If \(n (2 n +1) \int_{0}^{1}\left(1- x ^{ n }\right)^{2 n } dx =1177 \int_{0}^{1}\left(1- x ^{ n }\right)^{2 n +1} dx\), then \(n \in N\) is equal to \(\dots\dots\)

Let the foot of perpendicular of the point \(P (3,-2,-9)\) on the plane passing through the points \((-1,-2,-3),(9,3,4),(9,-2,1)\) be \(Q(\alpha, \beta, \gamma)\). Then the distance of \(Q\) from the origin is:

The integral \(\int_{\frac{\pi }{{12}}}^{\frac{\pi }{4}} {\frac{{8\,\cos \,2x}}{{{{\left( {\tan \,x + \cot \,x} \right)}^3}}}\,dx} \) equals

The least integral value \(\alpha \) of \(x\) such that \(\frac{{x - 5}}{{{x^2} + 5x - 14}} > 0\) , satisfies

Let \(P\) be an arbitrary point having sum of the squares of the distance from the planes \(x + y + z =0, l x - nz =0\) and \(x -2 y + z =0\) equal to \(9 .\) If the locus of the point \(P\) is \(x ^{2}+ y ^{2}+ z ^{2}=9,\) then the value of \(l- n\) is equal to ...... .

Consider an elIipse, whose centre is at the origin and its major axis is along the \(x-\) axis. If its eccentricity is \(\frac{3}{5}\) and the distance between its foci is \(6\), then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable \(x\) to be the number of rotten apples in a draw of two apples, the variance of \(x\) is

The remainder when \(428^{2024}\) is divided by \(21\) is ............

If the shortest distance between the lines \(\vec{r}_{1}=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0\) and \(\vec{r}_{2}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k}), \mu \in R\) is \(9\), then \(\alpha\) is equal to \(.....\)

The value of \(\int_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} d x\) is equal to :

Let \(z_1\) and \(z_2\) be two complex number such that \(z_1\) \(+z_2=5\) and \(z_1^3+z_2^3=20+15 i\). Then \(\left|z_1^4+z_2^4\right|\) equals-

\(\lim _{x \rightarrow \infty} \frac{\left(2 x^2-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^2+5 x+4\right) \sqrt{(3 x+2)^x}}\) is equal to :