Relations & Functions Questions - JEE

Let $A = \{1, 2, 3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is ___

Let $f : \mathbb{R} - \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x)f\!\left(\frac{1}{x}\right) = f(x) + f\!\left(\frac{1}{x}\right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is:

If $f(x) = \dfrac{2^x}{2^x + \sqrt{2}}$, $x \in \mathbb{R}$, then $\displaystyle\sum_{k=1}^{81} f\!\left(\frac{k}{82}\right)$ is equal to:

If a function $f$ satisfies $f(m+n)=f(m)+f(n)$ for all $m,n\in\mathbb{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that $\sum_{k=1}^{2022}f(\lambda+k)\leq(2022)^2$ is equal to ________.

Let $A=\{(x,y):2x+3y=23,\ x,y\in\mathbb{N}\}$ and $B=\{x:(x,y)\in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to ________.

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $\mathrm{m}$ and $\mathrm{n}$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to _____

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A\times B$ by $(a_1,b_1)\,R\,(a_2,b_2)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is _____

If g(x) = x^2 + x - 1 and (gof)(x) = 4x^2 - 10x + 5, then f\left( \frac{5}{4} \right) is equal to

137. If $g(x)$ and $h(x)$ are invertible functions and $h(x)=3g(x)+7$, then $h^{-1}(x)$ is equal to:

Let f : R - {0} \to (-\infty, 1) be a polynomial of degree 2, satisfying f (x)f ( 1 ) = f (x) + f ( 1 ) . If x x f (K) = -2K , then the sum of squares of all possible values of K is :

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

Let $X = \mathbb{R} \times \mathbb{R}$. Define a relation $R$ on $X$ as: $(a_1, b_1)\,R\,(a_2, b_2) \Leftrightarrow b_1 = b_2$. Statement I: $R$ is an equivalence relation. Statement II: For some $(a,b) \in X$, the set $S = \{(x,y) \in X : (x,y)\,R\,(a,b)\}$ represents a line parallel to $y = x$. Choose the correct option:

The function $f : (-\infty, \infty) \to (-\infty, 1)$, defined by $f(x) = \dfrac{2^x - 2^{-x}}{2^x + 2^{-x}}$ is:

The relation $R = \{(x,y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even}\}$ is:

We first draw the graphs of $x + 2y = 10,\ x + y = 6,\ 3x + y = 8$. The shaded region ABCD is the feasible region R determined by the above constraints. The feasible region is unbounded. Therefore, minimum of z may or may not occur. If it occurs, it will be on the corner point. Find the minimum value of z.

If P(S) denotes the set of all subsets of a given set S, then the number of one-to-one functions from the set S = {1, 2, 3} to the set P(S) is

A function $f:\mathbb{I}\to\mathbb{I}$ is: $f(n)=n+3$ (odd $n$), $f(n)=n/2$ (even $n$). $k$ is odd and $f(f(f(k)))=27$. Then sum of digits of $k$ is

If $f(x) = x^{11} + x^9 - x^7 + x^5 + x^3 + 1$ and $f(\arcsin(\sin 8)) = a$, where $a$ is a constant, then $f(\arctan(\tan 8))$ is equal to

Let $f:[1,\infty)\to[3,\infty)$, $f(x)=(\log_2 x)^2+2\log_2 x+3$. Which are correct?

Let $f(x)$ be a function such that $f(x+y) = f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1) = 3$ and $\displaystyle\sum_{k=1}^{n} f(k) = 3279$, then the value of $n$ is

The shaded region in the given figure is(The figure shows three overlapping circles A, B, C where the region inside A but outside B and C is shaded.)

Let $f(x) = 2x^n + \lambda$, $\lambda \in \mathbb{R}$, $n \in \mathbb{N}$, and $f(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is

Let $R = \{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:

The domain of $f(x)=\dfrac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer $\leq x$)

The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is

Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Relation $R=\{((a_1,b_1),(a_2,b_2)):\ a_1\leq b_2\ \text{and}\ b_1\leq a_2\}$ has how many elements?

