If $A$ and $B$ are two invertible matrices of the same order,then $adj \,(AB)$ is equal to :-

Let $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$,then:

Let $P$ and $Q$ be $3 \times 3$ matrices such that $P \neq Q$. If $P^3 = Q^3$ and $P^2Q = Q^2P$,then the determinant $\det(P^2 + Q^2)$ is equal to:

The maximum value of the determinant of the matrix $\left[\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+4 \sin 2 x \end{array}\right]$ is

Let $A_1, A_2, A_3$ be three $A$.$P$.s with the same common difference $d$ and having their first terms as $A, A+1, A+2$,respectively. Let $a, b, c$ be the $7^{\text{th}}, 9^{\text{th}}, 17^{\text{th}}$ terms of $A_1, A_2, A_3$,respectively,such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$. If $a=29$,then the sum of the first $20$ terms of an $A$.$P$. whose first term is $c-a-b$ and common difference is $\frac{d}{12}$,is equal to $........$.

If $\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \frac{9}{8}(103x+81)$,then $\lambda$ and $\frac{\lambda}{3}$ are the roots of the equation: