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New in IIT-JEE 2010 IITs, IIMs to dump grades (CGPA) for percentage? Which engineering field to choose�������part 1 Tips on how to choose your engineering college Top 10 engineering colleges with ...
Ques: 25 Show that
one can use a composition of trigonometry buttons such as,
and to replace the broken reciprocal button on a
we have for any x >0,
as desired. It is not difficult to check that will also do the trick.%
Ques: 26 Prove that
in a triangle ABC,
From the law of sines and the
sum-to-product formulas, we have
Ques: 27 Let a, b,
c be real numbers, all different from âˆ'1 and 1, such that a +b+c
= abc. Prove that
Let where for all integers k. The condition a +
b + c = abc translates to tan(x + y + z) = 0, as indicated in notes
Question 13(1). From the double-angle
formulas, it follows that
using a similar argument to the one in Question 13(1). This implies that
and the conclusion follows.
Ques: 28 Prove that
a triangle ABC is isosceles if and only
By the extended law of sines, a =
2R sin A, b = 2R sin B, and c = 2R sin C. The desired identity is
The last equality simplifies to
which in turn is equivalent to
by Question 7. The conclusion now follows.
Ques: 29 Prove that
is an irrational number.
Assume, for the sake of contradiction, that
is rational. Then so is Using the identity
we obtain by strong induction that is rational for all integers But this is clearly false, because, for example, is not rational, yielding a contradiction.
Note: For the reader not familiar with the idea of induction. We can reason in the following way. Under the assumption that both and are rational, relation (âˆ-) implies that is rational, by setting n = 2 in the relation (âˆ-). Similarly, by the assumption that both and are rational, relation (âˆ-) implies that is rational, by setting n = 4 in the relation (âˆ-). And so on.We conclude that is rational, for all positive integers n, under the assumption that is rational.
Ques: 30 Prove that
Multiplying the two sides of the inequality by
we obtain the equivalent form
But this follows from Cauchy-Schwarz inequality because according to this inequality, the left-hand side is greater than or equal to
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