The fossil bone has a ${}^{14}C:{}^{12}C$ ratio,which is $\left( \frac{1}{16} \right)$ of that in a living animal bone. If the half-life of ${}^{14}C$ is $5730 \text{ years}$,then the age of the fossil bone is ......... $\text{years}$.

Let $R_1$ be a relation defined by $R_1 = \{(a, b) | a \geq b, a, b \in \mathbb{R}\}$. Then $R_1$ is

The pulley and spring are massless,and friction is absent everywhere. The $5\,kg$ block is released from rest. The speed of the $5\,kg$ block when the $2\,kg$ block leaves contact with the ground is (take the force constant of the spring $K = 40\,N/m$ and $g = 10\,m/s^2$):

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

$\int_0^\pi \frac{x \tan x}{\sec x+\tan x} d x=$

Given below are two statements regarding silent heart attack.
Statement $I$: This type of heart attack shows symptoms like chest pain and shortness of breath.
Statement $II$: In this type of heart attack,usually person shows very mild symptoms and thus it is ignored.
In the light of above two statements,choose the correct answer from options given below.