Relations & Functions Questions (810)

If \(f: R \to R\) is defined as \(f(x) = \begin{cases} x+4, & x

The ellipse \(\frac{(a-6)^2}{3^2} + \frac{(b-5)^2}{2^2} = 1\) passes through \((4, 6)\). If \(-1

Let $A=\{2,3,4\}$, $B=\{8,9,12\}$. Let $R=\{((a_1,b_1),(a_2,b_2)):\ a_1|b_2\ \text{and}\ a_2|b_1\}$. Number of elements in $R$ is

In 60 athletes who won medals, $n(A)=48$, $n(B)=25$, $n(C)=18$, $n(A\cap B\cap C)=5$. Number winning exactly two events is

Let $A=\{1,2,3,4,5\}$ and $B=\{1,2,3,4,5,6\}$. Then the number of functions $f:A\to B$ satisfying $f(1)+f(2)=f(4)-1$ is equal to........

Let R be a relation on \mathbb{R}, given by R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number}\}. Then R is

The relation R = \{(a, b) : \gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\} is:

Let P(S) denote the power set of S = \{1, 2, 3, \ldots, 10\}. Define the relations R_1 and R_2 on P(S) as: A R_1 B if (A \cap B^c) \cup (B \cap A^c) = \varnothing, and A R_2 B if A \cup B^c = B \cup A^c, for all A, B \in P(S). Then:

Let $D$ be the domain of $f(x)=\sin^{-1}\!\left(\log_{3x}\dfrac{6+2\log_{3x}}{-5x}\right)$. If the range of $g:D\to\mathbb{R}$ defined by $g(x)=x-[x]$ is $(\alpha,\beta)$, then $\alpha^2+\dfrac{5}{\beta}$ is equal to

Given X = \{ n \in \mathbb{N} : 1 \leq n \leq 50 \}, A = \{ n \in X : n \text{ is multiple of } 2\} = \{2, 4, 6, 8, \ldots, 50\}, and B = \{ n \in X : n \text{ is multiple of } 7\} = \{7, 14, 21, 28, 35, 42, 49\}. Find the smallest subset of X containing elements of both A and B.

Let $A=\{0,3,4,6,7,8,9,10\}$ and $R=\{(x,y):\ x-y\text{ is odd positive}\text{ or }x-y=2\}$. Minimum elements to add to $R$ for it to be symmetric is equal to _________

Let $R=\{a,b,c,d,e\}$ and $S=\{1,2,3,4\}$. Total number of onto functions $f:R\to S$ such that $f(a)\neq1$ is equal to ________.

If $f(\tan z) = \cos 2z, z \neq \left(2n+1\right)\frac{\pi}{2}, n \in \mathbb{Z}$, then incorrect statement is

Let $A=\{1,2,3,4\}$ and $R=\{((a,b),(c,d)):\ 2a+3b=4c+5d\}$ on $A\times A$. Number of elements in $R$ is _________.

The statement \(p \to (q \to p)\) is equivalent to

Let \(f(2-x)=f(2+x)\) and \(f(20-x)=f(x)\) for all \(x\in\mathbb{R}\). If \(f(0)=5\), minimum solutions of \(f(x)=5\) on \([0,170]\):

If the domain of $\log_e\!\left(\dfrac{6x^2+5x+1}{2x-1}\right)+\cos^{-1}\!\left(\dfrac{2x^2-3x+4}{3x-5}\right)$ is $(\alpha,\beta)\cup(\gamma,\delta)$, then $18(\alpha^2+\beta^2+\gamma^2+\delta^2)$ is equal to

Let S = N \cup {0}. Define a relation R from S to R by : 2 R = {(x, y) : log y = x log ( ), x \in S, y \in R} e e 5 Then, the sum of all the elements in the range of R is equal to :

Which of the following is inverse to itself?

The function \(f:[0,3] \to [1,29]\), defined by \(f(x) = 2x^3 - 15x^2 + 36x + 1\), is:

Let $A$ and $B$ denote the domain and range respectively of $f(x)=\dfrac{1}{\sqrt{[x]-x}}$, where $[x]$ denotes the smallest integer $\geq x$. Among (S1): $A\cap B=(1,\infty)\setminus\mathbb{N}$ and (S2): $A\cup B=(1,\infty)$,

Let \(T = \{(x, y) : x - y \in I\}\). Then which of the following is true?

