The value of \(\cot \frac{\pi}{24}\) 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 : _______

Let \(f(x)\) be a positive function such that the area bounded by \(y=f(x), y=0\) from \(x=0\) to \(x=a>0\) is \(\mathrm{e}^{-\mathrm{a}}+4 \mathrm{a}^2+\mathrm{a}-1\). Then the differential equation, whose general solution is \(y=c_1 f(x)+c_2\), where \(c_1\) and \(c_2\) are arbitrary constants, is :

Let \(\mathrm{A}(4,-2), \mathrm{B}(1,1)\) and \(\mathrm{C}(9,-3)\) be the vertices of a triangle \(A B C\). Then the maximum area of the parallelogram AFDE , formed with vertices \(\mathrm{D}, \mathrm{E}\) and F on the sides \(\mathrm{BC}, \mathrm{CA}\) and AB of the triangle ABC respectively, is ______ .

Let the minimum value \(v_{0}\) of \(v=|=|^{2}+|z-3|^{2}+|z-60|^{2}\), \(z \in C\) is attained at \(z=z_{0}\). Then \(\left|2 z_{0}^{2}-z_{0}^{3}+3\right|^{2}+v_{0}^{2}\) is equal to.

The set of all real values of \(\lambda \) for which exactly two common tangents can be drawn to the circles \(x^2 + y^2 - 4x - 4y+ 6\, = 0\) and \(x^2 + y^2 - 10x - 10y + \lambda \, = 0\) is the interval:

Let \(P_n=\alpha^n+\beta^n, n \in \mathbf{N}\). If \(P_{10}=123, P_9=76\), \(P_8=47\) and \(P_1=1\), then the quadratic equation having roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) is :

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is:

If \(y^{2}+\log _{e}\left(\cos ^{2} x\right)=y, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right),\) then

Consider the parabola with vertex \(\left(\frac{1}{2}, \frac{3}{4}\right)\) and the directrix \(\mathrm{y}=\frac{1}{2}\). Let \(\mathrm{P}\) be the point where the parabola meets the line \(\mathrm{x}=-\frac{1}{2}\). If the normal to the parabola at \(\mathrm{P}\) intersects the parabola again at the point \(\mathrm{Q}\), then \((\mathrm{PQ})^{2}\) is equal to :

Let \(A=\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}:|\alpha-1| \leq 4 \text { and }|\beta-5| \leq 6\}\) and \(B=\{(\alpha, \beta) \in \mathbf{R} \times\) \(\mathbf{R}: 16(\alpha-2)^2+9(\beta-6)^2 \leq 144\}\)

Let \(f ( x )=\sum \limits_{ k =1}^{10} kx ^{ k }, x \in R\). If \(2 f(2)+f^{\prime}(2)=119(2)^{ n }+1\) then \(n\) is equal to \(..........\).

The sum \(1+2 \cdot 3+3 \cdot 3^{2}+\ldots . .+10 \cdot 3^{9}\) is equal to