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27 Mar 2023, 11:45
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A
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Difficulty:
95%
(hard)
Question Stats:
39% (03:18) correct
61%
(02:56)
wrong
based on 130
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If \(a\) is the integer portion of \(\sqrt{11+√72}\) and \(b\) is the decimal portion of \(\sqrt{11-√72}.\), what is the value of \(\frac{1}{a-2+\sqrt{2}}-\frac{1}{b}\)?
A. \(-\sqrt{2}\)
B. \(-\sqrt{3} \)
C. \(-1 \)
D. \(-1-\sqrt{5} \)
E. \(-2\)
A. \(-\sqrt{2}\)
B. \(-\sqrt{3} \)
C. \(-1 \)
D. \(-1-\sqrt{5} \)
E. \(-2\)
30 Mar 2023, 00:34
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Bunuel
- \(\sqrt{11+√72}\)
\(\sqrt{11+6\sqrt{2}}\)
\(\sqrt{(3 + \sqrt{2})^2}\)
\(3 + \sqrt{2}\)
As a is the integer portion of \(3 + \sqrt{2}\) \(\approx 3 + \sqrt{2} \approx 4.XXX\)
a = 4 - \(\sqrt{11-√72}\)
\(\sqrt{11-6\sqrt{2}}\)
\(\sqrt{(3 - \sqrt{2})^2}\)
\(3 - \sqrt{2} \approx 3 - 1.414 \approx 1.XXX\)
As b is the decimal portion, we can subtract the integer portion from the equation to get 'b'
b = \(3 - \sqrt{2} - 1 = 2 - \sqrt{2}\)
b = \(2 - \sqrt{2}\)
Substituting the value of 'a' and 'b' in the equation -
\(\frac{1}{a-2+\sqrt{2}}-\frac{1}{b}\)
\(\frac{1}{4-2+\sqrt{2}}-\frac{1}{(2 - \sqrt{2})}\)
\(\frac{1}{2+\sqrt{2}}-\frac{1}{(2 - \sqrt{2})}\)
We can take \((2 - \sqrt{2})*(2 + \sqrt{2})\) as the LCM
\(\frac{(2 - \sqrt{2})-(2 + \sqrt{2})}{(2 - \sqrt{2})*(2 + \sqrt{2})}\)
\(\frac{2 - \sqrt{2}- 2 - \sqrt{2}}{4-2}\)
\(\frac{-2*\sqrt{2}}{2}\)
\(-\sqrt{2}\)
Option A
