Binomial Theorem · NDA previous year questions

For the next three (03) items that follow: In the expansion of ${\left(x^3-\frac{1}{x^2}\right)}^n$ where $n$ is a positive integer, the sum of the coefficients of $x^5$ and $x^{10}$ is 0. What is the value of the independent term?

1. A.5005
2. B.7200
3. C.$-5005$
4. D.$-7200$

For the next three (03) items that follow: In the expansion of ${\left(x^3-\frac{1}{x^2}\right)}^n$ where $n$ is a positive integer, the sum of the coefficients of $x^5$ and $x^{10}$ is 0. What is the sum of the coefficients of the two middle terms?

1. A.0
2. B.1
3. C.$-1$
4. D.None of the above

How many terms are there in the expansion of $(1+2x+x^2{)}^{10}$?

1. A.11
2. B.20
3. C.21
4. D.30

Which one of the following may be the parameter of a binomial distribution?

1. A.$np=2,npq=4$
2. B.$n=4,p=\frac{3}{2}$
3. C.$n=8,p=1$
4. D.$np=10,npq=8$

What is the sum of the coefficients in the expansion of $(1+x{)}^n$ ?

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

If $P(77,31)=x$ and $C(77,31)=y$, then which one of the following is correct ?

1. A.$x=y$
2. B.$2x=y$
3. C.$77x=31y$
4. D.$x>y$

What is the differential coefficient of ${\log }_{x}x$ ?

1. A.0
2. B.1
3. C.$\frac{1}{x}$
4. D.x

What is ${\int }_0^2\frac{dx}{x^2+4}$ equal to ?

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{\pi }{8}$
4. D.None of the above

What is ${\int }_{-a}^a(x^3+\sin x)dx$ equal to ?

1. A.a
2. B.$2a$
3. C.0
4. D.1

What is ${\int }_0^{1}x{e}^{x}dx$ equal to ?

1. A.1
2. B.$-1$
3. C.0
4. D.e

The binomial distribution has

1. A.only one parameter
2. B.two parameters
3. C.three parameters
4. D.four parameters

In the expansion of $(1+x{)}^n$, what is the sum of even binomial coefficients?

1. A.$2^n$
2. B.$2^{n-1}$
3. C.$2^{n+1}$
4. D.None of the above

The value of the term independent of x in the expansion of ${\left(x^2-\frac{1}{x}\right)}^9$ is:

1. A.9
2. B.18
3. C.48
4. D.84

What is the ratio of coefficient of $x^{15}$ to the term independent of x in ${\left(x^2+\frac{2}{x}\right)}^{15}$?

1. A.$1/64$
2. B.$1/32$
3. C.$1/16$
4. D.$1/4$

1. A.$-abc$
2. B.$abc$
3. C.0
4. D.1

What is the middle term in the expansion of ${\left(1-\frac{x}{2}\right)}^8$?

1. A.$\frac{35x^4}{8}$
2. B.$\frac{17x^5}{8}$
3. C.$\frac{35x^5}{8}$
4. D.None of the above

What is the sum of all the coefficients in the expansion of $(1+x{)}^n$?

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

What is the coefficient of $x^4$ in the expansion of ${\left(\frac{1-x}{1+x}\right)}^2$?

1. A.-16
2. B.16
3. C.8
4. D.-8

What is the coefficient of $x^{17}$ in the expansion of ${\left(3x-\frac{x^3}{6}\right)}^9$ ?

1. A.$189/8$
2. B.$567/2$
3. C.$21/16$
4. D.None of the above

What is the number of terms in the expansion of $(a+b+c{)}^n,n\in N$ ?

1. A.$n+1$
2. B.$n+2$
3. C.$n(n+1)$
4. D.$(n+1)(n+2)/2$

What is the number of signals that can be sent by $6$ flags of different colours taking one or more at a time?

1. A.$21$
2. B.$63$
3. C.$720$
4. D.$1956$

What is the sum of the coefficients of all the terms in the expansion of $(45x-49{)}^4$?

1. A.$-256$
2. B.$-100$
3. C.100
4. D.256

Consider the following statements related to a variable X having a binomial distribution $b_X(n,p)$: 1. If $p=\frac{1}{2}$, then the distribution is symmetrical. 2. p remaining constant, $P(X=r)$ increases as n increases. Which of the statements given above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is the coefficient of $x^4$ in the expansion of $(1+2x+3x^2+4x^3+\dots {)}^{1/2}$ ?

1. A.$1/4$
2. B.$1/16$
3. C.1
4. D.$1/128$

If $x=2+2^{1/3}+2^{2/3}$, then what is the value of $x^3-6x^2+6x$ ?

1. A.1
2. B.2
3. C.3
4. D.$-2$