Binomial Theorem

Q.

The sum of the series ${}^n{C}_1{+}^n{C}_2{+}^n{C}_3+\dots {+}^n{C}_n$ equals

1. $2^n-1$

2. $2^n$

3. $2^n+1$

4. none of these

Q. The coefficient of $x^{256}$ in the expansion of $(1-x{)}^{101}(x^2+x+1{)}^{100}$ is

1. ${}^{100}{C}_{15}$
2. ${}^{100}{C}_{16}$
3. $-{}^{100}{C}_{16}$
4. $-{}^{100}{C}_{15}$

Q.

The value of $\sum _{r=0}^{20}{}^{50-r}C_6$ is equal to

1. ${}^{51}C_7-{}^{30}C_7$

2. ${}^{51}C_7+{}^{30}C_7$

3. ${}^{50}C_7-{}^{30}C_7$

4. ${}^{51}C_6-{}^{30}C_6$

Q.

Using binomial theorem, prove that $2^{3n}-7n-1$ is divisible by 49, where $n\in N.$

Q. If $(1+ax{)}^n=1+8x+24x^2+...,$ then which of the following is/are correct?

1. $a=1$
2. $a=2$
3. $n=8$
4. $n=4$

Q.

Using binomial theorem evaluate each of the following:

$\left(i\right)\left(96{)}^3\right(ii\left)\right(102{)}^5\left(iii\right)\left(101{)}^4\right(iv\left)\right(98{)}^5$

Q.

If n is a positive integer, prove $3^{3n}-26n-1$ is divisible by 676.

Q.

Using binomial theorem, write down the expansions of the following: $\left(i\right)(2x+3y{)}^5$

$\left(ii\right)(2x-3y{)}^4$

$\left(iii\right){(x-\frac{1}{x})}^6$

$\left(iv\right)(1-3x{)}^7$

$\left(v\right){(ax-\frac{b}{x})}^6$

$\left(vi\right){(\sqrt{\frac{x}{a}}-\sqrt{\frac{a}{x}})}^6$

$\left(vii\right)(\sqrt[3]{3x}-\sqrt[3]{3a}{)}^6$

$\left(viii\right)(1+2x-3x^2{)}^5$

$\left(ix\right){(x+1-\frac{1}{x})}^3$

$\left(x\right)(1-2x+3x^2{)}^3$

Q. If the coefficients of $r^{th}$ and $(r+1{)}^{th}$ terms in the expansion of $(3+7x{)}^{29}$ are equal, then the value of $r$ is

Q. If $\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$ is the A.M. between $a$ and $b$, then find the value of $n$.

Q. If ${}^{100}{C}_{50}$ can be prime factorised as $2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma }\cdot 7^{\delta }\dots$ where $\alpha ,\beta ,\gamma ,\delta ,\dots$ are non-negative integers, then the CORRECT relation(s) is (are)

1. $\alpha <\beta$
2. $\gamma <\delta$
3. $\alpha +\delta =\beta +\gamma -1$
4. $\gamma +\delta =0$

Q.

Using binomial theorem determine which number is larger $(1.2{)}^{4000}$ or 800?

Q. The value of $2^{2010}\frac{\underset{0}{\overset{1}{\int }}{x}^{1004}{(1-x)}^{1004}dx}{\underset{0}{\overset{1}{\int }}{x}^{1004}{(1-x^{2010})}^{1004}dx}$ is equal to

1. $1005$
2. $2010$
3. $4020$
4. $8040$

Q. The identity element for the binary operation $\ast$ defined on $Q-\left\{0\right\}$ as

$a\ast b=\frac{ab}{2}\forall a,b\in Q-\left\{0\right\}$ is

1. $2$
2. $0$
3. None of these
4. $1$

Q.

Find the cube root of $64$ by prime factorization method.

