge ~q)$ is

1. A contradiction

2. A tautology

3. Either $A$ or $B$

4. Neither $A$ nor $B$

Q.

Let A = {$x:xϵN$}, B = {$x:x=2n,nϵN$}, C = {$x:x=2n-1,nϵN$} and , D = {x: x is a prime natural number}. Find :

(i) $A\cap B$ (ii) $A\cap C$

(iii) $A\cap D$ (iv) $B\cap C$

(v) $B\cap D$ (vi) $C\cap D$

Q.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find :

(i) $A^{\prime }$ (ii) $B^{\prime }$ (iii) $(A\cap C{)}^{\prime }$

(iv) $(A\cup B{)}^{\prime }$ (v) ${\left(A^{\prime }\right)}^{\prime }$ (vi) $(B-C{)}^{\prime }$

Q.

Evaluate $\tan \left(\frac{\pi }{4}+\theta \right)-\tan \left(\frac{\pi }{4}-\theta \right)$

1. $2\tan \left(2\theta \right)$

2. $2\cot \left(\theta \right)$

3. $\tan \left(2\theta \right)$

4. $\cot \left(2\theta \right)$

Q.

If $\frac{(1-{\tan }^2\mathrm{\theta })}{{\sec }^2\mathrm{\theta }}=\frac{1}{2},$ then the general value of $\theta$ is

1. $\mathrm{n\pi }\pm \frac{\mathrm{\pi }}{6}$

2. $n\pi +\frac{\pi }{6}$

3. $2\mathrm{n\pi }\pm \frac{\mathrm{\pi }}{6}$

4. None of these

Q. Let $Z$ be the set of integers. If $A=\{x\in Z:{\log }_{10}(x^2-7x+13)=0\}$ and $B=\{x\in Z:(x+2\left)\right(x-3\left)\right(x^2-1)\le 0\}$ then the number of subsets of set $A\times B$ is

1. $2^{10}-1$
2. $2^8$
3. $2^{10}$
4. $2^8+1$

Q.

Ram secures $100$ marks in maths, then he will get a mobile. The converse is

1. If Ram gets a mobile, then he will not secures $100$ marks

2. If Ram does not get a mobile, then he will secure $100$ marks

3. If Ram will get a mobile, then he secures $100$ marks in maths

4. None of these

Q.

Let $A=\left\{a,b,c\right\}$, $B=\left\{b,c,d\right\}$, $C=\left\{a,b,d,e\right\}$, then $A\cap \left(B\cup C\right)$ is

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

2. $\left\{b,c,d\right\}$

3. $\left\{a,b,d,e\right\}$

4. $\left\{e\right\}$

Q.

(i) If A = {1, 2, 3, 4, 5},

B= {4, 5, 6, 7, 8},

C= {7, 8, 9, 10, 11} and

D= {10, 11, 12, 13, 14}. Find :

(i) $A\cup B$ (ii) $A\cup C$

(iii) $B\cup C$ (iv) $B\cup D$

(v) $A\cup B\cup C$ (vi) $A\cup B\cup D$

(vii) $B\cup C\cup D$ (viii) $A\cap (B\cup C)$

(ix) $(A\cap B)\cap (B\cap C)$

(x) $(A\cup D)\cap (B\cup C)$

Q. Let $A=\{x:x\text{is a natural number and a factor of}18\}$ and
$B=\{x:x\text{is a natural number and less than 6}\}$.
If $A\cap B=\{a,b,c\}$, then the value of $a+b+c$ is

Q.

All possible two factors products are fromed from numbers $1,2,3,4.....200$.The number of factors out of the total obtained which are multiplies of $5$ is ?

1. $5040$

2. $7180$

3. $8150$

4. $Noneofthese$

Q.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},

A = {2, 4, 6, 8} and B = {2, 3, 5, 7}.

Verify that :

(i) $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$

(ii) $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$.

Q.

The inverse of the point $(1,2)$ with respect to the circle $x^2+y^2-4x-6y+9=0$, is

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

2. $(2,1)$

3. $(0,1)$

4. $(1,0)$

Q.

Write the set in roster form

The set of all integers 'x' such that | x minus 3 | <8

Q.

show that the relation in the set A={x€Z:0<x<12} given by R={(a, b):|a-b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1.

Please explain the answer in detail

Q.

The area in the positive quadrant is enclosed by the circle $x^2+y^2=4$, the line $x=y\sqrt{3}$ and the $x$-axis is

1. $\frac{\mathrm{\pi }}{2}$

2. $\frac{\mathrm{\pi }}{4}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\mathrm{\pi }$

Q. Let $V=\{a,e,i,o,u\}$ and $B=\{a,i,k,u\}$.
Find $V-B$ and $B-V$.

Q.

The most general values of $\theta$ satisfying $\tan \theta +\tan \left(\frac{3\mathrm{\pi }}{4}+\theta \right)=2$ are given by

1. $\theta =2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{3},n\in \mathrm{\mathbb{Z}}$

2. $\theta =n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{3},n\in \mathrm{\mathbb{Z}}$

3. $\theta =2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6},n\in \mathrm{\mathbb{Z}}$

4. $\theta =n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6},n\in \mathbb{Z}$

Q.

If $A=\{8,9,10\}$ and $B=\{1,2,3,4,5\}$, then the number of elements in $A\times A\times B$are

1. $15$

2. $30$

3. $45$

4. $75$

Q. Explain Cartesian product.