The energy that should be added to an electron,to reduce its de-Broglie wavelength from $10^{-10} \ m$ to $0.5 \times 10^{-10} \ m$,will be

The gravitational force acting on a particle,due to a solid sphere of uniform density and radius $R$,at a distance of $3 R$ from the centre of the sphere is $F_1$. $A$ spherical hole of radius $(R / 2)$ is now made in the sphere as shown in the figure. The sphere with hole now exerts a force $F_2$ on the same particle. Ratio of $F_1$ and $F_2$ is

In a $\triangle ABC$,$\cos \left(\frac{B+2C+3A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$ is equal to

The graphs given below depict the dependence of two reactive impedances $X_1$ and $X_2$ on the frequency of the alternating $e.m.f.$ applied individually to them. We can then say that

The solution set of the inequality $\log_{10}(x^2 - 2x - 2) \le 0$ is

Equimolar solutions of the following were prepared in water separately. Which one of the solutions will record the highest $pH$?