Sets, Relations & Functions · NDA previous year questions

1. A.$A=\{x|x>a\}$ and $B=\{x|x>b\}$
2. B.$A=\{x|x<a\}$ and $B=\{x|x>b\}$
3. C.$A=\{x|x<a\}$ and $B=\{x|x<b\}$
4. D.$A=\{x|x>a\}$ and $B=\{x|x<b\}$

What is the value of $${\left[\frac{i+\sqrt{3}}{2}\right]}^{2019}+{\left[\frac{i-\sqrt{3}}{2}\right]}^{2019}?$$

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

If both $p$ and $q$ belong to the set $\{1,2,3,4\}$, then how many equations of the form $p{x}^2+qx+1=0$ will have real roots?

1. A.12
2. B.10
3. C.7
4. D.6

Directions for the following three (03) items : Read the following information and answer the three items that follow : Consider the function $f(x)=g(x)+h(x)$ where $g(x)=\sin \left(\frac{x}{4}\right)$ and $h(x)=\cos \left(\frac{4x}{5}\right)$ What is the period of the function $g(x)$?

1. A.$\pi$
2. B.$2\pi$
3. C.$4\pi$
4. D.$8\pi$

Directions for the following three (03) items : Read the following information and answer the three items that follow : Consider the function $f(x)=g(x)+h(x)$ where $g(x)=\sin \left(\frac{x}{4}\right)$ and $h(x)=\cos \left(\frac{4x}{5}\right)$ What is the period of the function $g(x)$? What is the period of the function $h(x)$?

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

Directions for the following three (03) items : Read the following information and answer the three items that follow : Consider the function $f(x)=g(x)+h(x)$ where $g(x)=\sin \left(\frac{x}{4}\right)$ and $h(x)=\cos \left(\frac{4x}{5}\right)$ What is the period of the function $g(x)$? What is the period of the function $f(x)$?

1. A.$10\pi$
2. B.$20\pi$
3. C.$40\pi$
4. D.$80\pi$

Directions for the following three (03) items : Read the following information and answer the three items that follow : Consider the function $f(x)=g(x)+h(x)$ where $g(x)=\sin \left(\frac{x}{4}\right)$ and $h(x)=\cos \left(\frac{4x}{5}\right)$ What is the period of the function $g(x)$? Consider the following statements : 1. $f[g(x)]$ is a polynomial of degree 3. 2. $g[g(x)]$ is a polynomial of degree 2. 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 the range of a set of observations on a variable $X$ is known to be 25 and if $Y=40+3X$, then what is the range of the set of corresponding observations on $Y$?

1. A.25
2. B.40
3. C.75
4. D.115

Which one of the following is the second degree polynomial function $f(x)$ where $f(0)=5$, $f(-1)=10$ and $f(1)=6$?

1. A.$5x^2-2x+5$
2. B.$3x^2-2x-5$
3. C.$3x^2-2x+5$
4. D.$3x^2-10x+5$

Suppose $X=\{1,2,3,4\}$ and R is a relation on X. If $R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)\}$, then which one of the following is correct?

1. A.R is reflexive and symmetric, but not transitive
2. B.R is symmetric and transitive, but not reflexive
3. C.R is reflexive and transitive, but not symmetric
4. D.R is neither reflexive nor transitive, but symmetric

A relation R is defined on the set N of natural numbers as $xRy\Rightarrow x^2-4xy+3y^2=0$. Then which one of the following is correct?

1. A.R is reflexive and symmetric, but not transitive
2. B.R is reflexive and transitive, but not symmetric
3. C.R is reflexive, symmetric and transitive
4. D.R is reflexive, but neither symmetric nor transitive

Consider the following statements for the two non-empty sets A and B: 1. $(A\cap B)\cup (A\cap \bar{B})\cup (\bar{A}\cap B)=A\cup B$ 2. $(A\cup (\bar{A}\cap \bar{B}))=A\cup B$ 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

Let X be a non-empty set and let A, B, C be subsets of X. Consider the following statements: 1. $A\subset C\Rightarrow (A\cap B)\subset (C\cap B)$, $(A\cup B)\subset (C\cup B)$ 2. $(A\cap B)\subset (C\cap B)$ for all sets $B\Rightarrow A\subset C$ 3. $(A\cup B)\subset (C\cup B)$ for all sets $B\Rightarrow A\subset C$ Which of the above statements are correct?

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 $A=\{\lambda ,\{\lambda ,\mu \}\}$, then the power set of A is

1. A.$\{\varphi ,\{\varphi \},\{\lambda \},\{\lambda ,\mu \}\}$
2. B.$\{\varphi ,\{\lambda \},\{\{\lambda ,\mu \}\},\{\lambda ,\{\lambda ,\mu \}\}\}$
3. C.$\{\varphi ,\{\lambda \},\{\lambda ,\mu \},\{\lambda ,\{\lambda ,\mu \}\}\}$
4. D.$\{\{\lambda \},\{\lambda ,\mu \},\{\lambda ,\{\lambda ,\mu \}\}\}$

A function f defined by $f(x)=\ln (\sqrt{x^2+1}-x)$ is

1. A.an even function
2. B.an odd function
3. C.Both even and odd function
4. D.Neither even nor odd function

The domain of the function f defined by $f(x)={\log }_{x}10$ is

1. A.$x>10$
2. B.$x>0$ excluding $x=10$
3. C.$x\ge 10$
4. D.$x>0$ excluding $x=1$

The domain of the function $f(x)=\sqrt{(2-x)(x-3)}$ is

1. A.$(0,\infty )$
2. B.$[0,\infty )$
3. C.$[2,3]$
4. D.$(2,3)$

If $f(x)=\frac{x-2}{x+2}$, $x\ne -2$, then what is $f^{-1}(x)$ equal to?

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

If $f(x)={\log }_{10}(1+x)$, then what is $4f(4)+5f(1)-{\log }_{10}2$ equal to ?

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

${\lim }_{x\to 0}\frac{1-{\cos }^{3}4x}{x^2}$ is equal to

1. A.0
2. B.12
3. C.24
4. D.36

If A, B and C are subsets of a Universal set, then which one of the following is not correct?

1. A.$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
2. B.$A^{\prime }\cup (A\cup B)=(B^{\prime }\cap A{)}^{\prime }\cup A$
3. C.$A^{\prime }\cup (B\cup C)=(C^{\prime }\cap B{)}^{\prime }\cap A^{\prime }$
4. D.$(A\cap B)\cup C=(A\cup C)\cap (B\cup C)$

How many students like to play all the three games?

1. A.14
2. B.21
3. C.28
4. D.35

How many students like to play exactly only one game?

1. A.196
2. B.228
3. C.254
4. D.268

Let $A=\{x\in \mathbf{R}:-1\le x\le 1\}$, $B=\{y\in \mathbf{R}:-1\le y\le 1\}$ and S be the subset of $A\times B$, defined by $S=\{(x,y)\in A\times B:x^2+y^2=1\}$. Which one of the following is correct?

1. A.S is a one-one function from A into B
2. B.S is a many-one function from A into B
3. C.S is a bijective mapping from A into B
4. D.S is not a function

What is the value of ${\left(\frac{-1+i\sqrt{3}}{2}\right)}^{3n}+{\left(\frac{-1-i\sqrt{3}}{2}\right)}^{3n}$, where $i=\sqrt{-1}$?

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