The direction cosines of a line which makes equal angles with the co-ordinate axes are____

If \(x, y\) are two positive integers such that \(x+y=20\) and the maximum value of \(x^3 y\) is \(k\) at \(x=\alpha, y=\beta\) then \(\frac{k}{\alpha^2 \beta^2}=\)

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)

\(\sin ^3 10^{\circ}+\sin ^3 50^{\circ}-\sin ^3 70^{\circ}=\)

A variable line passing through a fixed point \((\alpha, \beta)\) intersects the coordinate axes at \(A\) and \(B\). If \(O\) is the origin, then the locus of the centroid of the \(\triangle O A B\) is

Let \(\bar{u}, \bar{v}, \bar{w}\) be vectors such that \(|\bar{u}|=1,|\bar{v}|=2,|\bar{w}|=3\). If the projection of \(\bar{v}\) on \(\bar{u}\) is equal to that of \(\bar{w}\) on \(\bar{u}\), and the vectors \(\bar{v}, \bar{w}\) are perpendicular to each other, then \(|\bar{u}-\bar{v}+\bar{w}|=\)

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are three unit vectors such that \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c}\), then the angles between \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{a}, \mathbf{c}\) respectively are

The number of extensive and intensive properties in the following list is respectively Mass, temperature, pressure, enthalpy, heat capacity, internal energy, density

What is the angle of banking of a railway track of radius of curvature \(250 \mathrm{~m}\), if the maximum velocity of the train is \(90 \mathrm{~km} / \mathrm{h}\) ? \(\left(g=10 \mathrm{~ms}^{-2}\right)\)

If \(\int \frac{\sin ^2 \alpha-\sin ^2 x}{\cos x-\cos \alpha} d x=f(x)+A x+B\) and \(B \in R\), then

If the sum of the slopes of the lines given by \(4 x^2+2 \lambda x y-7 y^2=0\) is equal to the product of the slopes, then \(\lambda\) is equal to

A nucleus with atomic mass number ' \(A\) ' produces another nucleus by loosing 2 alpha particles. The volume of the new nucleus is 60 times that of the alpha particle. The atomic mass number A of the original nucleus is

In an adsorption experiment, a graph between $\log (\mathrm{x}/\mathrm{m})$ versus $\mathrm{logP}$ was found to be linear with a slope of $45°$. The intercept on $\log (\mathrm{x}/\mathrm{m})$ axis was found to be $0.3010$. The amount of gas adsorbed per gram of charcoal under a pressure of $0.5 \mathrm{atm}$ is