JEE AITS probs,need help!

1]If median AD of a triangle ABC makes angle 30 degrees with side BC then (cotBcotC)2(\cot B-\cot C)^2 is equal to?Options:6,9,12,15. 2]Tn=r=2n3n1rnr2+n2,Sn=r=2n+13nrnr2+n2T_n=\sum_{r={2n}}^{3n-1}\dfrac{rn}{r^2+n^2},S_n=\sum_{r={2n+1}}^{3n}\dfrac{rn}{r^2+n^2}.Then,Tn,Sn> or <ln22T_n,S_n >\ or\ < \dfrac{\ln 2}{2}.

Note by Adarsh Kumar
3 years, 6 months ago

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For number 1, this is what I thought of:

The way the question is worded implies that the result of (cotBcotC)2(\cot{B}-\cot{C})^2 is always the same regardless of what the values of mBm\angle{B} and mCm\angle{C} are. You can use this to your advantage.

Let mADC=30m\angle{ADC}=30^\circ, and let mC=90m\angle{C}=90^\circ. Now cotC=0\cot{C}=0, and you can use 30/60/90 triangle relationships to find cotB\cot{B}. I hope this helps!

Andy Hayes - 3 years, 6 months ago

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The second one is based on a Jee 2008 question . I'll post the solution next week if you want me to. :)

Keshav Tiwari - 3 years, 6 months ago

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Yep... Its a JEE problem based on Riemann Sums....

Rishabh Jain - 3 years, 6 months ago

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Best of luck for tmmrw!

Harsh Shrivastava - 3 years, 6 months ago

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@Harsh Shrivastava Thanks..... :-)

Rishabh Jain - 3 years, 6 months ago

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But how can we use that bcoz we don't know that n tends to infinity? And yes,best of luck!

Adarsh Kumar - 3 years, 6 months ago

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@Adarsh Kumar A definite integral is bounded between 2 summations

Suhas Sheikh - 1 year, 2 months ago

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Yeah please do!

Adarsh Kumar - 3 years, 6 months ago

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Is the ans. Tn> ln2/2 and Sn<ln2/2 ?. You should see this video https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/riemann-sums/v/simple-riemann-approximation-using-rectangles

Keshav Tiwari - 3 years, 6 months ago

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