Empty Set

Q. $A,B,C,D,E,F,G$ and $H$ are sitting around a circle facing at the centre. $E$ is second to the left of $F$ and third to the right of $A.$ $B$ is third to the right of $G$ who is not an immediate neighbour of either $E$ or $F.$ $C$ is second to the right of $B.$ $D$ is to the immediate left of $A$ and third to the left of $H.$ Who is the fifth to the right of $C$?

1. $H$
2. $G$
3. $B$
4. $E$

Q.

Simplify $\frac{64}{125}$ raised to the power $\frac{-2}{3}$.

Q. If $A$ and $B$ are disjoint non-empty sets, then $A–(A–B)$ is

1. $B$
2. $\varphi$
3. $A$
4. $A\cup B$

Q.

Is the sum of two prime numbers is always even?

Q.

Let $\mathrm{R}$ be an equivalence relation on a finite set $\mathrm{A}$ having $\mathrm{n}$ elements. Then the number of ordered pairs in $\mathrm{R}$ is

1. $>n$

2. $\le n$

3. $\ge n$

4. None of the above

Q.

Which of the follwing is correct?

1. $A$ intersection $\phi =A$

2. $A$ intersection $\phi =\varphi$

3. $A$ intersection $\phi =U$

4. $A$ intersection $\phi =A$

Q.

How many lines can pass through:

Two given points.

Q. If $\varphi$ denotes the empty set, then which of the folowing is correct

1. $\varphi \in \varphi$
2. $\varphi \in \left\{\varphi \right\}$
3. $\left\{\varphi \right\}\in \left\{\varphi \right\}$
4. $0\in \varphi$

Q.

Express the following numbers as sum of two odd prime numbers.

$48=\_\_\_\_+\_\_\_\_$

Q. Which of the following is a null set

1. $\{x:x$ is a real number and $x^2+1=0\}$
2. $\{x:x$ is a real number and $x^2-9=0\}$
3. $\{x:x$ is a real number and $x^2=5x+6\}$
4. $\{x:x$ is a real number and $x^2-1=0\}$

Q. There are 15 points in a plane , no three of which are in the same straight line with exception of 6, which are in the same straight line. Find the number of .a) straight lines formed by) number of triangles formed by joining these points.

Q.

The values of ‘a’ for which $(a^2-1)x^2+2(a-1)x+2$ is positive for any x are

1. 

2. 

3. >

4. < >

Q.

Find sets A, B and C such that $A\cap B$, $A\cap C$ and $B\cap C$ are non-empty sets and $A\cap B\cap C=\varphi$

Q.

If square of a number ends with $5$, then its cube ends with $25$. State whether statement is True or False.

1. True
2. False

Q.

How many points does a line contain

Q.

Define null set and give example.

Q.

If $p$ and $q$ are simple propositions, then $p\Rightarrow q$ is false when.

1. $p$ is true and $q$ is true

2. $p$is false and $q$ is true

3. $p$ is true and $q$ is false

4. Both $p$ and $q$ are false

Q.

Every group of order $7$ is

1. Not abelian

2. Not cyclic

3. Cyclic

4. None of these

Q.

State whether the following statement is true or false:

$\{a,b\}=\{a,a,b,b,a\}$

1. True
2. False

Q.

Which of the following are examples of empty set ?

(i) Set of all even numbers divisible by 5.

(ii) Set of all even prime numbers.

(iii) {$x:x^2=0$ and x is rational}.

(iv) {x : x is a natural number, x < 8 and simultaneously x > 12}.

(v) {x : x is a point common to any two parallel lines}.

Q. The contrapositive statement of the statement "If $x$ is prime number, then $x$ is odd" is

1. If $x$ is not a prime number, then $x$ is not odd
2. If $x$ is a prime number, then $x$ is not odd
3. If $x$ is not odd, then $x$ is not a prime number
4. If $x$ is not a prime number, then $x$ is odd

Q.

If A represents the set of natural numbers less than 5, then find the number of elements in A.

Also, write the number of subsets.

Q.

$a^m\times a^n$ is equal to

1. ${\left(a^2\right)}^{mn}$

2. $a^{m-n}$

3. $a^{m+n}$

4. $a^{mn}$

Q. What is the cardinality of power set of empty set?

1. $2$
2. $1$
3. $0$
4. Infinity

Q.

If A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then insert the appropriate symbol $ϵor\text{/}ϵ$ in each of the following blank spaces :

(i) 4 ..... A

(ii) -4 ..... A

(iii) 12 ....... A

(iv) 9 ....... A

(v) 0 ....... A

(vi) -2 ....... A

Q. Let $X=$ {Ram, Geeta, Akbar} be the set of students of class $XI$, who are in school hockey team.
Let $Y=$ {Geeta, David, Ashok} be the set of students of class $XI$, who are in school football team.
Find $X\cap Y$.

Q.

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {$(L_1,L_2):L_1$ is parallel to $L_2$}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Q. Find which of the following sets are singleton sets:

The set of points of intersection of two non-parallel straight lines on the same plane.

Q.

The least value of the sum of any positive real number and its reciprocal is

1. $1$

2. $2$

3. $3$

4. $4$

Q. If square root of 8 - 15i is $\pm$ (a - ib). The a + b is _____

1. Irrational Number

2. Rational Number

3. Integer

4. Complex number