A) \[2\sqrt{3}+2-\sqrt{5}\]
B) \[\sqrt{3}+2-2\sqrt{5}\]
C) \[\sqrt{3}+4-\sqrt{5}\]
D) \[2\sqrt{3}+2-2\sqrt{5}\]
Correct Answer: A
Solution :
| \[\sqrt{21-4\sqrt{5}+8\sqrt{3}+4\sqrt{15}}=\sqrt{x}+\sqrt{y}-\sqrt{z}\] ?(i) |
| \[\Rightarrow \] \[21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}\] |
| \[={{(\sqrt{x}+\sqrt{y}-\sqrt{z})}^{2}}=x+y+z+2\sqrt{xy}-2\sqrt{yz}\]\[-2\sqrt{zx}\] |
| Comparing LHS and RHS |
| \[x+y+z=21\] ?(ii) |
| \[2\sqrt{xy}=8\sqrt{3}\] |
| \[\Rightarrow \] \[\sqrt{xy}=4\sqrt{3}\] ?(iii) |
| \[2\sqrt{yz}=4\sqrt{5}\] |
| \[\Rightarrow \]\[\sqrt{yz}=2\sqrt{5}\] ?(iv) |
| \[2\sqrt{zx}=4\sqrt{15}\] |
| \[\Rightarrow \]\[\sqrt{zx}=2\sqrt{15}\] ?(v) |
| From Eq. (iii) \[\times \] (iv) \[\times \] (v), we get |
| \[x\times y\times z=16\times 3\times 5=240\] |
| \[\Rightarrow \]\[\sqrt{xyz}=4\sqrt{15}\] ?(vi) |
| From Eq. (vi) \[\div \] (iv), we get |
| \[\sqrt{x}=4\sqrt{15}\div 2\sqrt{5}=2\sqrt{3}\] |
| So that, |
| \[\sqrt{y}=4\sqrt{15}\div 2\sqrt{15}=2\] |
| \[\sqrt{z}=4\sqrt{15}\div 4\sqrt{3}=\sqrt{5}\] |
| Putting in Eq. (ii), we get |
| LHS \[x+y+z=12+4+5=21\]RHS |
| From Eq. (i), |
| Required value \[=\sqrt{x}+\sqrt{y}-\sqrt{z}=2\sqrt{3}+2-\sqrt{5}\] |
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