Let a line \(L: 2 x+y=k, k\,>\,0\) be a tangent to the hyperbola \(x^{2}-y^{2}=3 .\) If \(L\) is also a tangent to the parabola \(y^{2}=\alpha x\), then \(\alpha\) is equal to :

The sides of a rhombus \(ABCD\) are parallel to the lines, \(x - y + 2\, = 0\) and \(7x - y + 3\, = 0\). If the diagonals of the rhombus intersect at \(P( 1, 2)\) and the vertex \(A\) ( different from the origin) is on the \(y\) axis, then the ordinate of \(A\) is

If the coefficients of \(x^{7}\) in \(\left(x^{2}+\frac{1}{b x}\right)^{11}\) and \(x^{-7}\) in \(\left(x-\frac{1}{b x^{2}}\right)^{11}, b \neq 0\), are equal, then the value of \(b\) is equal to:

Let \( |A|=6 \) where A is a \( 3 \times 3 \) matrix. If \( |adj(3adj(A^{2} \cdot adj(2A)))|=2^{m} \cdot 3^{n} \), \( m, n \in N \), then \( m+n \) is equal to:

If \(\sum_{r=1}^{10} r !\left( r ^{3}+6 r ^{2}+2 r +5\right)=\alpha(11 !),\) then the value of \(\alpha\) is equal to ...... .

Three balls are drawn at random from a bag containing \(5\) blue and \(4\) yellow balls. Let the random variables \(\mathrm{X}\) and \(\mathrm{Y}\) respectively denote the number of blue and Yellow balls. If \(\bar{X}\) and \(\bar{Y}\) are the means of \(X\) and \(Y\) respectively, then \(7 \bar{X}+4 \bar{Y}\) is equal to ..........

A hyperbola whose transverse axis is along the major axis of then conic, \(\frac{{{x^2}}}{3} + \frac{{{y^2}}}{4} = 4\) and has vertices at the foci of this conic . If the eccentricity of the hyperbola is \(\frac{3}{2}\) , then which of the following points does \(NOT\) lie on it ?

Consider a circle \((x-\alpha)^2+(y-\beta)^2=50\), where \(\alpha, \beta>0\). If the circle touches the line \(y+x=0\) at the point \(P\), whose distance from the origin is \(4 \sqrt{2}\) , then \((\alpha+\beta)^2\) is equal to ................

Let \(z_1, z_2\) and \(z_3\) be three complex numbers on the circle \(|z|=1\) with \(\arg \left(z_1\right)=\frac{-\pi}{4}, \arg \left(z_2\right)=0\) and \(\arg \left(z_3\right)=\frac{\pi}{4}\). If \(\left|z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1\right|^2=\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}\), then the value of \(\alpha^2+\beta^2\) is :

\(\lim _{x \rightarrow 0} \operatorname{cosec} x\)\(\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)\) is:

Let \(g\left( x \right) = \cos {x^2},f\left( x \right) = \sqrt x \) and \(\alpha ,\beta (\alpha < \beta )\) be the roots of the quadratic equation \(18{x^2} - 9\pi x + {\pi ^2} = 0\). Then the area (in sq. units) bounded by the curve \(y = \left( {gof} \right)\left( x \right)\) and the lines \(x = \alpha ,x = \beta \) and \(y = 0\) is :

Let the solution curve of the differential equation \(x d y=\left(\sqrt{x^{2}+y^{2}}+y\right) d x, x>0\), intersect the line \(x =1\) at \(y =0\) and the line \(x =2\) at \(y =\alpha\). Then the value of \(\alpha\) is.

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is