Binomial Theorem · NDA previous year questions

In the expansion of ${\left(x+\frac{1}{x}\right)}^{2n}$, what is the $(n+1)$th term from the end (when arranged in descending powers of x)?

1. A.$C(2n,n)x$
2. B.$C(2n,n-1)x$
3. C.$C(2n,n)$
4. D.$C(2n,n-1)$

How many terms are there in the expansion of ${\left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)}^{21}$ where $a\ne 0,b\ne 0$?

1. A.21
2. B.22
3. C.42
4. D.43

For what values of k is the system of equations $2k^{2}x+3y-1=0,7x-2y+3=0,6kx+y+1=0$ consistent?

1. A.$\frac{3\pm \sqrt{11}}{10}$
2. B.$\frac{21\pm \sqrt{161}}{10}$
3. C.$\frac{3\pm \sqrt{7}}{10}$
4. D.$\frac{4\pm \sqrt{11}}{10}$

Consider the expansion of $(1+x{)}^n$. Let p, q, r and s be the coefficients of first, second, nth and (n + 1)th terms respectively. What is (ps + qr) equal to?

1. A.$1+2n$
2. B.$1+2n^2$
3. C.$1+n^2$
4. D.$1+4n$

What is C(n, 1) + C(n, 2) + _ _ _ _  _ + C(n, n) equal to

1. A.$2+2^2+2^3+\cdots +2^n$
2. B.$1+2+2^2+2^3+\cdots +2^n$
3. C.$1+2+2^2+2^3+\cdots +2^{n-1}$
4. D.$2+2^2+2^3+\cdots +2^{n-1}$

What is the sum of the coefficients of first and last terms in the expansion of $(1+x{)}^{2n}$, where n is a natural number?

1. A.1
2. B.2
3. C.n
4. D.2n

What is the coefficient of the middle term in the expansion of $(1+4x+4x^2{)}^5$?

1. A.8064
2. B.4032
3. C.2016
4. D.1008

If $C_0$, $C_1$, $C_2$, $\dots$, $C_n$ are the coefficients in the expansion of $(1+x{)}^n$, then what is the value of $C_1+C_2+C_3+\dots +C_n$?

1. A.$2^n$
2. B.$2^n-1$
3. C.$2^{n-1}$
4. D.$2^n-2$

The term independent of $x$ in the binomial expansion of ${\left(\frac{2}{x^2}-\sqrt{x}\right)}^{10}$ is equal to

1. A.180
2. B.120
3. C.90
4. D.72

If $(1+2x-x^2{)}^6=a_0+a_{1}x+a_2{x}^2+\dots +a_{12}{x}^{12}$, then what is $a_0-a_1+a_2-a_3+a_4-\dots +a_{12}$ equal to?

1. A.32
2. B.64
3. C.2048
4. D.4096

If $C(20,n+2)=C(20,n-2)$, then what is $n$ equal to?

1. A.18
2. B.25
3. C.10
4. D.12

What is the sum of the last five coefficients in the expansion of $(1+x{)}^9$ when it is expanded in ascending powers of $x$?

1. A.256
2. B.512
3. C.1024
4. D.2048

If $\sum x_i=20$, $\sum x_i^2=200$ and $n=10$ for an observed variable $x$, then what is the coefficient of variation?

1. A.80
2. B.100
3. C.150
4. D.200

How many terms are there in the expansion of $(1+2x+x^2{)}^5+(1+4y+4y^2{)}^5$ ?

1. A.12
2. B.20
3. C.21
4. D.22

If the middle term in the expansion of ${\left(x^2+\frac{1}{x}\right)}^{2n}$ is $184756x^{10}$, then what is the value of $n$?

1. A.10
2. B.8
3. C.5
4. D.4

If the constant term in the expansion of ${\left(\sqrt{x}-\frac{k}{x^2}\right)}^{10}$ is 405, then what can be the values of $k$?

1. A.$\pm 2$
2. B.$\pm 3$
3. C.$\pm 5$
4. D.$\pm 9$

If $x^{{\log }_{7}x}>7$ where $x>0$, then which one of the following is correct?

1. A.$x\in (0,\infty )$
2. B.$x\in \left(\frac{1}{7},7\right)$
3. C.$x\in \left(0,\frac{1}{7}\right)\cup (7,\infty )$
4. D.$x\in \left(\frac{1}{7},\infty \right)$

What is the number of terms in the expansion of $[(2x-3y{)}^2(2x+3y{)}^2{]}^2$?

1. A.$4$
2. B.$5$
3. C.$8$
4. D.$16$

In the expansion of $(1+ax{)}^n$, the first three terms are respectively 1, $12x$ and $64x^2$. What is n equal to?

1. A.$6$
2. B.$9$
3. C.$10$
4. D.$12$

What is the coefficient of the middle term in the binomial expansion of $(2+3x{)}^4$?

1. A.6
2. B.12
3. C.108
4. D.216

Let the coefficient of the middle term of the binomial expansion of $(1+x{)}^{2n}$ be $\alpha$ and those of two middle terms of the binomial expansion of $(1+x{)}^{2n-1}$ be $\beta$ and $\gamma$. Which one of the following relations is correct?

1. A.$\alpha >\beta +\gamma$
2. B.$\alpha <\beta +\gamma$
3. C.$\alpha =\beta +\gamma$
4. D.$\alpha =\beta \gamma$

What is the value of ${\log }_7{\log }_7\sqrt{7\sqrt{7\sqrt{7}}}$ equal to?

1. A.$3{\log }_{2}7$
2. B.$1-3{\log }_{2}7$
3. C.$1-3{\log }_{7}2$
4. D.$\frac{7}{8}$

What is the value of ${\log }_{9}27+{\log }_{8}32$?

1. A.$\frac{7}{2}$
2. B.$\frac{19}{6}$
3. C.4
4. D.7

If $(0.2{)}^x=2$ and ${\log }_{10}2=0.3010$, then what is the value of x to the nearest tenth?

1. A.$-10.0$
2. B.$-0.5$
3. C.$-0.4$
4. D.$-0.2$

In the expansion of $(1+x{)}^{43}$, if the coefficients of $(2r+1{)}^{\text{th}}$ and $(r+2{)}^{\text{th}}$ terms are equal, then what is the value of $r$ ($r\ne 1$) ?

1. A.5
2. B.14
3. C.21
4. D.22