$A$ centigrade and a Fahrenheit thermometer are dipped in boiling water. The temperature is lowered until the Fahrenheit thermometer registers $140^{\circ}F$. What is the fall in temperature in $^{\circ}C$ as registered by the centigrade thermometer?

The equation of the pair of lines passing through the point $(2, 1)$ and perpendicular to the pair of lines $4xy + 2x + 6y + 3 = 0$ is:

The angle between the lines represented by $y^2 \sin^2 \theta - xy \sin^2 \theta + x^2(\cos^2 \theta - 1) = 0$ is

The plates of a parallel plate capacitor are charged up to $100\, V$. $A$ $2\, mm$ thick plate is inserted between the plates. To maintain the same potential difference,the distance between the capacitor plates is increased by $1.6\, mm$. The dielectric constant of the plate is:

The number of integral values of $k$,for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution,is

The most general value of $\theta$ which satisfies both the equations $\tan \theta = -1$ and $\cos \theta = \frac{1}{\sqrt{2}}$ is