JEE AITS probs,need help!

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

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

For number 1, this is what I thought of:

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

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

Andy Hayes - 5 years, 2 months ago

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

Keshav Tiwari - 5 years, 2 months ago

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

Rishabh Jain - 5 years, 2 months ago

Best of luck for tmmrw!

Harsh Shrivastava - 5 years, 2 months ago

@Harsh Shrivastava – Thanks..... :-)

Rishabh Jain - 5 years, 2 months ago

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

Adarsh Kumar - 5 years, 2 months ago

@Adarsh Kumar – A definite integral is bounded between 2 summations

Suhas Sheikh - 2 years, 11 months ago

Yeah please do!

Adarsh Kumar - 5 years, 2 months ago

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 - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$