Among S = {(a, b) : a, b ∈R \ {0}, 2 + a/b > 0} and T = {(a, b) : a, b ∈R, a2 −b2 ∈Z}:

Find the range of $f(x) = \frac{2x}{1+x^2}$.

Let A = A1 ∪A2 ∪· · · ∪Ak where Ai ∩Aj = ∅for i ̸= j. Define R = {(x, y) : y ∈Ai ⇔ x ∈Ai}. Then R is:

The reflection about the line $x + y = 0$ of the inverse function $f^{-1}(x)$ of a function $f(x)$ is:

Let the domain of the function $f(x)=\log_3\log_5\!\left(7-\log_2(x^2-10x+85)\right)+\sin^{-1}\!\left(\left|\dfrac{3x-7}{17-x}\right|\right)$ be $(\alpha,\beta]$. Then $\alpha+\beta$ is equal to:

Let $f(g(x))=x+3-\sqrt{x}$ where $g(x)=\sqrt{x}+1$. Then $f(0)$ is equal to

The equation $x^2 - 4x + [x] + 3 = x[x]$, where $[x]$ denotes the greatest integer function, has:

If $ f(x) + 2f\left(\frac{1}{x}\right) = 3x, x \neq 0, $ and S = $ \{x \in \mathbb{R} : f(x) = f(-x)\} $; then S :

Let R be a relation on \mathbb{N} \times \mathbb{N} defined by (a, b) R (c, d) if and only if ad(b - c) = bc(a - d). Then R is

For $ x \in \mathbb{R} - \{0, 1\} $, let f_1(x) = \frac{1}{x}, f_2(x) = 1 - x and f_3(x) = \frac{1}{1 - x} \) be three given functions. If a function, J(x) satisfies $ f_2 \circ f_1 \circ f(x) = f_3(x) $ then J(x) is equal to :-

If R = {(x, y) : x, y ∈Z, x2 + 3y2 ≤8}, then the domain of R−1 is:

Let R1 = {(a, b) ∈R2 : a2 + b2 ∈Q} and R2 = {(a, b) ∈R2 : a2 + b2 /∈Q}, where Q is the set of rationals. Then:

Let R = {(a, b) : 3a −3b + √ 7 is irrational} on R. Then R is:

The domain of definition of the function $y = 3e^{x-1}\log(x-1)$ is

Let $ \mathbb{N} $ be the set of natural numbers and two functions f and g be defined as f,g : $ \mathbb{N} \to \mathbb{N} $ such that : $ f(n) = \frac{n + 1}{2} $ if n is odd and g(n) = n - (-1)n. The fog is :

Suppose $f(x) = x^3 + \log_2\left(x + \sqrt{x^2 + 1}\right)$. For any $a, b \in \mathbb{R}$ to satisfy $f(a) + f(b) \geq 0$, the condition $a + b \geq 0$ is:

Let $f(x) = ([a]^2 - 5[a] + a)x^3 - (8[a]^2 - 5[a] + 1)x - (\tan x)\operatorname{sgn}x$, be an even function for all $x \in \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$, then sum of all possible values of $a$ is: (where $[.]$ and $\{.\}$ denotes greatest integer function and fractional part functions, respectively)

The range of $f(x) = \tan x + \frac{1}{2}\sin^{-1} x$ is:

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

Let f(x) = x^2, x \in \mathbb{R}. For any A \subseteq \mathbb{R}, define g(A) = \{x \in \mathbb{R} : f(x) \in A\}. If S = [0, 4], then which one of the following statements is not true ?

The domain of the function $f(x) = \frac{3}{5-x} + \log_{10}(x^2 - 3x)$ is

If the domain of the function $f(x) = \dfrac{[x]}{1+x^2}$, where $[x]$ is greatest integer $\le x$, is $(2, 6)$, then its range is

The function $ f : \mathbb{R} \to \left[-\frac{1}{2}, \frac{1}{2}\right] $ defined as $ f(x) = \frac{x}{1+x^2} $ is :

Relations & Functions Questions (810)