Integer Part or Limit Involving Conjugate Surd Binomial Expansions

The question involves computing the integer part (floor), a limit, or an exact expression for (a+√b)^n by exploiting the conjugate (a−√b)^n, where the conjugate is small.

jee-main 2012 Q66 View

If $n$ is a positive integer, then $(\sqrt{3}+1)^{2n} - (\sqrt{3}-1)^{2n}$ is
(1) an irrational number
(2) an odd positive integer
(3) an even positive integer
(4) a rational number other than positive integers

jee-main 2020 Q55 View

If $\{ \mathrm { p } \}$ denotes the fractional part of the number p , then $\left\{ \frac { 3 ^ { 200 } } { 8 } \right\}$ is equal to
(1) $\frac { 5 } { 8 }$
(2) $\frac { 7 } { 8 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 8 }$

jee-main 2021 Q64 View

The lowest integer which is greater than $\left( 1 + \frac { 1 } { 10 ^ { 100 } } \right) ^ { 10 ^ { 100 } }$ is
(1) 3
(2) 4
(3) 2
(4) 1