Sets, Relations & Functions · NDA previous year questions

Let $Z$ be the set of integers and $aRb$, where $a,b\in Z$ if and only if $(a-b)$ is divisible by 5. Consider the following statements : 1. The relation $R$ partitions $Z$ into five equivalent classes. 2. Any two equivalent classes are either equal or disjoint. 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

If $A=\{x:x\text{ is a multiple of }3\}$ and $B=\{x:x\text{ is a multiple of }4\}$ and $C=\{x:x\text{ is a multiple of }12\}$, then which one of the following is a null set?

1. A.$(A\setminus B)\cup C$
2. B.$(A\setminus B)\setminus C$
3. C.$(A\cap B)\cap C$
4. D.$(A\cap B)\setminus C$

Let $A=\{x,y,z\}$ and $B=\{p,q,r,s\}$. What is the number of distinct relations from $B$ to $A$?

1. A.4096
2. B.4094
3. C.128
4. D.126

Let $A=\{1,2,3,4,5,6,7,8,9,10\}$. Then the number of subsets of $A$ containing exactly two elements is

1. A.20
2. B.40
3. C.45
4. D.90

How many words can be formed using all the letters of the word 'NATION' so that all the three vowels should never come together?

1. A.354
2. B.348
3. C.288
4. D.None of the above

What is ${\sum }_{r=0}^1{}^{n+r}{C}_n$ equal to?

1. A.${}^{n+2}{C}_1$
2. B.${}^{n+2}{C}_n$
3. C.${}^{n+3}{C}_n$
4. D.${}^{n+2}{C}_{n+1}$

For each non-zero real number $x$, let $f(x)=\frac{x}{|x|}$. The range of $f$ is

1. A.a null set
2. B.a set consisting of only one element
3. C.a set consisting of two elements
4. D.a set consisting of infinitely many elements

If $f(x)={\log }_e\left(\frac{1+x}{1-x}\right)$, $g(x)=\frac{3x+x^3}{1+3x^2}$ and $g\circ f(t)=g(f(t))$, then what is $g\circ f\left(\frac{e-1}{e+1}\right)$ equal to?

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

Consider the following functions : 1. $f(x)=x^3,x\in \mathbb{R}$ 2. $f(x)=\sin x,0<x<2\pi$ 3. $f(x)=e^x,x\in \mathbb{R}$ Which of the above functions have inverse defined on their ranges?

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

If $G(x)=\sqrt{25-x^2}$, then what is ${\lim }_{x\to 1}\frac{G(x)-G(1)}{x-1}$ equal to?

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

Consider the function $f(x)=\{\begin{array}{ll}-1& \text{if }x\le 0\\ ax+b& \text{if }0<x<1\\ 1& \text{if }x\ge 1\end{array}$ where $a,b$ are constants. The function is continuous everywhere. What is the value of $a$?

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

Consider the function $f(x)=\{\begin{array}{ll}\frac{\alpha \cos x}{\pi -2x}& \text{if }x\ne \frac{\pi }{2}\\ 3& \text{if }x=\frac{\pi }{2}\end{array}$ which is continuous at $x=\frac{\pi }{2}$, where $\alpha$ is a constant. What is ${\lim }_{x\to 0}f(x)$ equal to?

1. A.$0$
2. B.$3$
3. C.$\frac{3}{\pi }$
4. D.$\frac{6}{\pi }$

For the next two (2) items that follow : Given that ${\lim }_{x\to \infty }\left(\frac{2+x^2}{1+x}-Ax-B\right)=3$. What is the value of $A$?

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

Given that ${\lim }_{x\to \infty }\left(\frac{2+x^2}{1+x}-Ax-B\right)=3$. What is the value of $B$?

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

Consider the function $f(x)=0.75x^4-x^3-9x^2+7$. Consider the following statements : 1. The function attains local minima at $x=-2$ and $x=3$. 2. The function increases in the interval $(-2,0)$. 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

The integral $\int \frac{dx}{a\cos x+b\sin x}$ is of the form $\frac{1}{r}\ln \left[\tan \left(\frac{x+\alpha }{2}\right)\right]$. What is $\alpha$ equal to?

1. A.${\tan }^{-1}\left(\frac{a}{b}\right)$
2. B.${\tan }^{-1}\left(\frac{b}{a}\right)$
3. C.${\tan }^{-1}\left(\frac{a+b}{a-b}\right)$
4. D.${\tan }^{-1}\left(\frac{a-b}{a+b}\right)$

A and B are two sets having 3 elements in common. If $n(A)=5$, $n(B)=4$, then what is $n(A\times B)$ equal to ?

1. A.0
2. B.9
3. C.15
4. D.20

The relation S is defined on the set of integers Z as xSy if integer x divides integer y. Then

1. A.S is an equivalence relation.
2. B.S is only reflexive and symmetric.
3. C.S is only reflexive and transitive.
4. D.S is only symmetric and transitive.

What is $\frac{(1+i{)}^{4n+5}}{(1-i{)}^{4n+3}}$ equal to, where n is a natural number and $i=\sqrt{-1}$ ?

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

What is ${\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)}^6$ equal to, where $i=\sqrt{-1}$ ?

1. A.1
2. B.1/6
3. C.6
4. D.2

If $A=\left(\begin{array}{cc}4& x+2\\ 2x-3& x+1\end{array}\right)$ is symmetric, then what is x equal to ?

1. A.2
2. B.3
3. C.-1
4. D.5

For the next three (03) items that follow : Consider the function $f(x)=\frac{x-1}{x+1}$. What is $\frac{f(x)+1}{f(x)-1}+x$ equal to ?

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

For the next three (03) items that follow : Consider the function $f(x)=\frac{x-1}{x+1}$. What is $f(2x)$ equal to ?

1. A.$\frac{f(x)+1}{f(x)+3}$
2. B.$\frac{f(x)+1}{3f(x)+1}$
3. C.$\frac{3f(x)+1}{f(x)+3}$
4. D.$\frac{f(x)+3}{3f(x)+1}$

For the next three (03) items that follow : Consider the function $f(x)=\frac{x-1}{x+1}$. What is $f(f(x))$ equal to ?

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

The function $f:N\to N$, N being the set of natural numbers, defined by $f(x)=2x+3$ is

1. A.injective and surjective
2. B.injective but not surjective
3. C.not injective but surjective
4. D.neither injective nor surjective