Question
For each $$k\, \in \,N,$$ let $${C_k}$$ denote the circle whose equation is $${x^2} + {y^2} = {k^2}.$$ On the circle $${C_k},$$ a particle moves $$k$$ units in the anticlockwise 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 $$\left( {1,\,0} \right).$$ If the particle crosses the positive direction of the $$x$$-axis for the first time on the circle $${C_n}$$ then $$n$$ is :
A.
$$7$$
B.
$$6$$
C.
$$2$$
D.
none of these
Answer :
$$7$$
Solution :
The angle described anticlockwise before leaving for $${C_n}$$ is $$\left\{ {1 + 1 + .....{\text{to}}\,\left( {n - 1} \right){\text{times}}} \right\}$$ radians and that before leaving for $${C_{n + 1}}$$ is $$\left\{ {1 + 1 + .....{\text{to}}\,n\,{\text{times}}} \right\}$$ radians
$$\therefore \,\,\,\left( {n - 1} \right) < 2\pi < n\,\,\,\, \Rightarrow n - 1 < \frac{{44}}{7} < n$$
The angle described anticlockwise before leaving for $${C_n}$$ is $$\left\{ {1 + 1 + .....{\text{to}}\,\left( {n - 1} \right){\text{times}}} \right\}$$ radians and that before leaving for $${C_{n + 1}}$$ is $$\left\{ {1 + 1 + .....{\text{to}}\,n\,{\text{times}}} \right\}$$ radians
$$\therefore \,\,\,\left( {n - 1} \right) < 2\pi < n\,\,\,\, \Rightarrow n - 1 < \frac{{44}}{7} < n$$