Consistency of Linear System of Equations

Q. If the system of linear equations
$x+ay+z=3$
$x+2y+2z=6$
$x+5y+3z=b$
has no solution, then:

1. $a=-1,b\ne 9$
2. $a=1,b\ne 9$
3. $a=-1,b=9$
4. $a\ne -1,b=9$

Q. For the system of linear equations :
$x-2y=1,x-y+kz=-2,ky+4z=6,k\in R,$
consider the following statements :
$\text{(}A)$ The system has unique solution if $k\ne 2,k\ne -2.$
$\left(B\right)$ The system has unique solution if $k=-2.$
$\left(C\right)$ The system has unique solution if $k=2.$
$\left(D\right)$ The system has no-solution if $k=2.$
$\left(E\right)$ The system has infinite number of solutions if $k\ne -2.$

Which of the following statements are correct ?

1. $\left(B\right)$ and $\left(E\right)$ only
2. $\left(C\right)$ and $\left(D\right)$ only
3. $\left(A\right)$ and $\left(D\right)$ only
4. $\left(A\right)$ and $\left(E\right)$ only

Q.

If X and Y are two sets such that X has 40 elements $X\cup Y$ has 60 elements and $X\cap Y$ has 10 elements, how many elements does Y have?

Q.

If X and Y are two sets such that $X\cup Y$ has 18, X has 8 elements and Y has 15 elements; how many elements does $X\cap Y$ have?

Q. The number of values of $\theta \in (0,\pi )$ for which the system of linear equations $x+3y+7z=0$
$-x+4y+7z=0$
$\left(\sin 3\theta \right)x+\left(\cos 2\theta \right)y+2z=0$ has a non-trivial solution, is :

1. four
2. three
3. two
4. one

Q. The system of linear equations
$3x-2y-kz=10$
$2x-4y-2z=6$
$x+2y-z=5m$
is inconsistent if:

1. $k=3,m\ne \frac{4}{5}$
2. $k\ne 3,m\in R$
3. $k\ne 3,m\ne \frac{4}{5}$
4. $k=3,m=\frac{4}{5}$

Q. The system has no solution if:

1. $\lambda \ne 3$
2. $\lambda =3,\mu =10$
3. $\lambda =3,\mu \ne 10$
4. none of these

Q. Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{c}p\\ q\end{array}\right]\ne \left[\begin{array}{c}0\\ 0\end{array}\right]$. If $AB=B$ and $a+d=2$, then the value of $ad-bc$ is