Let $$a{x^3} + b{x^2} + cx + d = \left| {\matrix{ {x - 1} & {2x} & {3x} \cr {2x + 3} & {x + 1} & x \cr {2 - x} & {3x + 4} & {5x - 1} \cr } } \right|$$

What is the value of c?

Let $$a{x^3} + b{x^2} + cx + d = \left| {\matrix{ {x - 1} & {2x} & {3x} \cr {2x + 3} & {x + 1} & x \cr {2 - x} & {3x + 4} & {5x - 1} \cr } } \right|$$

What is the value of a + b + c + d ?

If $$m = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$ and $$n = \left[ {\matrix{ 0 & 1 \cr { - 1} & 0 \cr } } \right]$$, then what is the value of the determinant of $$m\cos \theta $$ $$-$$ $$n\sin \theta $$ ?

If $$f(x) = \left[ {\matrix{ {\cos x} & { - \sin x} & 0 \cr {\sin x} & {\cos x} & 0 \cr 0 & 0 & 1 \cr } } \right]$$, then which of the following are correct?

I. $$f(\theta ) \times f(\phi ) = f(\theta + \phi )$$.

II. The value of the determinant of the matrix $$f(\theta ) \times f(\phi )$$ is 1.

III. The determinant of $$f(x)$$ is an even function.

Select the correct answer using the code given below.