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

Q.

If $R=(6\sqrt{6}+14{)}^{2n+1}$ and f=R-[R], where [.] denotes the greatest integer function, then Rf equals:

1. 20²ⁿ⁺¹

2. None

3. 20ⁿ

4. 20²ⁿ

Q.

The integral part of $(5\sqrt{5}+11{)}^{2n+1}$ is an odd number.

1. True

2. False

Q.

For positive integers n₁, n₂ the value of the expression

is a real number if and only if ($1+i{)}^{n_1}$+($1+i^3{)}^{n_1}$+($1+i^5{)}^{n_2}$($1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$ is a real number if and only if

1. 

2. 

3. 

4. 

Q.

If T = $(5+2\sqrt{6}{)}^n$ = M + f , n \in N , 0 $\le$ f < 1 , Then M =

1. 

2. 

3. 

4. 

Q. If x and y are positive real numbers and m, n are any positive integers, then
$\frac{x^n.y^m}{(1+x^{2n})(1+y^{2m})}$ $>$$\frac{1}{4}$

1. True

2. False

Q.

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

1. 416

2. 414

3. 417

4. 415

Q. Find the square root of $\sqrt{2x}+\sqrt{-x^4-1}$

1. 
2. 
3. 
4. 

Q.

The maximum value if 2 (a-x)(x + $\sqrt{x^2+b^2})$ is ( x , a, b $\in$ R )

1. a-b

2. 

3. 

4. a+b

Q.

HM of two positive real numbers a & b is

1. 
2. 

3. 

4. 

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. $2^{2m}$