Determinants Te...

$$D\mathrm{e}\mathrm{t} \left\{\begin{array}{lll}
2 & 45 & 55\\
1 & 29 & 32\\
3 & 68 & 87
\end{array}\right\}=\ldots.$$ .

$$\mathrm{If}$$ $$\left|\begin{array}{lll}
1 & 0 & 0\\
2 & 3 & 4\\
5 & -6 & x
\end{array}\right|$$ $$= 45$$ $$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$$ $$\mathrm{x}=$$

$$\begin{vmatrix} 4\sin^{2}\theta  & \cos^{2}\theta \\ 3\sec^{2}\theta & \text{cosec}^{2}\theta \end{vmatrix}=$$

$$Adj \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1  \end{bmatrix}=\begin{bmatrix} 5 & a & -2 \\ 1 & 1 & 0 \\ -2 & -2 & b  \end{bmatrix}\Rightarrow \left [ a\     b\ \right ]=$$ ____

If $$A$$ is a $$3\times 3$$ matrix and $$\text{det}  (3A)=k(\text{det}  A)$$, then $$k=$$

If A and B are similar matrices such that $$det  (AB)=0, $$ then 

$$\displaystyle \begin{vmatrix}cos\: C &tan\: A  &0 \\ sin\: B &0  &-tan\: A \\ 0 &sin\: B  &cos\: C \end{vmatrix}$$ has the value

If$$\displaystyle \left | \begin{matrix}-12 &0   &\lambda  \\  0&  2& -1\\  2& 1 &15 \end{matrix} \right |=-360$$, then the value of $$\lambda$$,is