Q. In a library, there are $4$ science books, $4$ maths books and $3$ political science books all of which are on different topics. The number of ways in which at least one book of each subject is selected, is

1. $1575$
2. $1023$
3. $961$
4. $2047$

Q. The expression $x^5+10x^{4}a+40x^3{a}^2+80x^2{a}^3+80x{a}^4+32a^5$ is equal to

1. $(x+3a{)}^5$
2. $(x+2a{)}^5$
3. $(x+a{)}^5$
4. $(x+4a{)}^5$

Q. The coefficient of $x^3$ in the expansion of $(x+1{)}^5$ is

1. $10$
2. $5$
3. $1$
4. $15$

Q. The expression $a{x}^2+bx+c$ ($a,b$ and $c$ are real) has the same sign as that of a for all $x$ is

1. $b^2-4ac>0$
2. $b^2-4ac\ne 0$
3. $b^2-4ac\le 0$
4. $b$ and $c$ have the same sign as that of $a$

Q. The polynomial $(x+y{)}^9$ is expanded in decreasing powers of $x.$ The second and third terms have equal values when evaluated at $x=p$ and $y=q,$ where $p$ and $q$ are positive numbers whose sum is one. The value of $p$ is

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

Q. Let $N=2^{200}\left({}^{200}{C}_0\right)\left({}^{200}{C}_{100}\right)-2^{198}\left({}^{200}{C}_1\right)\left({}^{199}{C}_{99}\right)+2^{196}\left({}^{200}{C}_2\right)\left({}^{198}{C}_{98}\right)+\cdots +\left({}^{200}{C}_{100}\right)\left({}^{100}{C}_0\right)$
then -

1. $N$ is equal to the coefficient of $x^{100}$ in $(1-x{)}^{200}$
2. The number of zeros at the end of $N$ is only one.
3. $N=3^{101}\times k,$ where $k$ is an integer
4. $N$ is equal to the number of ways of distributing $200$ different objects among $2$ persons equally.

Q. Third term in the expression $(x+a{)}^7$ is

1. ${}^7{C}_2{x}^5{a}^2$
2. ${}^7{C}_3{x}^4{a}^3$
3. ${}^7{C}_4{x}^3{a}^4$
4. ${}^7{C}_6{x}^1{a}^6$

Q.

Prove that : 1 P(1, 1) + 2.P(2, 2) +3.P(3, 3)+...+ n.P (n, n)= P(n+1, n+1)-1.

Q. Let $N=2^{200}\left({}^{200}{C}_0\right)\left({}^{200}{C}_{100}\right)-2^{198}\left({}^{200}{C}_1\right)\left({}^{199}{C}_{99}\right)+2^{196}\left({}^{200}{C}_2\right)\left({}^{198}{C}_{98}\right)+\cdots +\left({}^{200}{C}_{100}\right)\left({}^{100}{C}_0\right)$
then -

1. $N$ is equal to the coefficient of $x^{100}$ in $(1-x{)}^{200}$
2. The number of zeros at the end of $N$ is only one.
3. $N=3^{101}\times k,$ where $k$ is an integer
4. $N$ is equal to the number of ways of distributing $200$ different objects among $2$ persons equally.

Q. The coefficient of $x^5$ in the expansion of ${(1+x)}^{21}+{(1+x)}^{22}+\dots +{(1+x)}^{30}$

1. 
2. 
3. 
4. 

Q. Prove the following by using the principle of mathematical induction for all $n\in N$.
$1\cdot 2+2\cdot 2^2+3\cdot 2^3+\cdots +n\cdot 2^n=(n-1)2^{n+1}+2$

Q. Let $f\left(x\right)=x^{12}+a_{11}{x}^{11}+a_{10}{x}^{10}+\cdots +a_0$ where $a_i$'s are integers. The number of polynomials such that all the roots of $f\left(x\right)$ are from the set $\{1,2,3,4\}$ is

1. $48$
2. $165$
3. $455$
4. $495$

Q.

Using binomial theorem, prove that $3^{3n+2}-8b-9$ is divisible by $64,n\in N.$

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

1. $\frac{19}{28}$
2. $\frac{17}{36}$
3. $\frac{17}{28}$
4. $\frac{19}{36}$

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

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