If a line \(l\) passes through \((k, 2 k),(3 k, 3 k)\) and \((3,1), k \neq 0\), then the distance from the origin to the line \(l\) is

In a $∆ABC$, medians, $AD$ and $BE$ are drawn. If $AD=4$, $\angle DAB=\frac{\pi }{6} \mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\angle ABE=\frac{\pi }{3}$, then the area of the $∆ABC$ is

If \(f\) is defined by \(f(x)=\left\{\begin{array}{l}x \text { for } 0 \leq x < 1 \\ 2-x \text { for } x \geq 1\end{array}\right.\), then at \(x=1, f(x)\) is

The system of linear equations \(x+2 y+z=-3,3 x+3 y-\) \(2 z=-1,2 x+7 y+7 z=-4\) has

In \(\triangle \grave{A B C}\), if \(4 r_1=5 r_2=6 r_3\), then \(\sin ^2 \frac{A}{2}+\sin ^2 \frac{B}{2}+\sin ^2 \frac{C}{2}=\)

Seven white balls and three black balls are randomly arranged in a row. The probability that no two black balls are placed adjacently is

If the latusrectum of an ellipse subtends a right angle at the centre of that ellipse, then the eccentricity of that ellipse is

\(\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots 16\) terms \(=\)

A \(100 \mathrm{~V}\) battery is connected across the series combination of the two capacitors of \(4 \mu \mathrm{F}\) and \(8 \mu \mathrm{F}\). The energy stored in the series combination is

Three charged particles of each mass \(0.1 \mathrm{~g}\) and charge ' \(\mathrm{q}\) ' are suspended from a common rigid point by insulated massless threads of each \(1 \mathrm{~m}\) long. If the three particles are in equilibrium and are located at the corners of an equilateral triangle of side \(3 \mathrm{~cm}\), the charge ' \(q\) ' on each particle is ____ \(\mathrm{nC}\). (The angle made by the line joining the centroid of the triangle and the point of suspension with the vertical is very small).
(Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) and \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\) )

An observer is standing \(500 \mathrm{~m}\) away from a vertical hill. Starting between the observer and the hill, a police van sounding a siren of frequency \(1000 \mathrm{~Hz}\) moves towards the hill with a uniform speed. If the frequency of the sound heard directly from the siren is \(970 \mathrm{~Hz}\), the frequency of the sound heard after reflection from the hill (in Hz) is about, (velocity of sound \(=330 \mathrm{~ms}^{-1}\) ) :

If two progressive sound waves represented by \(y_1=3 \sin 250 \pi t\) and \(y_2=2 \sin 260 \pi \mathrm{t}\) (where displacement is in metre and time is in second) superimpose, then the time interval between two successive maximum intensities is

If \(e_1, e_2\) are respectively the eccentricities of the curves \(9 x^2-16 y^2-144=0\) and \(9 x^2-16 y^2+144=0\), then \(\frac{e_1^2 e_2^2}{e_1^2+e_2^2}=\)