List of top Mathematics Questions on Relations and functions asked in JEE Main

If $f(x) = (x - 2)^2 (x - 3)^3$ and $x ∈ [1, 4]$ and If $M$ and $m$ denotes maximum and minimum values respectively, then $M - m$ is

Consider the relations $R_1$ and $R_2$ defined as \[a R_1 b \iff a^2 + b^2 = 1 \quad \text{for all } a, b \in \mathbb{R},\]and \[(a, b) R_2 (c, d) \iff a + d = b + c \quad \text{for all } (a, b), (c, d) \in \mathbb{N} \times \mathbb{N}.\]Then:

The interval in which the function $ f(x) = x^x, \, x>0 $, is strictly increasing is:

Let the relations $ R_1 $ and $ R_2 $ on the set
$ X = \{ 1, 2, 3, \dots, 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:

The function $ f(x) = \frac{x^2 + 2x - 15}{x^2 - 4x + 9} $, $ x \in \mathbb{R} $ is:

Let a relation $ R $ on $ \mathbb{N} \times \mathbb{N} $ be defined as: $$(x_1, y_1) \, R \, (x_2, y_2) \text{ if and only if } x_1 \leq x_2 \text{ or } y_1 \leq y_2.$$
Consider the two statements:
[(I)] $ R $ is reflexive but not symmetric.
[(II)] $ R $ is transitive.
Then which one of the following is true:

Let $ A = \{1, 2, 3, \ldots, 100\} $. Let $ R $ be a relation on $ A $ defined by $ (x, y) \in R $ if and only if $ 2x = 3y $. Let $ R_1 $ be a symmetric relation on $ A $ such that $ R \subset R_1 $ and the number of elements in $ R_1 $ is $ n $. Then, the minimum value of $ n $ is $\_\_\_\_\_$.

Let $ f : \mathbb{R} \rightarrow \mathbb{R} $ be a function defined by \[ f(x) = \frac{4^x}{4^x + 2} \] and \[ M = \int_{f(a)}^{f(1 - a)} x \sin^4 \left( x (1 - x) \right) \, dx, \] \[ N = \int_{f(a)}^{f(1 - a)} \sin^4 \left( x (1 - x) \right) \, dx; \quad a \neq \frac{1}{2}. \] If $ \alpha M = \beta N $, $ \alpha, \beta \in \mathbb{N} $, then the least value of $ \alpha^2 + \beta^2 $ is equal to _____

Let $ A = \{1, 2, 3, 4\} $ and $ R = \{(1, 2), (2, 3), (1, 4)\} $ be a relation on $ A $.Let $ S $ be the equivalence relation on $ A $ such that $ R \subseteq S $ and the number of elements in $ S $ is $ n $. Then, the minimum value of $ n $ is _____

Let $ A = \{ 1, 2, 3, \dots, 20 \} $. Let $ R_1 $ and $ R_2 $ be two relations on $ A $ such that $R_1 = \{(a, b) : b \text{ is divisible by } a\}$
and $R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}$.Then, the number of elements in $ R_1 - R_2 $ is equal to $\_\_\_\_.$

If $f(x) =ln(\frac {1-x^2}{1+x^2})$ then value of $225(f'(x) – f''(x))$ at $x=\frac 12$

If the domain of the function $ f(x) = \log_e \left( \frac{2x + 3}{4x^2 + x - 3} \right) + \cos^{-1} \left( \frac{2x - 1}{x + 2} \right) $ is $ (\alpha, \beta] $, then the value of $ 5\beta - 4\alpha $ is equal to

If $f(x)=\frac {4x+3}{6x-4}$, $x≠\frac 23 $and $(fof)(x)=g(x)$, where $g:R-[\frac 23→R→{\frac 23}]$. Then $(gogog)(4)$ is equal to

If $f(x)=\begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\[0.3em] 3x^2+2 & 2x & x^3+6 \\[0.3em] x^3-x & 4 & x^2-2 \end{vmatrix}$ for all $x∈R$, then $2f(0) + f'(0)$ is equal to

Let $f^1(x)=\frac{3 x+2}{2 x+3}, x \in R -\left\{\frac{-3}{2}\right\}$ For $n \geq 2$, define $f^{ n }(x)=f^1 o f^{ n -1}(x)$ if $f^5(x)=\frac{ a x+ b }{ b x+ a }, \operatorname{gcd}( a , b )=1$, then $a + b$ is equal to ______ .

Let A = {1, 2, 3, 4} and R be a relation on the set AxA defined by R = {((a, b), (c, d)): 2a+3b = 4c +5d}. Then the number of elements in R is

Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $a d(b-c)=b c(a-d)$ Then $R$ is

Let $f:(0,1) \rightarrow R$ be a function defined by $f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$ Consider two statements (I) $g$ is an increasing function in $(0,1)$ (II) $g$ is one-one in $(0,1)$Then,

Let $f:(0,1) \rightarrow R$ be a function defined by

$f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$ Consider two statements

(I) $g$ is an increasing function in $(0,1)$

(II) $g$ is one-one in $(0,1)$Then,

Let for $x \in R$ $f(x)=\frac{x+|x|}{2} \text { and } g(x)=\begin{cases}x, & x<0 \\x^2, & x \geq 0\end{cases} $

Then area bounded by the curve $y=(f \circ g)(x)$ and the lines $y=0,2 y-x=15$ is equal to

For $x \in \mathbb{R}$, two real‐valued functions $f(x)$ and $g(x)$ are such that

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) \neq 1 $, is equal to:

A has 5 elements and B has 2 elements. The number of subsets of A × B such that the number of elements in subset is more than or equal to 3 and less than 6, is?

For differentiable function f: R-{0}⟶ R, let 3f(x)+2f($\frac{1}{x}$)-$\frac{1}{x}$-10, then |f(3)+f($\frac{1}{4}$)| is equal to

Let $f(x)=\frac{sinx+cosx-\sqrt2}{sinx-cosx},x∈[0.\pi]-\left[\frac{\pi}{4}\right]. Then f\left(\frac{7\pi}{12}\right)f^n\left(\frac{7\pi}{12}\right)$is equal to