If $\underset{0}{\overset{k}{\int }}\frac{dx}{2+18x^2}=\frac{\pi }{24}$ , then the value of $k$ is

If \(A+B+C=180^{\circ}\), then the value of
\(\tan \left(\frac{\mathrm{A}}{2}\right) \tan \left(\frac{\mathrm{B}}{2}\right)+\tan \left(\frac{\mathrm{B}}{2}\right) \tan \left(\frac{\mathrm{C}}{2}\right)+\tan \left(\frac{\mathrm{C}}{2}\right) \tan \left(\frac{\mathrm{A}}{2}\right)\) is

The equation of tangent to the curve \(y^{2}=a x^{2}+b\) at point \((2,3)\) is \(y=4 x-5\), then the values of \(a\) and \(\bar{b}\) are

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors and \(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \quad\) then \(2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\)

If \(4 \hat{i}+7 \hat{j}+8 \hat{k}, 2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(2 \hat{i}+5 \hat{j}+7 \hat{k}\) are the position vectors of the vertices \(\mathrm{A}, \mathrm{B}\) and C , respectively, of triangle ABC , then the position vector of the point in which bisector of \(\angle B\) meets \(C A\) is

The general solution of the differential equation \(x^2+y^2-2 x y \frac{d y}{d x}=0\) is (where \(C\) is a constant of integration.)

If \(\cos ^{-1} x-\cos ^{-1} \frac{y}{3}=\alpha\), where \(-1 \leq x \leq 1\), \(-3 \leq y \leq 3, x \leq \frac{y}{3}\), then for all \(x, y\) \(9 x^2-6 x y \cos \alpha+y^2\) is equal to

Two small drops of mercury each of radius ' \(R\) ' coalesce to from a large single drop. The ratio of the total surface energies before and after the change is

The kinetic energy of an electron having de-Broglie wavelength $\lambda$ is ( $h=$ Planck's constant, $m=$ mass of electron)

An ice cube of edge \(1 \mathrm{~cm}\) melts in a gravity free container. The approximate surface area of water formed is (water is in the form of spherical drop)

The charge on each capacitor when a voltage source of \(15 \mathrm{~V}\) is connected in the circuit as shown, is

What is the correct relation between rate constant and half-life for zero order reaction

Consider the Doppler effect in two cases. In the first case, an observer moves towards a stationary source of sound with a speed of \(50 \mathrm{~m} / \mathrm{s}\). In the second case, the observer is at rest and the source moves towards the observer with the same speed of \(50 \mathrm{~m} / \mathrm{s}\). Then the frequency heard by the observer will be [velocity of sound in air \(=330 \mathrm{~m} / \mathrm{s}\).]