If the matrix \(A=\left(\begin{array}{cc}0 & 2 \\ K & -1\end{array}\right)\) satisfies \(A\left(A^{3}+3 I\right)=2 I\) then the value of \(\mathrm{K}\) is :

The straight lines \(l_1\) and \(l_2\) pass through the origin and trisect the line segment of the line \(L: 9 x+5 y=\) 45 between the axes. If \(m_1\) and \(m_2\) are the slopes of the lines \(l_1\) and \(1_2\),then the point of intersection of the line \(y =\left( m _1+ m _2\right) x\) with \(L\) lies on

Let the determinant of a square matrix \(A\) of order \(m\) be \(m-n\), where \(m\) and \(n\) satisfy \(4 m+n=22\) and \(17 m +4 n =93\). If \(\operatorname{det}(n \operatorname{adj}(\operatorname{adj}( mA )))=\) \(3^{ a } 5^{ b } 6^{ c }\). then \(a + b + c\) is equal to:

Let the vectors \(\vec{a}, \vec{b}, \vec{c}\) represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by \(\vec{a}, \vec{b}+\vec{c}\) and \(\vec{a}+2 \vec{b}+3 \vec{c}\) is equal to \(..........\,V\)

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function such that \(f\left(\dfrac{x+y}{3}\right) = \dfrac{f(x) + f(y)}{3}\) for all \(x, y \in \mathbb{R}\), and \(f'(0) = 3\). Then the minimum value of the function \(g(x) = 3 + e^x f(x)\), is:

Let the line \(\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}\) intersect the plane containing the lines \(\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}\) and \(4 a x-y+5 z-7 a=0=2 x -5 y - z -3, a \in R\) at the point \(P(\alpha, \beta, \gamma)\). Then the value of \(\alpha+\beta+\gamma\) equals\(...\)

In a class of \(60\) students, \(40\) opted for \(NCC,\,30\) opted for \(NSS\) and \(20\) opted for both \(NCC\) and \(NSS.\) If one of these students is selected at random, then the probability that the student selected has opted neither for \(NCC\) nor for \(NSS\) is

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

A point on the straight line, \(3x + 5y = 15\) which is equidistant from the coordinate, axes will lie only in

If \(\alpha, \beta\) are roots of the equation \(x^{2}+5 \sqrt{2} x+10=0, \alpha\,>\,\beta\) and \(P_{n}=\alpha^{n}-\beta^{n}\) for each positive integer \(\mathrm{n}\), then the value of \(\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{11} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)\) is equal to \(....\)

The sum of the infinite series \(1+\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+\ldots .\) is equal to

A vector \(\overrightarrow{ V }\) in the first octant is inclined to the \(x\) axis at \(60^{\circ}\), to the \(y\)-axis at \(45^{\circ}\) and to the z-axis at an acute angle. If a plane passing through the points \((\sqrt{2},-1,1)\) and \(( a , b , c )\), is normal to \(\overrightarrow{ v }\), then

The slope of the line touching both the parabolas \({y^2} = 4x\) and \({x^2} = - 32y\), is