Determinants Te...

If maximum and minimum values of $$D = \begin{vmatrix}1 & -\cos \theta & 1\\ \cos \theta & 1 & -\cos \theta\\ 1 & \cos \theta & 1\end{vmatrix}$$ are $$p$$ and $$q$$ respectively, then the value of $$2p + 3q$$ is _________.

If $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ satisfies the equation $$x^2 - (a + d) x + k = 0$$, then ?

For a positive numbers $$x, y$$ and $$z$$ the numerical value of the determinant $$\begin{bmatrix}1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 1 & \log_{y} z\\ \log_{z} x & \log_{z} y & 1\end{bmatrix}$$ is:

The value of the determinant 
$$\begin{vmatrix} \cos^2 \dfrac{\theta}{2}&\sin^2\dfrac{\theta}{2}\\ \sin^2\dfrac{\theta}{2} &\cos^2\dfrac{\theta}{2}  \end{vmatrix}$$ 
for all values of $$\theta $$, is

The adjoint of the matric $$A = \begin{bmatrix}1 & 0 & 2\\ 2 & 1 & 0\\ 0 & 3 & 1\end{bmatrix}$$ is

If $$A + B + C = \pi$$, then $$\begin{vmatrix} \sin (A + B + C)& \sin B & \cos C\\ -\sin B & 0 & \tan A\\ \cos (A + B) & -\tan A & 0\end{vmatrix}$$ is equal to

If the points $$A(-2,1),B(a,b)$$ and $$C(4,-1)$$ are collinear and $$a-b=1$$, find the values of $$a$$ and $$b$$.

The determinant $$\begin{vmatrix} xp+y & x & y\\ yp+z & y & z\\ 0 & xp+y & yp+z\end{vmatrix}=0$$. If.

If $$\triangle$$ =$$
\left |
\begin{array}{111}
x_1+y_1\omega  & x_1\omega^2+y_1 & x_1+y_1\omega+z_1\omega^2 \\
x_2+y_2\omega & x_2\omega^2+y_2 & x_2+y_2\omega+z_2\omega^2 \\
x_3+y_3\omega & x_3\omega^2+y_3 & x_3+y_3\omega+z_3\omega^2 \\
\end {array}
\right |
$$
where $$1,\omega,\omega^2$$ are cube roots of unity then $$\triangle$$ is equal to

The value of determinant $$\begin{vmatrix} x+1 & x+2 & x+4\\ x+3 & x+5 & x+8\\ x+7 & x+10 & x+14\end{vmatrix}$$ is?