If a = (21) - (20) and b = (18) - (17), then - Vidyadip

MCQ

If $a = \sqrt {(21)} - \sqrt {(20)} $ and $b = \sqrt {(18)} - \sqrt {(17),} $ then

A
$a = b$
B
$a + b = 0$
C
$a > b$
✓
$a < b$

✓

Answer

Correct option: D.

$a < b$

d
(d) $a - b = \sqrt {21} - \sqrt {20} - \sqrt {18} + \sqrt {17} $

= $(\sqrt {21} - \sqrt {18} ) - (\sqrt {20} - \sqrt {17} )$

= ${{(\sqrt {21} - \sqrt {18} )(\sqrt {21} + \sqrt {18} )} \over {\sqrt {21} + \sqrt {18} }} - {{20 - 17} \over {\sqrt {20} + \sqrt {17} }}$

= $3\,\left[ {{1 \over {\sqrt {21} + \sqrt {18} }} - {1 \over {\sqrt {20} + \sqrt {17} }}} \right]$

= ${{3\,[\sqrt {20} + \sqrt {17} - \sqrt {21} - \sqrt {18} ]} \over {(\sqrt {21} + \sqrt {18} )\,(\sqrt {20} + \sqrt {17} )}}$

= ${{3\,[(\sqrt {20} - \sqrt {21} ) + (\sqrt {17} - \sqrt {18)} ]} \over {(\sqrt {21} + \sqrt {18} )\,(\sqrt {20} + \sqrt {17} )}}$

= ${{ - 3\,[(\sqrt {21} - \sqrt {20} ) + (\sqrt {18} - \sqrt {17} )} \over {(\sqrt {21} + \sqrt {18} )\,(\sqrt {20} + \sqrt {17} )}} < 0$,

$\therefore a < b$.

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