Singleton Set

Q. Which of the following is a singleton set?

1. $\{x:|x|=1;x\in Z\}$
2. $\{x:|x|=5;x\in N\}$
3. $\{x:x$ is even prime number greater than $2\}$
4. $\{x:x=y+2$ where $y$ is natural number such that $2<y<5\}$

Q. The correct options among the following is/are

1. The set of even prime numbers greater than $2$ is a null set
2. The set of all good soccer players is a finite set
3. $A=\{x:x^2-3x+2=0\}$ and $B=\{x,x\in N\text{and}|x|\le 2\},$ then they are equal sets.
4. Set of number which are factors of $8$ and set of number which are factors of $10$ are equal sets.

Q. The correct options among the following is/are

1. The set of even prime numbers greater than $2$ is a null set
2. The set of all good soccer players is a finite set
3. $A=\{x:x^2-3x+2=0\}$ and $B=\{x,x\in N\text{and}|x|\le 2\},$ then they are equal sets.
4. Set of number which are factors of $8$ and set of number which are factors of $10$ are equal sets.

Q. If $A=\{x:x$ is a member of family of circles$\}$, $B=\{x:x$ is a vowel in English alphabet$\}$, then

1. $A$ is finite but $B$ is not
2. neither $A$ nor $B$ are finite
3. $A$ is infinite but $B$ is not
4. both $A$ and $B$ are finite

Q.

For each set, given below, state whether it is finite set, infinite set or the null set:

{Multiples of 8.}

Q. 3. if na=1 always and n goes to infinite, then the value of

Q. If $a{e}^x+b{e}^y=c,p{e}^x+q{e}^y=d$ and ${\Delta }_1=\left|\begin{array}{cc}a& b\\ p& q\end{array}\right|,{\Delta }_2=\left|\begin{array}{cc}c& b\\ d& q\end{array}\right|,{\Delta }_3=\left|\begin{array}{cc}a& c\\ p& d\end{array}\right|$ , where ${\Delta }_1,{\Delta }_2,{\Delta }_3$ are all positive, then the value of $(x,y)$ is:

1. $(\frac{{\Delta }_2}{{\Delta }_1},\frac{{\Delta }_3}{{\Delta }_1})$
2. $(\ln \frac{{\Delta }_2}{{\Delta }_1},\ln \frac{{\Delta }_3}{{\Delta }_1})$
3. $(\ln \frac{{\Delta }_1}{{\Delta }_3},\ln \frac{{\Delta }_1}{{\Delta }_2})$
4. $(\ln \frac{{\Delta }_1}{{\Delta }_2},\ln \frac{{\Delta }_1}{{\Delta }_3})$

Q. Which of the following is/are singleton sets?

1. $A=\{x:x\text{is H.C.F. of}12\text{and}18\}$
2. $B=\{x:x\text{is a common prime factor of}16\text{and}27\}$
3. $C=\{x:x\text{is a prime factor of}35\}$
4. $D=\{x:x\text{is L.C.M. of}6\text{and}11\}$

Q. The correct options among the following is/are

1. If $A=\{x:x\text{is a prime number less than}2\}$, then $A$ is a singleton set
2. If $B=\{x:x\in Z\text{and}|x-2|\le 5\}$, then $B$ is a finite set
3. If $C=\{x:x\in Z\text{and}|x|<0\}$, then $C$ is a null set
4. If $D=\{x:x\text{is a natural number greater than}100\}$, then $D$ is an infinite set

Q. Let $S$ denotes the sum of all the values of $\lambda$ for which the system of equations
$(1+\lambda )x_1+x_2+x_3=1$
$x_1+(1+\lambda )x_2+x_3=\lambda$
$x_1+x_2+(1+\lambda )x_3={\lambda }^2$
is inconsistent. Then |S| is

Q. The solution set of the system of equations $x+y-z=6;3x-2y+z=-5;x+3y-2z=14$ is $(x,y,z)$ then $x+y+z$ is equal to

Q. Assertion :Domain of $f\left(x\right)$ is singleton. Reason: Range of $f\left(x\right)$ is singleton.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. If the following system of linear equations
$2x+y+z=5x-y+z=3x+y+az=b$
has no solution, then :

1. $a\ne -\frac{1}{3},b=\frac{7}{3}$
2. $a\ne \frac{1}{3},b=\frac{7}{3}$
3. $a=-\frac{1}{3},b\ne \frac{7}{3}$
4. $a=\frac{1}{3},b\ne \frac{7}{3}$

Q. The solution set of the system of equations $x+y-z=6;3x-2y+z=-5;x+3y-2z=14$ is $(x,y,z)$ then $x+y+z$ is equal to

Q. $\left(1\right)$ ''I want to work for you, '' I said.
(Change it into indirect speech)

Q. Find the L.C.M. of $4,5,6$.

Q. Select the missing number from the given alternatives.

$3$	$4$	$2$	$14$
$6$	$5$	$4$	$44$
$5$	$2$	$7$	?

1. $58$
2. $14$
3. $4$
4. $49$

Q. Using integration find the area of the triangular region whose sides have the equations $y=2x+1,y=3x+1$ and $x=4$

Q. Let $R$ be a relation on $N$ defined by $x+2y=8$. The domain of $R$ is

1. $\{2,4,8\}$
2. $\{1,2,3,4\}$
3. $\{2,4,6,8\}$
4. $\{2,4,6\}$

Q. $\left\{\text{months of a year whose names begin with the letter F}\right\}$ is

1. empty set
2. singleton set
3. none of these
4. an infinite set

Q. The system of equation $ax+y+z=0,x+by+z=0;x+y+cz=0$ has a non-trivial solution then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

1. $1$
2. $-1$
3. $2$
4. $0$

Q. Area of the triangle formed by the lines $y^2-9xy+18x^2=0$ and $y=9$ is :

1. $\frac{27}{4}$
2. $0$
3. $\frac{9}{4}$
4. $27$