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

Q. If the constant term, in binomial expansion ${(2x^r+\frac{1}{x^2})}^{10}$ is $180$, then $r$ is equal to

Q. If $S=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\cdots ,$ then $160S$ is equal to

Q. The last two digits of the number $3^{400}$ is

1. $43$
2. $29$
3. $81$
4. $01$

Q. If $\left\{x\right\}$ represents the fractional part of $x,$ then $\left\{\frac{5^{200}}{8}\right\}$ is

1. $\frac{3}{8}$
2. $\frac{5}{8}$
3. $\frac{1}{4}$
4. $\frac{1}{8}$

Q. Consider the binomial expansion $R=(1+2x{)}^n=I+f,$ where $I$ is the integral part of $R$ and $f$ is the fractional part of $R$ for $n\in N$. If the sum of the coefficients of $R$ is $6561$ and $K=n+R-Rf,$ then which of the following is/are correct?

1. $K=9$ when $x=\frac{1}{\sqrt{2}}$
2. $K=7$ when $x=\frac{1}{\sqrt{2}}$
3. $5^{\text{th}}$ term is the greatest term when $x=\frac{1}{2}$
4. If $k^{\text{th}}$ term has the greatest coefficient, then the sum of all possible value(s) of $k$ is $13$

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. If $9^7+7^9$ is divided by $64$, then the remainder is

1. $1$
2. $11$
3. $0$
4. $61$

Q.

Find the square root of $1728$ in radical form

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. If $f\left(x\right)=x-x^2+x^3-x^4\cdots \infty$ and $|x|<1,$ then $f^{\prime }\left(x\right)=$

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

Q. Let $A$ is a non-singular matrix such that $A^2=I.$ Then the inverse of $\frac{A}{2}$ will be (where $I=$ identity matrix)

1. $2A$
2. $\frac{A^{-1}}{2}$
3. $\frac{A}{2}$
4. $A^2$

Q. Statement $1$:
If $p$ is a prime number $(p\ne 2)$, then $\left[\right(2+\sqrt{5}{)}^p]-2^{p+1}$ is always divisible by $p$ (where $[.]$ denotes the greatest integer function).

Statement $2$:
If $n$ is a prime, then ${}^n{C}_1,{}^n{C}_2{,}^n{C}_3,\dots {,}^n{C}_{n-1}$ must be divisible by $n$.

1. Both the statements are TRUE and STATEMENT $2$ is the correct explanation of STATEMENT $1$.
2. Both the statements are TRUE and STATEMENT $2$ is NOT the correct explanation of STATEMENT $1$.
3. STATEMENT $1$ is TRUE and STATEMENT $2$ is FALSE.
4. STATEMENT $1$ is FALSE and STATEMENT $2$ is TRUE.

Q. If ${}^{15}{C}_{3r}{=}^{15}{C}_{r+3}$, then the value of r is ........

1. 3

2. 4

3. 5

4. 8

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. The value of $\left\{\frac{2^{4n}}{15}\right\},n\in N$ is
(where {.} represents fractional part function)

1. $\frac{1}{15}$
2. $\frac{11}{15}$
3. $\frac{4}{15}$
4. $\frac{14}{15}$

Q. If n is positive integer and $(3\sqrt{3}+5{)}^{2n+1}=\alpha +\beta$ where $\alpha$ is an integer and $0<\beta <1$, then

1. 
2. 
3. 
4. 

Q.

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

1. Can't be determined.

2. 300!

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

4. 300! = $\sqrt{{300}^{300}}$

Q. In $I_{m,n}={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx,$ for $m,n\ge 1$ and ${\int }_0^1\frac{x^{m-1}+x^{n-1}}{(1+x{)}^{m+n}}dx=\alpha I_{m,n},\alpha \in R,$ then $\alpha$ equals

Q.

Which of the following is true about $(\sqrt{3}+1{)}^{2n}$, where n is a positive integer

1. The integer just above it is an even number

2. The integer above it is divisible by

3. The integer just above it is divisible by 3

4. The integer just above is divisible by

Q. If ${(5-\sqrt{21})}^{2n+1}$ and f = R –$[\left[R\right]$, where $\left[R\right]$denotes the greatest integer less than or equal to R, then R$(1-f)$ =

1. 
2. 
3. 
4. 

Q. If $\int \frac{x^2-x+1}{{(x^2+1)}^{\frac{3}{2}}}{e}^{x}dx=e^{x}f\left(x\right)+c,$ then which of the following is/are true:

1. $f\left(x\right)$ is an even function
2. $f\left(x\right)$ is an odd function
3. The range of $f\left(x\right)$ is $(0,1]$
4. $f\left(1\right)=\frac{1}{2}$

Q.

What is the value of $\sqrt{49}$ ?

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 $f$ is an even function defined on the interval $[-5,5]$, then the real values of $x$ satisfying the equation $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are

1. $\frac{-1\pm \sqrt{5}}{2}$
2. $\frac{-2\pm \sqrt{5}}{2}$
3. non of these
4. $\frac{-3\pm \sqrt{5}}{2}$

Q. $\int \frac{x^2{e}^x}{{(x+2)}^2}dx$ is equal to

1. $e^x\left[\frac{x-2}{x+2}\right]+C$
2. $e^x\left[\frac{4}{{(x+2)}^2}\right]+C$
3. $e^x\left[\frac{1}{x+2}\right]+C$
4. $e^x\left[\frac{x^2}{{(x+2)}^2}\right]+C$

Q.

The greatest integer less than or equal to $(\sqrt{2}+1{)}^6$ is

1. 196

2. 197

3. 198

4. 199

Q. $\int {\left(\frac{x+2}{x+4}\right)}^2{e}^{x}dx$ is equal to

1. $e^x\left(\frac{x}{x+4}\right)+C$
2. $e^x\left(\frac{x+2}{x+4}\right)+C$
3. $e^x\left(\frac{x-2}{x+4}\right)+C$
4. $\left(\frac{2x{e}^2}{x+4}\right)+C$

Q. If $6^{83}+8^{83}$ is divided by $49,$ then the remainder is

1. $35$
2. $5$
3. $1$
4. $0$

Q. A $7$-digit number is formed using the digits from $1,2,3,\cdots ,9$ such that exactly one digit is repeated thrice and exactly one digit is repeated twice. The total number of such $7$-digit number is

1. $\frac{{}^9{C}_1\times {}^8{C}_1\times {}^7{C}_5\times 7!}{3!\times 2!}$
   
2. $9\times 9\times 9\times 8\times 8\times 7\times 6$
3. $\frac{{}^9{C}_2\times {}^7{C}_5\times 7!}{5!}$
4. $\frac{7!}{3!\times 2!}$

Q.

Find $(5\sqrt{5}+11{)}^{2n+1}$ - $(5\sqrt{5}-11{)}^{2n+1}$

1. 2[ - ..............]

2. 2[ + ..............]

3. 2[ + ..............]

4. 2[ - ..............]