MAT Exam 2021 Paper

Concept:Probability is the ratio of the favourable area to the total area of the circle.Explanation:Let the radius of the circle be $r$.Let the distance of the selected point from the centre be $d$.The point is closer to the centre than the boundary when $d < r - d$.This simplifies to $2d < r$, so $d < \frac{r}{2}$.Thus, the favourable region is the inner circle of radius $\frac{r}{2}$.Favourable area $= \pi \left(\frac{r}{2}\right)^2 = \frac{\pi r^2}{4}$.Total area of the circle $= \pi r^2$.Required probability $= \frac{\frac{\pi r^2}{4}}{\pi r^2} = \frac{1}{4}$.Answer:$\frac{1}{4}$, i.e. option C.