Try this beautiful problem on Calculus, useful for ISI B.Stat Entrance.

Problem on Calculus | ISI B.Stat Entrance | Problem 696


If k is an integer such that lim \(\{{cos}^n(k\pi/4) – {cos}^n(k\pi/6)\} = 0\),
then

  • (a) k is divisible neither by 4 nor by 6
  • (b) k must be divisible by 12, but not necessarily by 24
  • (c) k must be divisible by 24
  • (d) either k is divisible by 24 or k is divisible neither by 4 not by 6

Key Concepts


Calculus

Limit

Trigonometry

Check the Answer


But try the problem first…

Answer: (d)

Source
Suggested Reading

TOMATO, Problem 694

Challenges and Thrills in Pre College Mathematics

Try with Hints


First hint

There are four options ,at first we have to check each options…..

If k is divisible by 24 then cos(kπ/4) = cos(kπ/6) = 1
\(\Rightarrow\) The limit exists and equal to RHS i.e. 0
If k is not divisible by 4 or 6 then cos(kπ/4), cos(kπ/6) both <1

Can you now finish the problem ……….

Final Step

Therefore ,

lim cosn(kπ/4), cosn(kπ/6) = 0. so we may say that
\(\Rightarrow \)The equation holds.

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