Circle 4 Question 13
13. For each natural number $k$. Let $C _k$ denotes the circle with radius $k$ centimetres and centre at 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 continue 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 $n=\ldots$
(1997, 2M)
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Answer:
Correct Answer: 13. $n=7$
Solution:
- It is given that, $C _1$ has centre $(0,0)$ and radius 1 .
Similarly, $C _2$ has centre $(0,0)$ and radius 2 and $C _k$ has centre $(0,0)$ and radius $k$.
Now, particle starts it motion from $(1,0)$ and moves 1 radian on first circle then particle shifts from $C _1$ to $C _2$. After that, particle moves 1 radian on $C _2$ and then particle shifts from $C _2$ to $C _3$. Similarly, particle move on $n$ circles.
Now, $n \geq 2 \pi$ because particle crosses the $X$-axis for the first time on $C _n$, then $n$ is least positive integer.
Therefore, $n=7$.