If the domain of $f(x)=\sec^{-1}\!\left(\dfrac{2x}{5x+3}\right)$ is $[\alpha,\beta)\cup(\gamma,\delta]$, then $|3\alpha+10(\beta+\gamma)+21\delta|$ is equal to __________

Find the range of \(f(x) = \log_2\left(\dfrac{4}{\sqrt{x+2}+\sqrt{2-x}}\right)\)

Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. Then the number of many-one functions $f : A \to B$ such that $1 \in f(A)$ is equal to:

The absolute minimum value of the function $f(x) = |x^2 - x + 1| + [x^2 - x + 1]$, where $[t]$ denotes the greatest integer function, in the interval $[-1, 2]$, is:

Let $A = \{(x,y) \in \mathbb{R} \times \mathbb{R} : |x+y| \geq 3\}$ and $B = \{(x,y) \in \mathbb{R} \times \mathbb{R} : |x|+|y| \leq 3\}$. If $C = \{(x,y) \in A \cap B : x = 0 \text{ or } y = 0\}$, then $\sum_{(x,y) \in C} |x+y|$ is:

If the domain of the function $f(x)=\dfrac{\sqrt{x^2-25}}{(4-x^2)}+\log_{10}(x^2+2x-15)$ is $(-\infty,\alpha)\cup[\beta,\infty)$, then $\alpha^2+\beta^3$ is equal to:

The number of elements in the relation $R=\{(x,y):4x^2+y^2<52,\,x,y\in\mathbb{Z}\}$ is

Let $S$ be the set of the first 11 natural numbers. Then the number of elements in $A=\{B\subseteq S:n(B)\geq2\text{ and the product of all elements of }B\text{ is even}\}$ is _____.

Let the relation $R$ on the set $M=\{1,2,3,\ldots,16\}$ be given by $R=\{(x,y):4y=5x-3,\,x,y\in M\}$. Then the minimum number of elements required to be added in $R$, in order to make the relation symmetric, is equal to

If f (x) = x x 2 ,x \in R , then \sum 81 k=1 f ( k 82 ) is equal to 2 +\sqrt2

Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisements is:

The equation $\sin(\cos x) = x$ has only one root $x_1$ in $(0, \pi/2)$ and the equation $\cos(\sin x) = x$ has also only one root $x_2$ in $(0, \pi/2)$. Then:

Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. Then the number of many-one functions f : A \to B such that 1 \in f ( A) is equal to :

Let $A = \{1, 2, 3, \ldots, 10\}$ and $B = \left\{\frac{m}{n} : m, n \in A,\, m < n \text{ and } \gcd(m,n) = 1\right\}$. Then $n(B)$ is equal to:

Let $S = \mathbb{N} \cup \{0\}$. Define a relation $R$ from $S$ to $\mathbb{R}$ by $R = \left\{(x,y) : \log_e y = x \log_e\left(\frac{2}{5}\right),\, x \in S,\, y \in \mathbb{R}\right\}$. Then, the sum of all elements in the range of $R$ is equal to:

Let $R$ be a relation defined on the set $\{1,2,3,4\}\times\{1,2,3,4\}$ by $R=\{((a,b),(c,d)):2a+3b=3c+4d\}$. Then the number of elements in $R$ is

Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3,\ldots,20\}$ be given by $R_1=\{(x,y):2x-3y=2\}$ and $R_2=\{(x,y):-5x+4y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals

Number of functions $f: \{1, 2, \ldots, 100\} \to \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ___

Find the number of real solutions of \([x]^2+2[x+2]-7=0\).

The minimum number of elements that must be added to the relation R = \{(a,b),(b,c)\} on the set \{a, b, c\} so that it becomes symmetric and transitive is:

The number of non-empty equivalence relations on the set {1, 2, 3} is :

Let $A=\{1,2,3,4,5\}$. Let $R$ be a relation on $A$ defined by $x\,R\,y$ if and only if $4x\leq5y$. Let $\mathrm{m}$ be the number of elements in $R$ and $\mathrm{n}$ be the minimum number of elements from $A\times A$ that are required to be added to $R$ to make it a symmetric relation. Then $\mathrm{m}+\mathrm{n}$ is equal to:

Number of functions f : {1, 2, \ldots , 100} \to {0, 1}, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ________.

If f(x) = 1/x and g(x) = {x²}, then the number of positive roots satisfying the equations f(x) = g(x) such that 2 ≤ x² ≤ 3

Let A = {(x, y) \in R \times R : |x + y| \ge 3} and B = {(x, y) \in R \times R : |x| + |y| \le 3}. If C = {(x, y) \in A \cap B : x = 0 or y = 0}, then \sum (x,y)\inC |x + y| is :

Let the set $S=\{2,4,8,16,\ldots,512\}$ be partitioned into 3 sets $A,B,C$ with equal number of elements such that $A\cup B\cup C=S$ and $A\cap B=B\cap C=A\cap C=\phi$. The maximum number of such possible partitions of $S$ is equal to:

If $f(x)=\begin{cases}2+2x,&-1\leq x<0\\1-\dfrac{x}{3},&0\leq x\leq3\end{cases}$; $g(x)=\begin{cases}-x,&-3\leq x\leq0\\x,&0<x\leq1\end{cases}$, then range of $(f\circ g)(x)$ is

Let \(g(x)\) be a function defined on \([-1, 1]\). If the area of the equilateral triangle with two of its vertices at \((0, 0)\) and \(\left(x, g(x)\right)\) is \(\frac{\sqrt{3}}{4}\), then the function \(g(x)\) is