Relations & Functions Questions - JEE

A survey shows that 63% of the people watch a news channel whereas, 76% watch an entertainment channel at a particular time. If $x\%$ of the people watch both types of channels, then

Let f : R - {0} \to R be a function such that f (x) - 6f ( 1 ) = 35 - 5 . x 3x 2 If the lim x\to0 ( 1 \alphax + f (x)) = \beta; \alpha, \beta \in R , then \alpha + 2\beta is equal to

If g is the inverse of a function f and $f'(x) = \dfrac{1}{1+x^5}$, then $g'(x)$ is equal to

If roots of the equation $|x-1| = 2[x] - 3\{x\}$ is $x_i$, then find $\left[\dfrac{\sum x_i}{5}\right]$ here (where $[x]$ & $\{x\}$ denotes integral and fractional part of $x$)

If $f(x) = \dfrac{2^{2x}}{2^{2x}+2}$, $x \in \mathbb{R}$, then $f\!\left(\dfrac{1}{2023}\right) + f\!\left(\dfrac{2}{2023}\right) + \cdots + f\!\left(\dfrac{2022}{2023}\right)$ is equal to

Let R1 = {(p, pn) : p prime, n ≥0, pn ≤50} and R2 = {(p, pn) : p prime, n = 0 or 1} on {1, . . . , 50}. The number of elements in R1 −R2 is:

The domain of the real-valued function $f(x) = \frac{x-2}{(x-1)\sqrt{x^2-4}}$ is

The number of real roots of the equation 5 + |2x - 1| = 2x (2x - 2) is :

Find the inverse of $y=5^{\log x}$.

Consider the following statements P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statement "Suman is brilliant and dishonest if any only if Suman is rich" can be expressed as

Let $f$ be a function defined on the set of real numbers such that for $x \geq 0$, $f(x) = 3\sin x + 4\cos x$. Then $f(x)$ at $x = -\dfrac{11\pi}{6}$ is equal to

Let $f : \mathbb{N} \to Y$ be a function defined as $f(x) = 4x + 3$, where $Y = \{y \in \mathbb{N} : y = 4x + 3 \text{ for some } x \in \mathbb{N}\}$. Show that f is invertible and its inverse is

JM Q20.

Let $S = \{1,2,3,4,5,6\}$. Then the number of one-one functions $f: S \to P(S)$, where $P(S)$ denotes the power set of $S$, such that $f(n) \subset f(m)$ where $n < m$ is ______.

If $f(x)$ be a function satisfying $f(x) \cdot f\!\left(\dfrac{1}{x}\right) = f(x) + f\!\left(\dfrac{1}{x}\right)$ and $f(4) = 65$, then find $f(6)$.

In a town of 10,000 families, it was found that 40% families buy newspaper A, 20% buy newspaper B and 10% buy newspaper C. Also, 5% families buy newspapers A and B, 3% buy newspapers B and C and 4% buy newspapers A and C. If 2% families buy all the three newspapers, then number of families which buy newspaper A only is

Let $f(x) = 9^x/(9^x + 3)$ for all $x \in \mathbb{R}$. Then the value of $\displaystyle\sum_{r=1}^{2008} f(r/2009)$ is ______.

If $f : A \to B$, $f(x) = \sin^{-1}\left(\dfrac{[x]}{\{x\}}\right)$ and $g : C \to D$, $g(x) = \cos^{-1}\left(\dfrac{[x]}{\{x\}}\right)$, then which of the following is always correct?[Note: $[\cdot]$ and $\{\cdot\}$ denotes greatest integer and fractional part function respectively.]

Let Z be the set of integers. If $A = \{x \in \mathbb{Z} : 2^{(x+2)(x^2-5x+6)} = 1\}$ and $B = \{x \in \mathbb{Z} : -3 , then the number of subsets of the set \(A \times B$ is

JM Q25.

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D be $\vec{a}-\vec{b}+\vec{c}$, $\lambda\vec{a}-3\vec{b}+4\vec{c}$, $-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectively. If $\overrightarrow{AB}$, $\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar, then $\lambda$ is:

If $f(x) = 3x + |x|$, $g(x) = \dfrac{3x}{4} - \dfrac{|x|}{4}$, then:

${\sim}(p \lor q) \lor ({\sim}p \land q)$ is equivalent to:

Find the domain and range of the function $f(x) = \sqrt{2 - x} + \sqrt{1 + x}$.

