Let \(A\) be a \(2 \times 2\) real matrix with entries from \(\{0,1\}\) and \(|\mathrm{A}| \neq 0 .\) Consider the following two statements : \((P)\) If \(A \neq I_{2},\) then \(|A|=-1\) \((\mathrm{Q})\) If \(|\mathrm{A}|=1,\) then \(\operatorname{tr}(\mathrm{A})=2\) where \(I_{2}\) denotes \(2 \times 2\) identity matrix and \(\operatorname{tr}(A)\) denotes the sum of the diagonal entries of \(A\) Then

The value of \(\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|\) is 

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

Let \(\mu\) be the mean and \(\sigma\) be the standard deviation of the distribution 

\(X_i\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(f_i\)	\(k+2\)	\(2k\)	\(K^{2}-1\)	\(K^{2}-1\)	\(K^{2}-1\)	\(k-3\)

where \(\sum f_i=62\). if \([x]\) denotes the greatest integer \(\leq x\), then \(\left[\mu^2+\sigma^2\right]\) is equal \(.........\).

Let \(A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]\), where \(a, c \in R\). If \(A^3=A\) and the positive value of a belongs to the interval ( \(n -1, n\) ], where \(n \in N\), then \(n\) is equal to \(..........\).

If the line \(l_1: 3 y -2 x =3\) is the angular bisector of the lines \(l_2: x - y +1=0\) and \(l_3: \alpha x +\beta y +17=0\), then \(\alpha^2+\beta^2-\alpha-\beta\) is equal to

The shortest distance between the lines \(x+1=2 y=-\) \(12 z\) and \(x=y+2=6 z-6\) is

If a hyperbola has length of its conjugate axis equal to \(5\) and the distance between its foci is \(13\), then the eccentricity of the hyperbola is

If the circles \(x^{2}+y^{2}+6 x+8 y+16=0\) and \(x^{2}+y^{2}+2(3-\sqrt{3}) x+x+2(4-\sqrt{6}) y\) \(= k +6 \sqrt{3}+8 \sqrt{6}, k >0\), touch internally at the point \(P(\alpha, \beta)\), then \((\alpha+\sqrt{3})^{2}+(\beta+\sqrt{6})^{2}\) is equal to \(\dots\dots\)

If the system of linear equations \(x+ ay+z\,= 3\) ; \(x + 2y+ 2z\, = 6\) ; \(x+5y+ 3z\, = b\) has no solution, then

If \(\left| {\begin{array}{*{20}{c}}{x - 4}&{2x}&{2x}\\{2x}&{x - 4}&{2x}\\{2x}&{2x}&{x - 4}\end{array}} \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2},\) then the ordered pair \(\left( {A,B} \right) = \). . . . .

Let \(S=\{1,2,3,5,7,10,11\}\). The number of nonempty subsets of \(S\) that have the sum of all elements a multiple of \(3\) , is \(........\)

Let \(K\) be the set of all real values of \(x\) where the function \(f\left( x \right) = \sin \,\left| x \right| - \left| x \right| + 2\,\left( {x - \pi } \right)\,\cos \,\left| x \right|\) is not differentiable. Then the set \(K\) is equal to