Sets, Relations & Functions · NDA previous year questions

If $1-{\log }_{10}2={\log }_{10}(5^x+4^x+3^x+2^x+1)$, then what is a value of $x$ ?

1. A.10
2. B.5
3. C.1
4. D.0

Passage: Consider the curves $y=x^2$ and $y=2|x|$. Question: What is the number of points of intersection of the curves ?

1. A.4
2. B.3
3. C.2
4. D.None

If $2f(x)+f(1-x)=x$, then what is $f(x)$ equal to ?

1. A.$x-1$
2. B.$x-\left(\frac{1}{2}\right)$
3. C.$2x$
4. D.$2x-1$

A set $S$ contains $(2n+1)$ elements. If the number of subsets of $S$ which contain at most $n$ elements is 1024, then what is the value of $n$ ?

1. A.10
2. B.8
3. C.6
4. D.5

Let $R$ be a relation on the set $N$ of natural numbers defined by $R=\{(x,y):x,y\in N\text{ and }x=y^3\}$. Which of the following statements is/are correct ? I. $R$ is symmetric relation II. $R$ is transitive relation Select the answer using the code given below :

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

Let $A$ and $B$ be two sets. For some set $C$, both $A\cap C$ and $B\cap C$ are empty sets and $A\cup C=B\cup C$. Which of the following is/are true ? I. $C=\phi$ II. $A=B$ III. $A\cup B=C$ Select the answer using the code given below :

1. A.I only
2. B.II only
3. C.I and II only
4. D.I, II and III

What is $(((A\cap B)\cup (A-B))-((A\cap B)\cup (B-A)))\cup A$ equal to ?

1. A.$\phi$
2. B.$A$
3. C.$B$
4. D.$A\cup B$

What is $\tan \left[2{\tan }^{-1}\frac{1}{2}-\frac{\pi }{4}\right]$ equal to ?

1. A.$-7$
2. B.0
3. C.$\frac{1}{5}$
4. D.$\frac{1}{7}$

Consider the following in respect of inverse circular functions : I. ${\sin }^{-1}(-x)=-{\sin }^{-1}x$ II. ${\cos }^{-1}(-x)={\cos }^{-1}x$ III. ${\tan }^{-1}(-x)=\pi -{\tan }^{-1}x$ IV. ${\cot }^{-1}(-x)=\pi -{\cot }^{-1}x$ How many of the above are correct ?

1. A.One
2. B.Two
3. C.Three
4. D.All the four

Passage: Let $(e^y{)}^x-y=0$, where $y$ is a function of $x$ whose domain is $(0,10]$. Question: What is $\frac{dy}{dx}$ equal to, given that $y=y_0$ when $x=1$ ?

1. A.$-\frac{y_0}{1+e^{y_0}}$
2. B.$-\frac{y_0{e}^{y_0}}{1+e^{y_0}}$
3. C.$\frac{y_0{e}^{y_0}}{1+e^{y_0}}$
4. D.$\frac{y_0{e}^{y_0}}{1-e^{y_0}}$

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

If $\sqrt[4]{y}=x+\sqrt{x^2+4}$, then what is $\sqrt{x^2+4}\frac{dy}{dx}$ equal to ?

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

1. A.$\{x\le 2\}\cup \{x\ge 3\}$
2. B.$\{x<2\}\cup \{x>3\}$
3. C.$(2,3)$
4. D.$[2,3]$

1. A.$\{x\le 2\}\cup \{x\ge 3\}$
2. B.$\{x<2\}\cup \{x>3\}$
3. C.$(2,3)$
4. D.$[2,3]$

If A and B are non-empty subsets of a set, and $A^c$ and $B^c$ represent their complements, then which of the following is/are correct? I. $A-B=B^c-A^c$ II. $A-B^c=A^c-B$ Select the answer using the code given below.

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

Let y=x! and z = (2x)! If (z / y) = 120 then what is the value of (3x)!?

1. A.362880
2. B.181440
3. C.90720
4. D.45360

What is the relation between m and n?

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

How many values of m are possible?

1. A.None
2. B.One
3. C.Two
4. D.More than two

Set X contains 3n elements and set Y contains 2n elements, and they have n elements in common. How many elements does (X - Y) × (Y-X) have?

1. A.$5n^2$
2. B.$4n^2$
3. C.$3n^2$
4. D.$2n^2$

Let A = {- 3, - 2, - 1, 0, 1, 2, 3} and B = {0, 1, 4, 9} How many elements does the subset of AB corresponding to the relation R = {(x, y) : |x| < y} have, where $x\in \text{A}$ and $y\in \text{B}$?

1. A.9
2. B.12
3. C.15
4. D.16

For the following two (02) items: Consider the function $f(x)=\frac{x}{1-x}(x>0,x\ne 1)$ What is $\frac{f(x)}{f(x+1)}$ equal to?

1. A.$-f(x^2)$
2. B.$-f(\sqrt{x})$
3. C.$f(x^2)$
4. D.$f(x-1)$

For the following two (02) items: Consider the function $f(x)=\frac{x}{1-x}(x>0,x\ne 1)$ What is $(1-x)f(\sqrt{x})+xf(\sqrt{x}+1)$ equal to?

1. A.-f(x)
2. B.f(x)
3. C.x
4. D.0

For the following two (02) items: Consider the function $f(x)=1-\sqrt[3]{(x-1{)}^2}$ What is the domain of the function?

1. A.$(1,\infty )$
2. B.$(-\infty ,\infty )$
3. C.$(0,\infty )$
4. D.$(-\infty ,\infty )\setminus \{1\}$

For the following two (02) items: Consider the function $f(x)=\{\begin{array}{ll}4(5^x),& x<0\\ 8k+x,& x\ge 0\end{array}$ What is f'(-1) equal to?

1. A.$\frac{2}{5}\ln 5$
2. B.$\frac{3}{5}\ln 5$
3. C.$\frac{4}{5}\ln 5$
4. D.$20\ln 5$

For the following two (02) items: Let $u=\int e^x\cos xdx$ and $v=\int e^x\sin xdx$. What is u + v equal to?

1. A.$-\frac{du}{dx}$
2. B.$-\frac{dv}{dx}$
3. C.$\frac{du}{dx}$
4. D.$\frac{dv}{dx}$