For \(p\,>\,0\), a vector \(\vec{v}_{2}=2 \hat{i}+(p+1) \hat{j}\) is obtained by rotating the vector \(\vec{v}_{1}=\sqrt{3} p \hat{i}+\hat{j}\) by an angle \(\theta\) about origin in counter clockwise direction. If \(\tan \theta=\frac{(\alpha \sqrt{3}-2)}{4 \sqrt{3}+3}\), then the value of \(\alpha\) is equal to \(....\)

If \(\sin(\tan^{-1}(x\sqrt{2})) = \cot(\sin^{-1}\sqrt{1-x^2})\), \(x \in (0,1)\), then the value of \(x\) is :

If \(\sum_{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \times 2^{29}\), then \(\alpha\) is equal to ______

The square of the distance of the point of intersection of the lines \(\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})\), \(a \neq 0\) and \(\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})\) from the origin is:

If \([ t ]\) denotes the greatest integer \(\leq t\), then the value of \(\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] d x\) is.

Let \(A\) be a \(3 \times 3\) matrix of non-negative real elements such that \(A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\). Then the maximum value of \(\operatorname{det}(\mathrm{A})\) is ...........

An aeroplane flying at a constant speed, parallel to the horizontal ground, \(\sqrt 3\, km\) above it, is observed at an elevation of \(60^o\) from a point on the ground. If, after five seconds, its elevation from the same point, is \(30^o\), then the speed (in \(km/hr\)) of the aeroplane is

Let \(S\) be the set of all functions \(f:[0,1] \rightarrow \mathrm{R}\) which are continuous on \([0,1]\) and differentiable on \((0,1) .\) Then for every \(f\) in \(\mathrm{S},\) there exists a \(\mathrm{c} \in(0,1),\) depending on \(f,\) such that

The equation of circle described on the chord \(3x + y+ 5\, = 0\) of the circle \(x^2 + y^2\, = 16\) as diameter is

If \(a, b\) and \(c\) are the greatest value of \(^{19} \mathrm{C}_{\mathrm{p}},^{20} \mathrm{C}_{\mathrm{q}}\) and \(^{21 }\mathrm{C}_{\mathrm{r}}\) respectively, then

Let \(\vec{a_k} = (\tan\theta_k)\hat{i} + \hat{j}\) and \(\vec{b_k} = \hat{i} - (\cot\theta_k)\hat{j}\), where \(\theta_k = \dfrac{2^{k-1}\pi}{2^n + 1}\), for some \(n \in \mathbb{N}\), \(n > 5\). Then the value of \(\dfrac{\displaystyle\sum_{k=1}^{n}|\vec{a_k}|^2}{\displaystyle\sum_{k=1}^{n}|\vec{b_k}|^2}\) is _____.

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

If all the six digit numbers \(x_1 x_2 x_3 x_4 x_5 x_6\) with \(0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6\) are arranged in the increasing order, then the sum of the digits in the \(72^{\text {th }}\) number is \(............\).