If x=16−15, y=22−21, z=31−30 , then

x=16−15, y=22−21, z=31−30x=(16−15)(16+15)(16+15); =(16)2−(15)216+15; =116+15y=(22−21)(22+21)(22+21); y=(22)2−(21)222+21; y=22−2122+21; y=122+21z=(31−30)(31+30)(31+30); z=(31)2−(30)231+30; z=131+30∴116+15>122+21>131+3016−15>22−21>31−30⇒x>y>z⇒z<y<x

If x=16−15, y=22−21, z=31−30 , then

x=16−15, y=22−21, z=31−30x=(16−15)(16+15)(16+15); =(16)2−(15)216+15; =116+15y=(22−21)(22+21)(22+21); y=(22)2−(21)222+21; y=22−2122+21; y=122+21z=(31−30)(31+30)(31+30); z=(31)2−(30)231+30; z=131+30∴116+15>122+21>131+3016−15>22−21>31−30⇒x>y>z⇒z<y<x

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If $x = \sqrt{16} - \sqrt{15}, y = \sqrt{22} - \sqrt{21}, z = \sqrt{31} - \sqrt{30}$ , then

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$x < y < z$

$y < z < x$

$z < x < y$

$z < y < x$

answer is D.

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Detailed Solution

$\begin{array}{l} x = \sqrt{16} - \sqrt{15}, y = \sqrt{22} - \sqrt{21}, z = \sqrt{31} - \sqrt{30} \\ x = \frac{(\sqrt{16} - \sqrt{15}) (\sqrt{16} + \sqrt{15})}{(\sqrt{16} + \sqrt{15})}; = \frac{{(\sqrt{16})}^{2} - {(\sqrt{15})}^{2}}{\sqrt{16} + \sqrt{15}}; = \frac{1}{\sqrt{16} + \sqrt{15}} \\ y = \frac{(\sqrt{22} - \sqrt{21}) (\sqrt{22} + \sqrt{21})}{(\sqrt{22} + \sqrt{21})}; y = \frac{{(\sqrt{22})}^{2} - {(\sqrt{21})}^{2}}{\sqrt{22} + \sqrt{21}}; y = \frac{22 - 21}{\sqrt{22} + \sqrt{21}}; y = \frac{1}{\sqrt{22} + \sqrt{21}} \\ z = \frac{(\sqrt{31} - \sqrt{30}) (\sqrt{31} + \sqrt{30})}{(\sqrt{31} + \sqrt{30})}; z = \frac{{(\sqrt{31})}^{2} - {(\sqrt{30})}^{2}}{\sqrt{31} + \sqrt{30}}; z = \frac{1}{\sqrt{31} + \sqrt{30}} \\ ∴ \frac{1}{\sqrt{16} + \sqrt{15}} > \frac{1}{\sqrt{22} + \sqrt{21}} > \frac{1}{\sqrt{31} + \sqrt{30}} \\ \sqrt{16} - \sqrt{15} > \sqrt{22} - \sqrt{21} > \sqrt{31} - \sqrt{30} \\ \Rightarrow x > y > z \Rightarrow z < y < x \end{array}$