Q. If $a,b,c$ are in A.P, then the determinant
$\left|\begin{array}{ccc}x+2& x+3& x+2a\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$ is

1. $0$
2. $1$
3. $x$
4. $2x$

Q. Evaluate the determinants
(i)$\left|\begin{array}{ccc}3& -1& -2\\ 0& 0& -1\\ 3& -5& -0\end{array}\right|$ (ii)$\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

Q. Let $A=\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$, where $0\le \theta \le 2\pi$. Then $|A|$ lies between

Q. $\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \cos (\alpha +\delta )\\ \sin \beta & \cos \beta & \cos (\beta +\delta )\\ \sin \gamma & \cos \gamma & \cos (\gamma +\delta )\end{array}\right|=0$

Q. If $\left|\begin{array}{cc}x& 2\\ 18& x\end{array}\right|=\left|\begin{array}{cc}6& 2\\ 3x& 6\end{array}\right|$, then $x$ is equal to

1. $-6$
2. $\pm 6$
3. $6$
4. $0$

Q. $\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|=-2(x+y)(x^2+y^2-xy)$

Q. If $x,y,z$ are non zero real numbers, then the inverse of matrix $A=$ $\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$ is

1. $\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& y^{-1}& 0\\ 0& 0& z^{-1}\end{array}\right]$
2. $xyz\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& y^{-1}& 0\\ 0& 0& z^{-1}\end{array}\right]$
3. $\frac{1}{xyz}\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$
4. $\frac{1}{xyz}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Q. Prove that $\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{array}\right|$ $=1$

Q. Let $A=\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$, where $0\le \theta \le 2\pi$. Then

1. $Det\left(A\right)=0$
2. $Det\left(A\right)\in (2,\infty )$
3. $Det\left(A\right)\in (2,4)$
4. $Det\left(A\right)\in [2,4]$

Q. If $A=\left[\begin{array}{ccc}1& 1& -2\\ 2& 1& -3\\ 5& 4& -9\end{array}\right]$, find |A|.

Q. Evaluate $\left|\begin{array}{ccc}1& x& y\\ 1& x+y& y\\ x+y& x& y\end{array}\right|$

Q.

Q. Find the value of determinant.
(i) $\left|\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right|$
(ii) $\left|\begin{array}{cc}{x}^2-x+1& x-1\\ x+1& x+1\end{array}\right|$

Q. If A=$\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\\ 0& 0& 4\end{array}\right]$ then show that |3A|=27|A|

Q. Find the value of determinant $\left|\begin{array}{cc}2& 4\\ -5& -1\end{array}\right|$.

Q. If A=$\left|\begin{array}{cc}1& 2\\ 4& 2\end{array}\right|$, then show that $|2A|=4|A|$

Q. Find the values of $x$, if
(i)$\left|\begin{array}{cc}2& 4\\ 5& 1\end{array}\right|=\left|\begin{array}{cc}2x& 4\\ 6& x\end{array}\right|$ (ii)$\left|\begin{array}{cc}2& 3\\ 4& 5\end{array}\right|=\left|\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right|$

Q.

Q. Using the property of determinants and without expanding, prove that
$\left|\begin{array}{ccc}x& a& x+a\\ y& b& y+b\\ z& c& z+c\end{array}\right|=0$

Q.