The greatest integer less than or equal to -3-16 is

The correct option is D 415 (√(3) + 1)^6 + (√(3) 1)^6 = 2[ ^6C_0 (√(3))^6 + ^6C_2 (√(3))^4 + ^6C_4 (√(3))^2 + ^6C_6] = 416 Let (√(3) + 1)^6 = I + F where I ...

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Byju's Answer

Standard XI

Mathematics

Finding Integral part of numbers of the form a^b where a is irrational

The greatest ...

Question

The greatest integer less than or equal to $(\sqrt{3} + 1)^{6}$ is

416

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414

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417

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415

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Solution

The correct option is D
415

$(\sqrt{3} + 1)^{6}$ + $(\sqrt{3} - 1)^{6}$ = 2 $[^{6} C_{0} (\sqrt{3})^{6} +^{6} C_{2} (\sqrt{3})^{4} +^{6} C_{4} (\sqrt{3})^{2} +^{6} C_{6}]$ = 416

Let $(\sqrt{3} + 1)^{6}$ = I + F where I is integral part and F is functional part

Let $(\sqrt{3} - 1)^{6}$ = G

0 < F < 1;0 < G < 1 $\Rightarrow$ 0 < F + G < 2 $\Rightarrow$ F + G = 1

I + F + G = 416 $\Rightarrow$ I + 1 = 416 $\Rightarrow$ I = 415

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The greatest integer less than or equal to (√3+1)6 is

The correct option is D 415 (√3+1)6 + (√3−1)6 = 2[6C0(√3)6+6C2(√3)4+6C4(√3)2+6C6] = 416 Let (√3+1)6 = I + F where I is integral part and F is functional part Let (√3−1)6 = G 0 < F < 1;0 < G < 1 ⇒ 0 < F + G < 2 ⇒ F + G = 1 I + F + G = 416 ⇒ I + 1 = 416 ⇒ I = 415

The greatest integer less than or equal to $(\sqrt{3} + 1)^{6}$ is