IIT-JEE 1997Question 1
For each natural number $k$, let $C_k$ denote the circle with radius $k$ centimetres and centre at the origin. On the circle $C_k$, 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. The motion of the particle continues in the 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 $n=$ ______.
Sequence and Series · Miscellaneous
Answer$7$
Solution
Formulas used. (1) Arc length $=$ radius $\times$ angle, so angle $=\frac{\text{arc}}{\text{radius}}$(2) $2\pi\approx6.28$Step 1 - Angle on each circle. On $C_k$ the particle moves $k$ cm on a circle of radius $k$: angle $=\dfrac kk=1$ radian. The radial steps do not change the angle. Step 2 - The path.After finishing $C_n$, the total angle turned is $n$ radians. Step 3 - First crossing. The positive $x$-axis is reached again at angle $2\pi\approx6.28$. After $C_6$ the angle is $6<2\pi$; on $C_7$ it goes from $6$ to $7$, passing $2\pi$. So the crossing happens on $C_7$. Answer: $7$