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For each natural number k, let Ck denote...

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

Text Solution

Verified by Experts

The correct Answer is:
n = 7
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)` ahs 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 paricle shifts from `C_(2) " to " C_(3)`. Similarly, particle move on n circles.
Now, `n ge 2pi` because particle crosses the X-axis for the first time on `C_(n)`, then n is least positive integer.
therefore, n = 7.
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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 centrimetres 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 ="__________________" .

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Knowledge Check

  • For each positve real number k, let C _(k) denotes the circle with centre at origin and radius k units. On a circle C _(k) particle alpha moves k units in the counter-clockwise direction. After completing its motion on C_(k), the particle moves onto the circle C _(k+1) in same well defined manner, where r gt 0. The motion of particle continues in this manner. Let the particle starts from the point A (2,0) and moves pi/2 units on circle C_(2) in the counter clockwise direction, then moves on circle C_(3) along tangential peth, let this straight the (tangential path traced by particle) intersect the circle C_(3) at points A and B. then tangents at A and B intersect at

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  • For each positve real number k, let C _(k) denotes the circle with centre at origin and radius k units. On a circle C _(k) particle alpha moves k units in the counter-clockwise direction. After completing its motion on C_(k), the particle moves onto the circle C _(k+1) in same well defined manner, where r gt 0. The motion of particle continues in this manner. Let k in l' and r=1, particle moves in the radial direction from C _(k) to C_(k+1). If particle starts from the point (-1, 0), then

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  • For each positve real number k, let C _(k) denotes the circle with centre at origin and radius k units. On a circle C _(k) particle alpha moves k units in the counter-clockwise direction. After completing its motion on C_(k), the particle moves onto the circle C _(k+1) in same well defined manner, where r gt 0. The motion of particle continues in this manner. Let k in l^(+) and r in R, particle moves tangentially from C_(k) to C_(k+r') such that length of tangent is equal to k units itself. If particle starts from teh point (2,0), then

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