The acute angle between the planes \(P_{1}\) and \(P_{2}\), when \(P_{1}\) and \(P_{2}\) are the planes passing through the intersection of the planes \(5 x+8 y+13 z-29=0\) and \(8 x-7 y+z-20=0\) and the points \((2,1,3)\) and \((0,1,2)\), respectively, is

If \((a, b, c)\) is the image of the point \((1,2,-3)\) in the line, \(\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1},\) then \(a+b+c\) is equal to

Let \(\quad P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) and \(Q=P Q P^{ T }\). If \(P ^{ T } Q ^{2007} P =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\), then \(2 a+b-3 c-4 d\) equal to \(...................\).

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is \(3\) units and after \(5\) seconds, it becomes \(7\) units, then its radius after \(9\) seconds is

Let \([\cdot]\) denote the greatest integer function. If the domain of the function \(f(x) = \sin^{-1}\left(\dfrac{x+[x]}{3}\right)\) is \([\alpha, \beta)\), then \(\alpha^2 + \beta^2\) is equal to:

Let the product of the focal distances of the point \(\mathrm{P}(4,2 \sqrt{3})\) on the hyperbola \(\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1\) be 32 .
Let the length of the conjugate axis of \(H\) be \(p\) and the length of its latus rectum be q . Then \(\mathrm{p}^2+\mathrm{q}^2\) is equal to _______

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{ dy }{ dx }+\frac{1}{ x ^{2}-1} y =\left(\frac{ x -1}{ x +1}\right)^{\frac{1}{2}}\), \(x>1\) passing through the point \(\left(2, \sqrt{\frac{1}{3}}\right)\). Then \(\sqrt{7} y (8)\) is equal to.

If \(a_n\) is the greatest term in the sequence \(a _{ n }=\frac{ n ^3}{ n ^4+147}, n =1,2,3 \ldots \ldots\). , then \(\alpha\) is equal to \(..........\).

If the angle between the lines, \(\frac{x}{2} = \frac{y}{2} = \frac{z}{1}\) and \(\frac{{5 - x}}{{ - 2}} = \frac{{7y - 14}}{p} = \frac{{z - 3}}{4}\)  is \({\cos ^{ - 1}}\,\left( {\frac{2}{3}} \right),\)  then \(p\)  is equal to

Let \(f(x)=x^{2025}-x^{2000}, x\in[0,1]\)and the minimum value of the function \(f(x)\) in the interval [0, 1] be \((80)^{80}(n)^{-81}\). Then n is equal to

Let the mean and variance of the frequency distribution

\(\mathrm{x}\)	\(\mathrm{x}_{1}=2\)	\(\mathrm{x}_{2}=6\)	\(\mathrm{x}_{3}=8\)	\(\mathrm{x}_{4}=9\)
\(\mathrm{f}\)	\(4\)	\(4\)	\(\alpha\)	\(\beta\)

be \(6\) and \(6.8\) respectively. If \(x_{3}\) is changed from \(8\) to \(7 ,\) then the mean for the new data will be:

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 _______