For each natural number k, let `C_k` denotes the circle radius k centimeters 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 the positive direction of the x-axis for first time on the circle `C_n`,then n equal to

Explanation

It is given that on the circle` C_k`​ of radius kcm, the particle moves kcm.

Therefore, angular displacement (theta) on kth circle is given by-

θ=(distance travelled by the particle​)/(radius)=k/k​ radian=1 radian

Thus angular displacement on each circle is 1 radian.

If the particle crosses the x-axis for the first time on circle Cn​, then-

Total angular displacement =n radians,
i.e.,
We have to move to the circle with radius n to cover the angle of 2(pi)n​.

∴n>2pi

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