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.

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Which of the following are correct in respect of the system of equations $$x + y + z = 8$$, $$x - y + 2z = 6$$ and $$3x - y + 5z = k$$ ?

I. They have no solution, if k = 15.

II. They have infinitely many solutions, if k = 20.

III. They have unique solution, if k = 25.

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Which of the following determinants have value 'zero'?

1. $$\left| {\matrix{ {41} & 1 & 5 \cr {79} & 7 & 9 \cr {29} & 5 & 3 \cr } } \right|$$

2. $$\left| {\matrix{ 1 & a & {b + c} \cr 1 & b & {c + a} \cr 1 & c & {a + b} \cr } } \right|$$

3. $$\left| {\matrix{ 0 & c & b \cr { - c} & 0 & a \cr { - b} & { - a} & 0 \cr } } \right|$$

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