Determinants Te...

Let $$A=\begin{bmatrix}5 & 5\alpha  &\alpha  \\ 0 & \alpha  &5\alpha  \\ 0 & 0 &5 \end{bmatrix}$$ If $$\left | A^{2} \right |=25$$, then $$\left |\alpha  \right |$$ equals

The number of distinct real roots of the equation, $$\begin{vmatrix} \cos x& \sin x & \sin x\\ \sin x & \cos x & \sin x\\ \sin x & \sin x & \cos x\end{vmatrix} = 0$$ 

The points $$\displaystyle \left( 0, \frac{8}{3} \right), (1, 3)$$ and $$(82, 30)$$ :

If $$A=\begin {bmatrix} 2 & -3\\ -4 & 1\end{bmatrix}$$, then adj $$(3A^2+12A)$$ is  equal to.

$$\mathrm{If}\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}=-2$$ and $$\mathrm{f}(\mathrm{x})=$$ $$\begin{vmatrix}1+a^{2}x &(1+b^{2})x &(1+c^{2})x \\(1+a^{2})x&1+b^{2}x &(1+c^{2})x\\(1+a^{2})x&(1+b^{2})x & 1+c^2x\end{vmatrix}$$  then $$f(x)$$ is a polynomial of degree 

If $$\Delta_1 = \begin{vmatrix}x & \sin \theta & \cos \theta\\-\sin \theta & -x & 1\\\cos \theta & 1 &x\end{vmatrix}$$ and $$\Delta_2 = \begin{vmatrix}x & \sin 2\theta & \cos 2\theta\\-\sin 2\theta & -x & 1\\ \cos 2\theta & 1 & x\end{vmatrix}, x \neq 0$$; then for all $$\theta \in \left(0, \dfrac{\pi}{2}\right):$$

Let $$P = [a_{ij}]$$ be a 3 $$\times$$ 3 matrix and let $$Q = [b_{ij}]$$, where $$b_{ij} = 2^{i + j} a_{ij}$$ for $$1 \leq i, j \leq 3$$. If the determinant of P is 2, then the determinant of the matrix Q is

Consider three points $${P}=(-\sin(\beta-\alpha), -\cos\beta) , {Q}=(\cos(\beta-\alpha), \sin\beta)$$ and $${R}=(\cos(\beta-\alpha +\theta), \sin(\beta-\theta))$$ , where $$0< \alpha,\ \beta,\ \theta <\displaystyle \frac{\pi}{4}$$. Then

The number of $$A$$ in $$T_p$$ such that $$A$$ is either symmetric or skew-symmetric or both, and det $$(A)$$ divisible by $$p$$, is

The value of $$|\mathrm{U}|$$ is