Two steel wires of same length $L$ but radii $r$ and $2r$ are connected together end to end and tied to a wall as shown. $A$ force $F$ stretches the combination by $10 \ mm$. How far does the midpoint $A$ move in $mm$?

$A$ thin liquid film formed between a $U$-shaped wire and a light slider supports a weight of $1.5 \times 10^{-2} \ N$ (see figure). The length of the slider is $30 \ cm$ and its weight is negligible. The surface tension of the liquid film is (in $N/m$)

The equation $x^2-5xy+py^2+3x-8y+2=0$ represents a pair of straight lines. If $\theta$ is the angle between them,then $\sin \theta$ is equal to

The half-life of a radioactive isotope $X$ is $50 \, years$. It decays to another element $Y$ which is stable. The two elements $X$ and $Y$ were found to be in the ratio of $1 : 15$ in a sample of a given rock. The age of the rock was estimated to be ........ $years$.

If $arg(z) < 0$,then $arg(-z) - arg(z)$ is equal to

Six faces of an unbiased die are numbered with $2, 3, 5, 7, 11$ and $13$. If two such dice are thrown,then the probability that the sum on the uppermost faces of the dice is an odd number is