Let \(N\) be the set of natural numbers and a relation \(R\) on \(N\) be defined by \(R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} .\) Then the relation \(R\) is:

The value of \(b>3\) for which \(12 \int \limits_{3}^{b} \frac{1}{\left(x^{2}-1\right)\left(x^{2}-4\right)} d x=\log _{e}\left(\frac{49}{40}\right)\), is equal to

Let the image of the point \(P (1,2,6)\) in the plane passing through the points \(A (1,2,0), B (1,4,1)\) and \(C(0,5,1)\) be \(Q(\alpha, \beta, \gamma)\). Then \(\left(\alpha^2+\beta^2+\gamma^2\right)\) is equal to :

Let \(\quad f(x)=x+\frac{a}{\pi^2-4} \sin x+\frac{b}{\pi^2-4} \cos x\) \(x \in R\) be a function which satisfies \(f(x)=x+\int \limits_0^{\pi / 2} \sin (x+y) f(y) d y\). Then \(( a + b )\) is equal to \(............\)

Let \(\int_\alpha^{\log _e^4} \frac{\mathrm{dx}}{\sqrt{\mathrm{e}^{\mathrm{x}}-1}}=\frac{\pi}{6}\). Then \(\mathrm{e}^\alpha\) and \(\mathrm{e}^{-\alpha}\) are the roots of the equation :

If the mean of the following probability distribution of a random variable \(X\);

\(X\)	\(0\)	\(2\)	\(4\)	\(6\)	\(8\)
\(P(X)\)	\(a\)	\(2a\)	\(a+b\)	\(2b\)	\(3b\)

is \(\frac{46}{9}\) , then the variance of the distribution is

The value of \(\int \limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x\) is equal to

If \(f\left( x \right)\left\{ {\begin{array}{*{20}{c}}
  {\frac{{\sin \,\left( {p + 1} \right)x + \sin \,x}}{x},\,\,}&{x < 0} \\ 
  {q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x = 0} \\ 
  {\frac{{\sqrt {x + {x^2}}  - \sqrt x }}{{x/2}},}&{x > 0} 
\end{array}} \right.\) Is continuous at \(x = 0\), then the ordered pair \((p, q)\) is equal to

Let P be the foot of the perpendicular from the point \((1,2,2)\) on the line \(\mathrm{L}: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}\). Let the line \(\vec{r}=(-\hat{i}+\hat{j}-2 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}), \lambda \in \mathbf{R}\), intersect the line L at Q . Then \(2(\mathrm{PQ})^2\) is equal to:

Let \(A\) and \(B\) be two events such that \(P ( B \mid A )=\frac{2}{5}\), \(P ( A \mid B )=\frac{1}{7}\) and \(P ( A \cap B )=\frac{1}{9} .\) Consider \(( S 1) P \left( A ^{\prime} \cup B \right)=\frac{5}{6}\) \(( S 2) P \left( A ^{\prime} \cap B ^{\prime}\right)=\frac{1}{18}\). Then.

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 \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

\(\mathop {\lim }\limits_{x \to 0} \,\frac{{x\,\cot \,\left( {4x} \right)}}{{{{\sin }^2}\,x\,{{\cot }^2}\,\left( {2x} \right)}}\) is equal to

Let the maximum value of \( (sin^{-1}x)^{2} + (cos^{-1}x)^{2} \) for \( x\in [-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}] \) be \( \frac{m}{n}\pi^{2} \), where gcd (m, n) = 1. Then \( m+n \) is equal to ........... .