Let \(f(x)=\lim _{\mathrm{n} \rightarrow \infty} \sum_{\mathrm{r}=0}^{\mathrm{n}}\left(\frac{\tan \left(x / 2^{r+1}\right)+\tan ^3\left(x / 2^{r+1}\right)}{1-\tan ^2\left(x / 2^{r+1}\right)}\right)\). Then \(\lim _{x \rightarrow 0} \frac{\mathrm{e}^x-\mathrm{e}^{f(x)}}{(x-f(x))}\) is equal to

The normal to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1\) at the point \((8,3 \sqrt{3})\) on it passes through the point

The circle passing through the intersection of the circles, \(x^{2}+y^{2}-6 x=0\) and \(x^{2}+y^{2}-4 y=0\) having its centre on the line, \(2 x-3 y+12=0\), also passes through the point

Let \(\vec \alpha =(\lambda -2) \vec a + \vec b\) and \(\vec \beta = (4\lambda -2)\vec a + 3\vec b\) be two given vectors where \(\vec a\) and \(\vec b\) are non collinear. The value of \(\lambda \) for which vectors and \(\vec \alpha \) and \(\vec \beta \) are collinear, is

The number of solutions of the equation \(2 x+3 \tan x=\pi, x \in[-2 \pi, 2 \pi]-\left\{ \pm \frac{\pi}{2}, \pm \frac{3 \pi}{2}\right\}\) is

The set of all \(\alpha\), for which the vector \(\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k} \quad\) and \(\quad \vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k} \quad\) are inclined at an obtuse angle for all \(t \in \mathbb{R}\) is :

Let \(a, b \in \mathbb{C}\). Let \(\alpha, \beta\) be the roots of the equation \(x^2 + ax + b = 0\). If \(\beta - \alpha = \sqrt{11}\) and \(\beta^2 - \alpha^2 = 3i\sqrt{11}\), then \((\beta^3 - \alpha^3)^2\) is equal to:

Let a curve \(y=f(x)\) pass through the points \((0,5)\) and \(\left(\log _e 2, k\right)\). If the curve satisfies the differential equation \(2(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d y=0\), then \(k\) is equal to

Let \(S\) be the mirror image of the point \(Q(1,3,4)\) with respect to the plane \(2 x-y+z+3=0\) and let \(\mathrm{R}(3,5, \gamma)\) be a point of this plane. Then the square of the length of the line segment \(SR\) is ..... .

Let \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) be a solution curve of the differential equation \((y+1) \tan ^{2} x d x+\tan x d y+y d x=0\) \(x \in\left(0, \frac{\pi}{2}\right)\). If \(\lim _{x \rightarrow 0+} x y(x)=1\), then the value of \(\mathrm{y}\left(\frac{\pi}{4}\right)\) is :

A value of \(x\) for which \(\sin \,\left( {{{\cot }^{ - 1}}\,\left( {1 + x} \right)} \right) = \cos \,\left( {{{\tan }^{ - 1}}\,x} \right)\), is

Let \(a \in R\) and let \(\alpha, \beta\) be the roots of the equation \(x^2+60^{\frac{1}{4}} x+a=0\). If \(\alpha^4+\beta^4=-30\), then the product of all possible values of \(a\) is \(......\)

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______