If the pair of lines $ax^2 + 2(a + b)xy + by^2 = 0$ represent the diameters of a circle and divide the circle into four sectors such that the area of one sector is three times the area of the other,then:

Two containers of equal volume contain the same gas at pressure $P_1$ and $P_2$ and absolute temperature $T_1$ and $T_2$ respectively. On joining the vessels,the gas reaches a common pressure $P$ and common temperature $T$. The ratio $P/T$ is equal to

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-coplanar vectors,then the vector equation $\overrightarrow{r}=(1-p-q) \overrightarrow{a}+p \overrightarrow{b}+q \overrightarrow{c}$ represents a :

The ratio of the de-Broglie wavelength of an $\alpha$-particle to that of a proton,when both are subjected to the same magnetic field such that the radii of their paths are equal,assuming the magnetic field induction vector $\overrightarrow{B}$ is perpendicular to the velocity vectors of both particles,is:

An aluminium ring $B$ faces an electromagnet $A$. The current $I$ through $A$ can be altered. Which of the following statements is correct?

The diameter of a plano-convex lens is $6 \ cm$ and its thickness at the center is $3 \ mm$. If the speed of light in the material of the lens is $2 \times 10^8 \ m/s$,the focal length of the lens is.......$cm$.