The area of the region enclosed by the parabolas \(y=x^2-5 x\) and \(y=7 x-x^2\) is ....................

The value of \(\tan \left(2 \tan ^{-1}\left(\frac{3}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)\right)\) is equal to:

An angle between the plane, \(x + y + z = 5\) and the line of intersection of the planes, \(3x + 4y + z- 1 = 0\) and \(5x + 8y + 2z+ 14 = 0\) ,  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 \(\alpha \in(0, \infty)\) and \(A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]\). If \(\operatorname{det}\left(\operatorname{adj}\left(2 \mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right) \cdot \operatorname{adj}\left(\mathrm{A}-2 \mathrm{~A}^{\mathrm{T}}\right)\right)=2^8\), then \((\operatorname{det}(\mathrm{A}))^2\) is equal to :

Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\) and R be a relation on \(A\) defined by \(x R y\) if and only if \(2 x-y \in\{0,1\}\). Let \(l\) be the number of elements in R. Let \(m\) and \(n\) be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m} \mathrm{n}\) is equal to :-

A circle passes through the points \((2, 3)\) and \((4, 5)\). If its centre lies on the line, \(y- 4x + 3 = 0\) , then its radius is equal to

The domain of the function \(f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)\) is.

If \(y=m x+4\) is a tangent to both the parabolas, \(\mathrm{y}^{2}=4 \mathrm{x}\) and \(\mathrm{x}^{2}=2 \mathrm{by},\) then \(\mathrm{b}\) is equal to 

Let \(f :R \to R\) be defined by \(f(x)\,\, = \,\,\frac{x}{{1 + {x^2}}},\,x\, \in \,R.\) Then the range of \(f\) is

Let \(y = y(x)\) be the solution of the differential equation \(x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx\), \(y(1) = \dfrac{\pi}{2}\) and let \(\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)\). Then the number of integral values of \(p\), for which the equation \(x^2 + y^2 - 2px + 2py + \alpha + 2 = 0\) represents a circle of radius \(r \leq 6\), is __________.

The minimum area of a triangle formed by any tangent to the ellipse \(\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{81}} = 1\) and the co-ordinate axes is

Let \(A=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right)\). Then the sum of the diagonal elements of the matrix \(( A + I )^{11}\) is equal to: