Rationalise the denominators of the following:
(i) \(\frac{1 }{ \sqrt{7 }}\)
(ii) \(\frac{1 }{ \sqrt{7 }-\sqrt6}\)
(iii) \(\frac{1 }{ \sqrt{5}+\sqrt2}\)
(iv) \(\frac{1 }{ \sqrt{7}-2}\)
(i) \(\frac{1 }{ \sqrt{7 }}\) =\(\frac{1}{\sqrt7}=\frac{1}{\sqrt7} ×\frac{ √7 }{√7}\)
= \(\frac{ √7 }{√7}\)
(ii) \(\frac{1 }{ \sqrt{7 }-\sqrt6}\)
\(= \frac{1}{ (√7 + √6)} ×\frac{(√7 - √6)}{(√7 + √6)}\)
= \(\frac{(√7 - √6)}{7 - 6}=\sqrt7+\sqrt6\)
(iii) \(\frac{1 }{ \sqrt{7}-2}\)
\(\frac{\sqrt5+\sqrt2}{3}\)
(iv) \(\frac{1 }{ \sqrt{7}-2}\)
\(=\frac{\sqrt7+2}{7-2}=\frac{\sqrt7+2}{3}\)
For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)2 is equal to _____.
Use these adverbs to fill in the blanks in the sentences below.
awfully sorrowfully completely loftily carefully differently quickly nonchalantly
(i) The report must be read ________ so that performance can be improved.
(ii) At the interview, Sameer answered our questions _________, shrugging his shoulders.
(iii) We all behave _________ when we are tired or hungry.
(iv) The teacher shook her head ________ when Ravi lied to her.
(v) I ________ forgot about it.
(vi) When I complimented Revathi on her success, she just smiled ________ and turned away.
(vii) The President of the Company is ________ busy and will not be able to meet you.
(viii) I finished my work ________ so that I could go out to play