Relations & Functions Questions - JEE

If $f(x) = \{x\} + \left\{x + \left[\dfrac{x}{1+x^2}\right]\right\} + \left\{x + \left[\dfrac{x}{1+2x^2}\right]\right\} + \ldots + \left\{x + \left[\dfrac{x}{1+99x^2}\right]\right\}$ then value of $f(\sqrt{2})$ isNote: $[k]$ and $\{k\}$ denote greatest integer and fractional part functions of $k$ respectively.

The Boolean expression $\sim(p \vee q) \vee (\sim p \vee q)$ is equivalent to

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as a R b if a is congruent to b for all a, b ∈ T. Then R is

For any two statements $p$ and $q$, the negation of the expression $p \vee (\sim p \wedge q)$ is:

Let $f(x) = (x+2)^2 - 2,\ x \geq -2$. If $g(x)$ is a function whose graph is reflection of the graph of $y = f(x)$ in the line $y = x$, then $g(x)$ is equal to:

Which of the following sets is empty?

Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{-2, -1, 0, 1, 2, 3, 4, 5\}$. The number of increasing functions from A to B is

The statement ~$(p \leftrightarrow\ \sim q)$ is

Given $N$ is the set of all natural numbers. Let $R_1 = \{(x, y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x, y) \in N \times N : x + 2y = 10\}$. The range of $R_2$ 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.]

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

Given relation is $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ on set $A = \{3, 6, 9, 12\}$. The relation $R$ is:

The statement $\sim(p \leftrightarrow \sim q)$ is

If $f(g(x)) = \ln\left(\dfrac{(1+x)^3}{(1-x)^3}\right) = 3f(x)$, then which of the following is true?

Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{-2, -1, 0, 1, 2, 3, 4, 5\}$. The number of non-decreasing functions from A to B is

If the function $ f(x) $ on the domain $ \left[\dfrac{1}{2}, \infty\right) $ is defined by $ f(x) = 2^{x(x-1)} $, then $ f^{-1}(x) $ equals:

Find the domain of the function $f(x) = \sqrt{3 - 2^x - 2^{1-x}} + \sqrt{\sin^{-1}x}$.

Let $A = \{1, 2, 3, 4, 5\}$. The number of onto functions from A to A such that $f(i) \neq i$ for all $i$, is

Two finite sets have m and n $(m > n)$ elements. The number of subsets of the first set is 112 more than that of the second set. The value of mn is

The domain of the definition of the function $f(x) = \dfrac{1}{4-x^2} + \log_{10}(x^3 - x)$ is:

If $ f(x) $ is an odd function, then $ f(0) $ equals:

Let S(K) = 1 + 3 + 5 + ⋯ + (2K − 1) = 3 + K2. Then which of the following is true?

Given $f(x) = x^2,\ x \in \mathbb{R}$, $g(A) = \{x \in \mathbb{R} : f(x) \in A\}$, $S \equiv [0, 4]$. If $g(s) = \{x \in \mathbb{R} : 0 \le x^2 \le 4\}$, which of the following is correct?

Let $A=\{2,3,5,7,9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x\leq3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l+m$ is equal to:

The domain of the real-valued function $f(x) = \log_{10}\frac{(3-x)(x+2)}{(x+1)(x-2)(x-4)}$ does not contain the intervals

The function $f(x)$ satisfies the functional equation $3f(x) + 2f\!\left(\dfrac{x+59}{x-1}\right) = 10x + 30$ for all real $x \neq 1$. The value of $f(7)$ is:

Let $A = \{x \in R : x\text{ is not a positive integer}\}$. Define a function $f : A \to R$ as $f(x) = \dfrac{2x}{x-1}$, then f is:

Let $f:\{1,2,3,4,5\}\to\{1,2,3,4,5\}$ be one-one with $f(x)=x+1\iff x$ is even. $f^{-1}(2)$ can be:

For $x\in\mathbb{R}\setminus\{0,1\}$, $f_1(x)=1/x, f_2(x)=1-x, f_3(x)=1/(1-x)$. If $(f_2\circ I\circ f_1)(x)=f_3(x)$, find $I(x)$.

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

If x = 2 + √3, and f(x) = (x² − 4x + 1)² + 2x − 2√3, find f(2 + √3).

Let A, B and C be sets such that $\phi \neq A \cap B \subseteq C$. Then which of the following statements is not true?

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

If the function $ f(x) $ on the domain $ \left[\dfrac{1}{2}, \infty\right) $ is defined by $ f(x) = 2^{x(x-1)} $, then $ f^{-1}(x) $ equals:

Which of the following is not a proposition?

If $f(3x+1) + f(3x-10) = 10$, find the period of $f(x)$.

$(A)$ one

In statistical survey of 1003 families of Kolkata, it was found that 63 families has neither a radio nor a TV, 794 families has a radio and 187 has TV. The number of families in that group having both a radio and a TV is

Given $\sim s \vee (\sim r \wedge s)$. The negation of this statement is:

Let $f(x) = a^x$ ($a > 0$) be written as $f(x) = f_1(x) + f_2(x)$, where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x+y) + f_1(x-y)$ equals:

If $x \in (0, \frac{\pi}{2})$, $\tan x \in (0, \infty)$, find the minimum value of the function $f(x) = 3\tan x + \cot x$.

If $f(x)$ satisfies the relation $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$ and $f(1) = 5$, then find $\displaystyle\sum_{n=1}^{m} f(n)$. Also prove that $f(x)$ is odd.

Consider the following two statements:P: If 7 is an odd number, then 7 is divisible by 2.Q: If 7 is a prime number, then 7 is an odd number.If $V_1$ is the truth value of the contrapositive of $P$ and $V_2$ is the truth value of contrapositive of $Q$, then the ordered pair $(V_1, V_2)$ equals

For the function with domain $D: [-2,-1] \cup [1,2]$ and $-1 \leq \log_2\left(\dfrac{x^2}{2}\right) \leq 1$, find the range.

Let $R$ be a relation defined on the set of all natural numbers as $R = \{(x, y) : x \in \mathbb{N}, 2x + y = 41\}$. Find the number of elements in the set domain of this relation.

If $n(A) = 3$, $n(B) = 6$ and $A \subseteq B$. Then the number of elements in $A \cup B$ is equal to

Let $f$ satisfy $f(10+x)=f(10-x)$ and $f(20+x)=-f(20-x)$ for all $x\in\mathbb{R}$. Which statement is correct?

Suppose A1, A2, ..., A30 are thirty sets each having 5 elements and B1, B2, ..., Bn are n sets each having 3 elements. Let $\bigcup_{i=1}^{30} A_i = \bigcup_{j=1}^{n} B_j = S$ and each element of S belongs to exactly 10 of the Ai's and exactly 9 of the Bj's, then find the value of n.

If the functions $f(x) = e^x/a$ and $g(x) = \ln(ax)$ are inverse of each other, then the value of $[a]$ (where $[\cdot]$ denotes the greatest integer function) is:

Given that $ \dfrac{1}{|x| - 3} \leq \dfrac{1}{2} $, the solution set is:

Relations & Functions Questions (810)