Subject: Mathematics — Circles
Question: For each natural number $k$, let $C_{k}$ denote the
circle with radius $k$ centimetres and centre at the origin. On the
circle $C_{k},\alpha$-particle moves $k$ centimetres in the
counter-clockwise direction. After completing its motion on $C_{k}$, the
particle moves to $C_{k + 1}$ in the radial direction. The motion of the
particle continues in this manner. The particle starts at $(1,0)$. If
the particle crosses the positive direction of the x-axis for the first
time on the circle $C_{n}$ then $n =$ $\_\_\_\_$
Answer: $n = 7$
Solution: [IMAGE] The radius of circle $C_{1}$ is $1\text{ }cm,C_{2}$ is 2 cm and so on.
It starts from $A_{1}(1,0)$ on $C_{1}$, moves a distance of 1 cm on
$C_{1}$ to come to $B_{1}$. The angle subtended by $A_{1}B_{1}$ at the
centre will be $\frac{\mathcal{l}}{r} = \frac{1}{1} = 1$ radian.\
From $B_{1}$ it moves along radius, $OB_{1}$ and comes to $A_{2}$ on
circle $C_{2}$ of radius 2. From $A_{2}$ it moves on $C_{2}$ a
distance 2 cm and comes to $B_{2}$. The angle subtended by
$A_{2}B_{2}$ is again as before 1 radian. The total angle subtended at
the centre is 2 radians. The process continues. In order to cross the
$x$-axis again, it must describe $2\pi$ radians i.e.
$2,\frac{22}{7} = 6.7$ radians. Hence it must be moving on circle
$C_{7}$.
$$\therefore n = 7$$