Let \( a_{1}, a_{2}, a_{3},..... \) be a G.P. of increasing positive terms such that \( a_{2} . a_{3} . a_{4}=64 \) and \( a_{1}+a_{3}+a_{5}=\frac{813}{7} \).
Then \( a_{3}+a_{5}+a_{7} \) is equal to:

The least positive integral value of \(\alpha\), for which the angle between the vectors \(\alpha \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \mathrm{k}\) and \(\alpha \hat{\mathrm{i}}+2 \alpha \hat{\mathrm{j}}-2 \mathrm{k}\) is acute, is

A bag contains \(4\) red and \(6\) black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are re­ turned to the bag. Ifnow a ball is drawn at random from the bag, then the probability that this drawn ball is red, is :

If \(A=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]\) and \(M=A+A^{2}+A^{3}+\ldots .+A^{20}\), then the sum of all the elements of the matrix \(\mathrm{M}\) is equal to \(.....\)

Let \([\cdot]\) denote the greatest integer function. Then the value of \(\displaystyle\int_0^3 \left(\dfrac{e^x + e^{-x}}{[x]!}\right) dx\) is :

Two tangents are drawn from the point \(\mathrm{P}(-1,1)\) to the circle \(\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}-6 \mathrm{y}+6=0\). If these tangents touch the circle at points \(A\) and \(B\), and if \(D\) is a point on the circle such that length of the segments \(A B\) and \(A D\) are equal, then the area of the triangle \(A B D\) is eqaul to:

\( (\frac{1}{3}+\frac{4}{7})+(\frac{1}{3^{2}}+\frac{1}{3}\times\frac{4}{7}+\frac{4^{2}}{7^{2}})+(\frac{1}{3^{3}}+\frac{1}{3^{2}}\times\frac{4}{7}+\frac{1}{3}\times\frac{4^{2}}{7^{2}}+\frac{4^{3}}{7^{3}}) + \dots \) upto infinite terms is equal to -

Let \(f(x)\) be a polynomial of degree \(5\), and have extrema at \(x = 1\) and \(x = -1\). If \(\displaystyle\lim_{x \to 0} \left(\dfrac{f(x)}{x^3}\right) = -5\), then \(f(2) - f(-2)\) is equal to:

Let P be a point in the plane of the vectors \(\overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}\) such that P is equidistant from the lines AB and AC. If \({|\overrightarrow{AP}|}=\frac{\sqrt{5}}{2}\) then the area of the triangle ABP is :

The Coefficient of \(x ^{-6}\), in the expansion of \(\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9\), is \(........\).

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

\(\lim \limits_{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{\frac{1}{x}}\) is equal to