Binomial Theorem for Any Index

Q. The coefficient of $x^{18}$ in the product $(1+x)(1-x{)}^{10}(1+x+x^2{)}^9$

1. $84$
2. $126$
3. $-126$
4. $-84$

Q. The coefficient of $t^4$ in the expansion of ${\left(\frac{1-t^6}{1-t}\right)}^3$ is :

1. $12$
2. $10$
3. $15$
4. $14$

Q. If the expansion of ${(x^2+\frac{2}{x})}^n$ for positive n has a term independent of x, then n can be

1. 23
2. 0
3. 16
4. 18

Q.

The value of $\left\{\frac{3^{2003}}{28}\right\}$ , where {.} denotes the fractional part, is equal to :

1. 19/28

2. 15/28

3. 5/28

4. 9/28

Q. If ${10}^m$ divides ${101}^{100}-1$, then the greatest value of $m$ is

Q. The last digit of ${17}^{256}$ is

Q. Find the remainder when $5^{99}$ is divided by $8$ is

Q. If $|x|<1$, then the coefficient of $x^n$ in the expansion of $(1+x+x^2+x^3+....{)}^2$ is

1. $n-1$
2. $n+1$
3. $n$
4. $n+2$

Q. The number of triplets $(a,b,c)$ of positive integers satisfying the equation $\left|\begin{array}{ccc}{a}^3+1& a^{2}b& a^{2}c\\ a{b}^2& b^3+1& b^{2}c\\ a{c}^2& b{c}^2& c^3+1\end{array}\right|=11$ is equal to

1. $0$
2. $3$
3. $6$
4. $12$

Q.

If $A$ and $B$ are coefficients of $x^n$ in the expansion of${(1+x)}^{2n}$ and${(1+x)}^{2n-1}$ respectively, then$\frac{A}{B}$ is equal

1. $4$

2. $2$

3. $9$

4. $6$

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 value of $(1-\beta )(p+\beta )$ is

1. $3$
2. $4$
3. $1$
4. $2$

Q. Let $R=(5\sqrt{5}+11{)}^{2n+1}$ and $f=R-\left[R\right]$ where $[.]$ denotes the greatest integer function, then the remainder when $R\times f$, for $n=2$, is divided by 15 is

Q.

The ninth term of the expansion ${\left(3x-\frac{1}{2x}\right)}^8$ is

1. $\frac{1}{512x^9}$

2. $\frac{-1}{512x^9}$

3. $\frac{-1}{256x^8}$

4. $\frac{1}{256x^8}$

Q.

If ${\mathrm{a}}^2+{\mathrm{b}}^2=\mathrm{z}$ and $\mathrm{ab}=\mathrm{y}$, what will be the equivalent of $4\mathrm{z}+8\mathrm{y}$?

Q. The fractional part of a real number $x$ is $x-\left[x\right],$ where $\left[x\right]$ is the greatest integer less than or equal to $x.$ Let $F_1$ and $F_2$ be the fractional parts of ${(44-\sqrt{2017})}^{2017}\text{and}{(44+\sqrt{2017})}^{2017}$ respectively. Then $F_1+F_2$ lies between the numbers

1. $0\text{and}0.45$
2. $0.45\text{and}0.9$
3. $0.9\text{and}1.35$
4. $1.35\text{and}1.8$

Q.

If the coefficient of (2r + 4)th term is equal to the coefficient of (r – 2)th term in the expansion of $(1+x{)}^{18}$ then r =

1. 1

2. 2

3. 3

4. 4

Q. The number of different four-digit numbers which can be made out using one $1$, two $2^{\prime }s$, three $3^{\prime }s$ and four $4^{\prime }s$, is

1. $192$
2. $126$
3. $164$
4. $175$

Q. If $(1-x{)}^{-n}=a_0+a_{1}x+a_2{x}^2+........+a_r{x}^r+.....$, then $a_0+a_1+a_2+......+a_r$ is equal to

1. $\frac{n(n+1)(n+2)....(n+r-1)}{r!}$
2. None of the above
3. $\frac{n(n+1)(n+2)....(n+r)}{r!}$
4. $\frac{(n+1)(n+2)....(n+r)}{r!}$

Q. If the coefficient of $3^{\text{rd}},4^{\text{th}}$ and $5^{\text{th}}$ terms in the expansion of $(1+x{)}^n,n\in N$ are in A.P., then the number of possible value(s) of $n$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q.

The expansion ${(1+x)}^n$ = 1 + nx + $\frac{n(n-1)}{2!}$${\left(x\right)}^2$........... is valid if |x| > 1.

1. True

2. False

Q. The coefficient of $t^8$ in ${(1+t)}^2{(1+t+t^2+....+t^9)}^3$ is

1. $146$
2. $147$
3. $144$
4. $145$

Q.

If $\frac{(1-3x{)}^{\frac{1}{2}}+(1-x{)}^{\frac{5}{3}}}{\sqrt{4-x}}$ is approximately equal to a + bx for small values of x, then (a, b) =

1. (1, $\frac{35}{24}$)

2. (1, - $\frac{35}{24}$)

3. (2, $\frac{35}{12}$)

4. (2, - $\frac{35}{12}$)

Q.

The expansion ${(1+x)}^n$ = 1 + nx + $\frac{n(n-1)}{2!}$${\left(x\right)}^2$........... is valid if |x| > 1.

1. True

2. False

Q. The approximate value of $(1.0002{)}^{3000}$ is

1. $1.8$
2. $1.2$
3. $1.6$
4. $1.4$

Q. Given:

$A=\text{x : x is a multiple of 2 and is less than 25}$

$B=\text{x : x is a square of a natural number}\text{and is less than 25}$

$C=\text{x : x is a multiple of 3 and is less than 25}$

$C=\text{x : x is a prime number less than 25}$

Write the set A, B, C and D in roster form.

Q. Number of terms in the expansion of $(a+b+c+d+e{)}^5$ is

1. 25

2. 63

3. 126

4. 252

Q. The function $f\left(x\right)=\frac{x^2}{e^x}$ is monotonically increasing if

1. $x<0$ only
2. $0<x<2$
3. $x\in (-\infty ,0)\cup (2,\infty )$
4. $x>2$ only

Q. The value of remainder when $3^{2n+1}+2^{n+2}$, where $n\ge 10$ is divided by $7$ is

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

Q.

In the term $-6a{b}^{2}c$, what are the coefficients of the following?

$a{b}^2$

Q. In the expansion of ${({2.2}^{\frac{2x}{3}}+2^{\frac{-x}{3}})}^n$ the sum of the last four binomial coefficients exceeds the sum of the first three binomial coefficients by 20 and if the middle term is-the numerically largest term, then x belongs to

1. $(-2,\infty )$
2. $(2,lo{g}_2\frac{1}{3})$
3. $(-2,lo{g}_2\frac{1}{3})$
4. $(-2,2lo{g}_2\frac{2}{3})$