Statement I:The integral part of ${(8 + 3 \sqrt{7})}^{20}$ is an even integer.
Statement II: ${(8 + 3 \sqrt{7})}^{20} + {(8 - 3 \sqrt{7})}^{20}$ is an even integer.

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Statement-I is true, statement-II is true and statement-II is NOT the correct explanation for statement-I

Statement-I is true, statement-II is true and statement-II is correct explanation for statement-I

Statement-I is false, statement-II is true

Statement-I is true, statement-II is false

answer is D.

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Detailed Solution

Let ${(8 + 3 \sqrt{7})}^{20} = l + f,$ where f= fractional part and I = integral part.Also let ${(8 - 3 \sqrt{7})}^{20} = g$
Then 0 < g<l
Here $I + f + g = {(8 + 3 \sqrt{7})}^{20} + {(8 - 3 \sqrt{7})}^{20}$ $= 2 [8^{20} + 20_{C_{2}} {.8}^{18} . {(3 \sqrt{7})}^{2}] + ......$
$\Rightarrow l + f + g$
= even integer
But 0 < f+g<2, So,I+1=even integer( f+g=1)
Therefore I is odd integer.