The mean and variance of a binomial distribution are \(\alpha\) and \(\frac{\alpha}{3}\) respectively. If \(P(X=1)=\frac{4}{243}\), then \(P ( X =4\) or \(5)\) is equal to.

If \(\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}\) is the solution of \(4 \cos \theta+5 \sin \theta=1\), then the value of \(\tan \alpha\) is

Let  \( \lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right. \) \( \left.+\ldots \ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right) \text { be } \frac{\pi}{k},\) using only the principal values of the inverse trigonometric functions. Then \(\mathrm{k}^2\) is equal to ..............

The coefficient of \(x^{2012}\) in the expansion of \((1-x)^{2008}\left(1+x+x^2\right)^{2007}\) is equal to

If \(A\) and \(B\) are two events such that \(P(A)=0.7\), \(\mathrm{P}(\mathrm{B})=0.4\) and \(\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})=0.5\), where \(\overline{\mathrm{B}}\) denotes the complement of \(B\), then \(P(B \mid(A \cup \bar{B}))\) is equal:-

The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola \(\frac{{{x^2}}}{4} - \frac{{{y^2}}}{5} = 1\) , meet \(x-\) axis and \(y-\) axis at \(A\) and \(B\) respectively. Then \((OA)^2 - (OB)^2\) , where \(O\) is the origin, equals

If the function \(g\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{k\sqrt {x + 1} ,\;\;0 \le x \le 3}\\{mx + 2,\;\;3 < x \le 5}\end{array}} \right.\) is differentiable, then the value of \(k + m\) is :

If \(\int \limits_{-0.15}^{0.15}\left|100 x ^2-1\right| dx =\frac{ k }{3000}\), then \(k\) is equal to \(..........\).

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(A\) and \(B\) respectively. Then the distance of the mid-point of the line segment \(A B\) from the plane \(2 x-2 y+z=14\) is

Let the domain of the function
\(f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right)\) be \([\alpha, \beta]\) and the domain of \(\mathrm{g}(\mathrm{x})=\log _2\left(2-6 \log _{27}(2 \mathrm{x}+5)\right)\) be \((\gamma, \delta)\).
Then \(|7(\alpha+\beta)+4(\gamma+\delta)|\) is equal to ________

Let \( A(1,2) \) and \( C(-3,-6) \) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line \( 7x-y=14 \). If \( B(\alpha, \beta) \) and \( D(\gamma, \delta) \) are the other two vertices, then \( |\alpha+\beta+\gamma+\delta| \) is equal to:

Let \(A = \left[ {\begin{array}{*{20}{c}}
  2&b&1 \\ 
  b&{{b^2} + 1}&b \\ 
  1&b&2 
\end{array}} \right]\)  where \(b > 0\). Then the minimum value of \(\frac{{\det \left( A \right)}}{b}\) is