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2017-02-06 Advanced

         

A uniform solid right-circular cone of mass \(M\) and radius \(R\) is kept on a rough horizontal floor (coefficient of friction \(\mu\)) on its circular base. It is spun with initial angular velocity \(\omega_{0}\) about its symmetry axis. Neglecting toppling effects (if any), find the time after which it stops spinning.

\(\)
Details and Assumptions:

  • Take the moment of inertia of the cone about its symmetry axis as \(I\).

If you are looking for more such twisted questions, Twisted problems for JEE aspirants is for you!

\(x,y,z,t\) are all integers satisfying the following system of equations: \[ \begin{cases} \begin{align} xz-2yt&=3 \\ xt+yz&=1. \end{align} \end{cases}\] Find \( x^2+y^2+z^2+ t^2 \).

A uniform wire of resistance \(R\) is cut into four circular rings of equal radius. Rings are then connected such that their centers lie on the vertices of a square (as illustrated in the figure above).

If the equivalent resistance between \(A\) and \(B\) can be expressed as \(\frac{aR}{b}\), where \(a\) and \(b\) are coprime positive integers, determine the value of \(a+b\).

\[S= \frac{1^{2}}{10}+\frac{2 \times 1^{2}+2^{2}}{10^{2}}+\frac{3 \times 1^{2}+2 \times 2^{2}+3^{2}}{10^{3}}+\cdots \]

The sum \(S\) defined above is an infinite sum whose \(n^\text{th}\) term is \[\dfrac{n \times 1^{2}+(n-1) \times 2^{2}+(n-2) \times 3^{2} + \cdots + n^{2}}{10^{n}}\] for \(n=1,2,\ldots \).

If \(S\) can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


Inspiration.

What is the maximum number of regions a plane may be cut into by 2017 ellipses?

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