If \({ }^{2 n+1} P_{n-1}:{ }^{2 n-1} P_n=11: 21\), then \(n^2+n+15\) is equal to:

If \(\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)\) and \(\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}\) are the roots of the quadratic equation \(a x^2+b x-\sqrt{e}=0\), then 12 \(\log _e(a+b)\) is equal to .............

The number of integral values of \(k\), for which one root of the equation \(2 x^2-8 x+k=0\) lies in the interval \((1,2)\) and its other root lies in the interval \((2,3)\), is :

Let the length of the latus rectum of an ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \( (a>b) \), be 30. If its eccentricity is the maximum value of the function \( f(t)=-\frac{3}{4}+2t-t^{2}, \) then \( (a^{2}+b^{2}) \) is equal to -

If \(\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan B=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}\) and \(\tan C=\left(x^{-3}+x^{-2}+x^{-1}\right)^{\frac{1}{2}}, 0 < A, B, C < \frac{\pi}{2}\) then \(A+B\) is equal to :

If the probability that the random variable X takes the value \(x\) is given by \(P(X=x)=k(x+1) 3^{-x}\), \(\mathrm{x}=0,1,2,3 \ldots \ldots\), where k is a constant, then \(\mathrm{P}(\mathrm{X} \geq 3)\) is equal to

The angle of elevation of the top of a tower from a point \(A\) due north of it is \(\alpha\) and from a point \(B\) at a distance of \(9\) units due west of \(A\) is \(\cos ^{-1}\left(\frac{3}{\sqrt{13}}\right)\). If the distance of the point \(B\) from the tower is \(15\) units, then \(\cot \alpha\) is equal to.

If \(\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-x+1}-a x\right)=b\), then the ordered pair \((a, b)\) is:

Let \(\mathrm{A}=\{\mathrm{n} \in[100,700] \cap \mathrm{N}: \mathrm{n}\) is neither a multiple of \(3\) nor a multiple of 4\(\}\). Then the number of elements in \(\mathrm{A}\) is

Let \(\mathrm{E}\) be an ellipse whose axes are parallel to the co-ordinates axes, having its center at \((3,-4)\), one focus at \((4,-4)\) and one vertex at \((5,-4) .\) If \(m x-y=4, m\,>\,0\) is a tangent to the ellipse \(\mathrm{E}\), then the value of \(5 \mathrm{~m}^{2}\) is equal to \(.....\)

If  \(2\int_0^1 {{{\tan }^{ - 1}}}\,xdx = \int_0^1 {{{\cot }^{ - 1}}}\,(1 - x + {x^2})dx,\) then \(\int_0^1 {{{\tan }^{ - 1}}}\, (1 - x + {x^2})dx\)  is equal to 

Let \(3,7,11,15, \ldots, 403\) and \(2,5,8,11, \ldots, 404\) be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ...........

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,