Question

For each natural number $k$ ,let $C_k$ denote the circle with radius $k$ centimeters and centre at the origin $O$. on the circle $C_k$ a 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 (away from centre). The motion of the particle continues in this manner. The particle starts at $(1,0)$. If the particle crosses the positive direction of x-axis for the first time on the circle $C_n$, then $n=$

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

The correct option is D $7$

The motion of the particle on the first four circles is shown with bold line in the figure $16.22$.
Note that on every circle the particle travels just one radian.
The particle crosses the positive direction of x-axis first time on $C_n$,.
where $n$ is the least positive integer such that $n>2\pi \Rightarrow n=7[note:2\pi \approx 6.28]$.