All the springs in figures $(a)$,$(b)$,and $(c)$ are identical,each having a force constant $K$. The mass attached to each system is $m$. If $T_a, T_b$,and $T_c$ are the time periods of oscillations of the three systems respectively,then:

The displacement of a particle from its mean position (in metre) is given by $y = 0.2 \sin(10\pi t + 1.5\pi) \cos(10\pi t + 1.5\pi)$. The motion of the particle is:

Let $\overrightarrow{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$.
Assertion $(A)$: The identity $|\overrightarrow{a} \times \hat{i}|^2+|\overrightarrow{a} \times \hat{j}|^2+|\overrightarrow{a} \times \hat{k}|^2=2|\overrightarrow{a}|^2$ holds for $\overrightarrow{a}$.
Reason $(R)$: $\overrightarrow{a} \times \hat{i}=a_3 \hat{j}-a_2 \hat{k}$, $\overrightarrow{a} \times \hat{j}=a_1 \hat{k}-a_3 \hat{i}$, and $\overrightarrow{a} \times \hat{k}=a_2 \hat{i}-a_1 \hat{j}$.
Which of the following is correct?

The ratio of the magnetic field at the centre of a current-carrying circular wire and the magnetic field at the centre of a square coil made from the same length of wire will be

The product of the perpendicular distances from the origin to the pair of straight lines $12x^2+25xy+12y^2+10x+11y+2=0$ is

For a ray of light,the critical angle is minimum when it travels from