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

A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a possible value of x can be

The domain of $f(x) = \dfrac{\log_{(x+1)}(x-2)}{e^{2\log_e x} - (2x+3)}$, $x \in \mathbb{R}$ is

If $f(x) = \log\frac{1+x}{1-x}$, then

Let R1 = {(a, b) ∈N × N : |a −b| ≤13} and R2 = {(a, b) ∈N × N : |a −b| ̸= 13}. Then:

Let $z$ be the set of integers. If $A = \{x \in z : 2^{(x+2)(x^2 - 5x + 6)} = 1\}$ and \(B = \{x \in z : -3

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

$f(x+y)=f(x)+f(y)$, $f(1)=2$. $g(n)=\sum_{k=1}^{n-1}f(k)$. Find $n$ with $g(n)=20$.

The number of functions f from $\{1, 2, 3, \ldots, 20\}$ onto $\{1, 2, 3, \ldots, 20\}$ such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :-

JM Q29.

For suitable $a$, $f(x)=\dfrac{a-x}{a+x}$ with $(f\circ f)(x)=x$. Find $f\!\left(-\frac{1}{2}\right)$.

For some $a, b, c \in \mathbb{N}$, let $f(x) = ax - 3$ and $g(x) = x^b + c$, $x \in \mathbb{R}$. If $(f \circ g)^{-1}(x) = \left(\dfrac{x-7}{2}\right)^{1/3}$, then $(f \circ g)(ac) + (g \circ f)(b)$ is equal to ____.

If $p$, $q$ and $r$ are simple propositions with truth values T, F and T, respectively, then the truth value of $(\sim p \vee q) \wedge \sim r \rightarrow p$ is

Consider a function $f: \mathbb{N} \to \mathbb{R}$, satisfying $f(1) + 2f(2) + 3f(3) + \cdots + xf(x) = x(x+1)f(x)$; $x \ge 2$ with $f(1)=1$. Then $\dfrac{1}{f(2022)} + \dfrac{1}{f(2028)}$ is equal to

JM Q34.

If the statements $(p \wedge \sim r) \rightarrow (q \vee r)$, $q$ and $r$ are all false, then $p$

Let a relation $R$ on $\mathbb{N}\times\mathbb{N}$ be defined as: $(x_1,y_1)\,R\,(x_2,y_2)$ if and only if $x_1\leq x_2$ 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 X, Y be two sets and X has 40 elements, $X \cup Y$ has 60 elements and $X \cap Y$ has 10 elements. Then the number of elements in Y is

The real function $f(x) = \cos^{-1}\sqrt{x^2 + 3x + 1} + \cos^{-1}\sqrt{x^2 + 3x}$ is defined on the set

If $g(x)=x^2+x-1$ and $(g\circ f)(x)=4x^2-10x+5$, find $f\!\left(\frac{5}{4}\right)$.

Let P(S) be the power set of S = {1, 2, . . . , 10}. Define A R1 B iff (A ∩Bc) ∪(B ∩Ac) = ∅, and A R2 B iff A ∪Bc = B ∪Ac. Then:

Let $f:(-\infty,-2]\to[3,\infty)$, $f(x)=x^2+2x+3$. Find $f^{-1}(x)$.

Let $f^1(x) = \dfrac{3x+2}{2x+3}$, $x \in \mathbb{R} - \left\{-\dfrac{3}{2}\right\}$. For $n \ge 2$, define $f^n(x) = f^1 \circ f^{n-1}(x)$. If $f^5(x) = \dfrac{ax+b}{bx+a}$, $\gcd(a,b)=1$, then $a+b$ is equal to ______.

If $f(\ln(1+|x|)) = (1 - \ln(1+|x|))^{\frac{1}{7}}$, then $f(f(\cos x))$ is equal to:

JM Q23.

JM Q18 -- functional equation.

Relations & Functions Questions (810)