Binomial Theorem · NDA previous year questions

If the coefficients of $a^m$ and $a^n$ in the expansion of $(1+a{)}^{m+n}$ are $\alpha$ and $\beta$, then which one of the following is correct ?

1. A.$\alpha =2\beta$
2. B.$\alpha =\beta$
3. C.$2\alpha =\beta$
4. D.$\alpha =(m+n)\beta$

What is the number of non-zero terms in the expansion of $(1+2\sqrt{3}x{)}^{11}+(1-2\sqrt{3}x{)}^{11}$ (after simplification) ?

1. A.4
2. B.5
3. C.6
4. D.11

If $n=(2017)!$, then what is $$\frac{1}{{\log }_{2}n}+\frac{1}{{\log }_{3}n}+\frac{1}{{\log }_{4}n}+\cdots +\frac{1}{{\log }_{2017}n}$$ equal to ?

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

If $x+{\log }_{15}(1+3^x)=x{\log }_{15}5+{\log }_{15}12$, where x is an integer, then what is x equal to ?

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

If $0<a<1$, the value of ${\log }_{10}a$ is negative. This is justified by

1. A.Negative power of 10 is less than 1
2. B.Negative power of 10 is between 0 and 1
3. C.Negative power of 10 is positive
4. D.Negative power of 10 is negative

The number of terms in the expansion of $(x+a{)}^{100}+(x-a{)}^{100}$ after simplification is

1. A.202
2. B.101
3. C.51
4. D.50

In the expansion of $(1+x{)}^{50}$, the sum of the coefficients of odd powers of $x$ is

1. A.$2^{26}$
2. B.$2^{49}$
3. C.$2^{50}$
4. D.$2^{51}$

If $x+{\log }_{10}(1+2^x)=x{\log }_{10}5+{\log }_{10}6$ then $x$ is equal to

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

If $A=\left(\begin{array}{cc}-2& 2\\ 2& -2\end{array}\right)$, then which one of the following is correct?

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

The expansion of $(x-y{)}^n,n\ge 5$ is done in the descending powers of $x$. If the sum of the fifth and sixth terms is zero, then $\frac{x}{y}$ is equal to

1. A.$\frac{n-5}{6}$
2. B.$\frac{n-4}{5}$
3. C.$\frac{5}{n-4}$
4. D.$\frac{6}{n-5}$

Consider the following for the next three (03) items that follow: Let $f(x)=\{\begin{array}{ll}-2,& -3\le x\le 0\\ x-2,& 0<x\le 3\end{array}$ and $g(x)=f(|x|)+|f(x)|$ What is the value of the differential coefficient of $g(x)$ at $x=-2$?

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

In the expansion of ${\left(\sqrt{x}+\frac{1}{3x^2}\right)}^{10}$ the value of constant term (independent of $x$) is

1. A.5
2. B.8
3. C.45
4. D.90

For the next three (03) items that follow: Consider the expansion of $(1+x{)}^{2n+1}$ If the coefficients of $x^r$ and $x^{r+1}$ are equal in the expansion, then $r$ is equal to

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

Consider the expansion of $(1+x{)}^{2n+1}$ The average of the coefficients of the two middle terms in the expansion is

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

Consider the expansion of $(1+x{)}^{2n+1}$ The sum of the coefficients of all the terms in the expansion is

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

The coefficient of $x^{99}$ in the expansion of $(x-1)(x-2)(x-3)\dots (x-100)$ is

1. A.5050
2. B.5000
3. C.-5050
4. D.-5000

If ${\log }_{8}m+{\log }_8\frac{1}{6}=\frac{2}{3}$, then $m$ is equal to

1. A.24
2. B.18
3. C.12
4. D.4

In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds $-9$ and $-1$ for roots. The correct equation is

1. A.$x^2-10x+9=0$
2. B.$x^2+10x+9=0$
3. C.$x^2-10x+16=0$
4. D.$x^2-8x-9=0$

For the next five (05) items that follow : Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$. What is the independent term in the given expansion ?

1. A.2103
2. B.3003
3. C.4503
4. D.None of the above

For the next five (05) items that follow : Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$. What is the ratio of coefficient of $x^{15}$ to the term independent of x in the given expansion ?

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

For the next five (05) items that follow : Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$. Consider the following statements : 1. There are 15 terms in the given expansion. 2. The coefficient of $x^{12}$ is equal to that of $x^3$. Which of the above statements is/are correct ?

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

For the next five (05) items that follow : Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$. Consider the following statements : 1. The term containing $x^2$ does not exist in the given expansion. 2. The sum of the coefficients of all the terms in the given expansion is $2^{15}$. Which of the above statements is/are correct ?

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

For the next five (05) items that follow : Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$. What is the sum of the coefficients of the middle terms in the given expansion ?

1. A.$C(15,9)$
2. B.$C(16,9)$
3. C.$C(16,8)$
4. D.None of the above

For the next three (03) items that follow : Consider the function $f(x)=\{\begin{array}{ll}{x}^2-5,& x\le 3\\ \sqrt{x+13},& x>3\end{array}$ What is the differential coefficient of $f(x)$ at $x=12$?

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

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 $n$ equal to?

1. A.5
2. B.10
3. C.15
4. D.None of the above