The path length of oscillation of a simple pendulum of length $1 \ m$ is $16 \ cm$. Its maximum velocity is $(g = \pi^2 \ m/s^2)$

Suppose $A$ and $B$ are two events such that $P(A \cap B) = \frac{3}{25}$ and $P(B - A) = \frac{8}{25}$. Then,$P(B)$ is equal to

In a playground,there is a merry-go-round of mass $120 \ kg$ and radius $4 \ m$. The radius of gyration is $3 \ m$. $A$ child of mass $30 \ kg$ runs at a speed of $5 \ m/s$ tangent to the rim of the merry-go-round when it is at rest and then jumps on it. Neglecting friction,find the angular velocity of the merry-go-round and child in $rad/s$.

On which of the following lines does the point of intersection of the line $\frac{x - 4}{2} = \frac{y - 5}{2} = \frac{z - 3}{1}$ and the plane $x + y + z = 2$ lie?

The standard entropies of $X_2$,$Y_2$,and $XY_3$ are $60$,$40$,and $50 \ J \ K^{-1} \ mol^{-1}$ respectively. For the reaction $\frac{1}{2}X_2 + \frac{3}{2}Y_2 \to XY_3$,$\Delta H = -30 \ kJ$. What will be the temperature in $K$ at equilibrium?

Particles of masses $m, 2m, 3m, \dots, nm$ grams are placed on the same line at distances $l, 2l, 3l, \dots, nl$ cm from a fixed point. The distance of the centre of mass of the particles from the fixed point (in cm) is