Finding Integral Part of Numbers of the Form a^b Where a Is Irrational

Q.

The coefficient of $x^4$ in ${\left[\left(\frac{x}{2}\right)-\left(\frac{3}{x^2}\right)\right]}^{10}$ is

1. $\frac{405}{256}$

2. $\frac{504}{259}$

3. $\frac{450}{263}$

4. None of these

Q.

When $9$ Superscript two-thirds is written in the simplest radical form, which value remains under the radical?

Q. The value of $\frac{1}{1!(n-1)!}+\frac{1}{3!(n-3)!}+\dots +\frac{1}{(n-1)!\left(1\right)!}$, where $n$ is even positive integer, is

1. $\frac{2^n}{(n-1)!}$
2. $\frac{2^{n-1}}{n!}$
3. $\frac{2^{n+1}}{n!}$
4. $\frac{2^n}{(n+1)!}$

Q. If $n$ be a positive integer and $(7+4\sqrt{3}{)}^n=p+\beta$, where $p$ is a positive integer and $\beta$ is a proper fraction, then

1. $(1-\beta )(p+\beta )=2$
2. $(1-\beta {)}^{1/n}+(p+\beta {)}^{1/n}=14$
3. $(1-\beta )(p+\beta )=1$
4. $(1-\beta {)}^{1/n}+(p+\beta {)}^{1/n}=8\sqrt{3}$

Q.

If m is a positive integer, then $\left[\right(\sqrt{3}+1{)}^{2m}]+1$, where [x] denotes greatest integer $\le x$, is divisible by

1. $2^m$
2. $2^{m+1}$
3. $2^{m+1-2}$
4. $2^{2m}$

Q.

The number of terms which are free from radical signs in the expansion of $(y^{\frac{1}{5}}+x^{\frac{1}{10}}{)}^{55}$ is

1. 5

2. 6

3. 7

4. None of these

Q. The last two digits of the number $(23{)}^{14}$ are

1. $01$
2. $05$
3. $03$
4. $09$

Q. If $p=(8+3\sqrt{7}{)}^n$ and $f=p-\left[p\right]$, then the value of $p(1-f)$ is
(where [.] denotes the greatest integer function)

1. $1$
2. $2$
3. $2^n$
4. $2^{2n}$

Q.

Find the greater number in 300! and $\sqrt{{300}^{300}}$

1. 300!

2. $\sqrt{{300}^{300}}$

3. 300! = $\sqrt{{300}^{300}}$

4. Can't be determined.