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For each natural number k,let Ck denote the circle with radius k centimetres and centre at the origin .on the circle Ck ,a particle moves k centimetres in the counter clockwise direction .after completing it`s motion on Ck ,the particle moves on CK+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 positive direction of the x axis for the first time on the circle Cn , then the value of n is equal to(a) 6. (b)5. (c)7. (d)3.
The path of the particle is shown by bold line segments and arcs. It is given that on the circle Ckof radiuskcentimetres the particle moveskcentimetres. Therefore, angular displacement onkth circle is given byThus, angular displacement on each circle is 1 radian.If the particle crosses thex-axis for the first time on circle Cn, thenTotal angular displacement =nradians.Clearly,
(C)7Th path of the particle is shown by bold line segments and arcs . It is given that on the circle c_k of radius k cm the particle moves k cm. Therefore angular displacement on kth circle is given by the circle already drawn in the previous answer . Therefore:Theta=k/k radians or 1 radian. Thus,angular displacement on eaxh circle is 1 radian.If the particle crosses the x-axis for the first time on circle c_n then;Total angular displacement=n radiansAs the particle crosses the positive direction of the X axis for the first time on the nth circle c_n.total angular displacement>2πradians Thus n>2π and so n= 7(C)is the correct answer
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