Relations & Functions Questions - JEE

Let Z be the set of integers. If A = {x ∈Z : 2(x+2)(x2−5x+6) = 1} and B = {x ∈Z : −3

Let A = {2, 3, 4, . . . , 30} and “∼” be defined on A × A by (a, b) ∼(c, d) iff ad = bc. The number of ordered pairs which are related to (4, 3) is:

Let A = {2, 3, 6, 8, 9, 11}, B = {1, 4, 5, 10, 15}. R on A × B: (a, b) R (c, d) iff 3ad −7bc is even. Then R is:

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 f : $\mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \frac{x}{1 + x^2}, x \in \mathbb{R}$. Then the range of f is :

Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) = \dfrac{x^2 + 2x + 1}{x^2 + 1}$. Then

On X = {1, . . . , 20}: R1 = {(x, y) : 2x −3y = 2} and R2 = {(x, y) : −5x + 4y = 0}. Let M, N be minimum elements to add for symmetry. M + N =?

Given below are two statements: **Statement I:** The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{x}{1+|x|}$ is one-one. **Statement II:** The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{x^2+4x-30}{x^2-8x+18}$ is many-one. In the light of the above statements, choose the correct answer:

If the domain of the function $f(x)=\log_{(10x^2-17x+7)}(18x^2-11x+1)$ is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then $90(a+b+c+d+e)$ equals:

The number of relations, defined on the set $\{a,b,c,d\}$, which are both reflexive and symmetric, is equal to:

Among the relations $ S = \left\{(a, b) : a, b \in \mathbb{R} - \{0\},\ 2 + \frac{a}{b} > 0\right\} $ and $ T = \{(a, b) : a, b \in \mathbb{R},\ a^{2} - b^{2} \in \mathbb{Z}\} $,

A function $f: \mathbb{R} \to \mathbb{R}$ has property $f(x+y) = f(x) \cdot e^{f(y)-1}$, for every $x, y \in \mathbb{R}$ then positive value of $f(4)$ is:

The function $f : [0,7] \to [0,70)$ where $f(x) = x^3 - 12x^2 + 45x$ is

Let $f: \mathbb{R} - \{2,6\} \to \mathbb{R}$ be real valued function defined as $f(x) = \dfrac{x^2+2x+1}{x^2-8x+12}$. Then range of $f$ is

Let R1 and R2 be defined on R by a R1 b ⇔ab ≥0 and a R2 b ⇔a ≥b. Then:

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?

Let $A=\{-2,-1,0,1,2,3,4\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x+y\leq2$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric respectively. Then $l+m+n$ is equal to:

If $f(x)$ is symmetric about the line $x = -2$, find the values of $a$ and $b$.

Let $A=\{0,1,2,\ldots,9\}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $|x-y|$ is a multiple of 3. Given below are two statements: **Statement I:** $n(R)=36$. **Statement II:** $R$ is an equivalence relation. Choose the correct answer:

Let $A=\{x:|x^2-10|\leq6\}$ and $B=\{x:|x-2|>1\}$. Then

If $f(x)$ is a real valued function such that $f(x+6) - f(x+3) + f(x) = 0, \forall x \in \mathbb{R}$, then period of $f(x)$ is

Let R be a relation defined on \mathbb{N} as a R b if 2a + 3b is a multiple of 5, a, b \in \mathbb{N}. Then R is

For x \in \left(0, \frac{3}{2}\right), let f(x) = \sqrt{x}, g(x) = \tan x and h(x) = \frac{1 - x^2}{1 + x^2}. If f(x) = (hof)og(x), then $\phi \left( \frac{\pi}{3} \right)$ is equal to :

Let $P(A)$ be defined as power set of set $A=\{1,2,3,\ldots,2025\}$. Let $R$ be a relation defined on $P(A)$ as $(B,C)\in R$ if $B$ is superset of $C$, then $R$ is

Let $R=\{(a,b): a/b$ is a prime number$\}$ on first twenty natural numbers. Consider: (I) $R$ is reflexive and symmetric but not transitive; (II) Range of $R^{-1}$ has 20 elements; (III) Domain of $R^{-1}$ has 10 elements. Which statements are true?

If a $ \in \mathbb{R} $ and the equation $ -3(x - \lfloor x \rfloor)^2 + 2(x - \lfloor x \rfloor) + a^2 = 0 $ (where $ \lfloor x \rfloor $ denotes the greatest integer $ \leq x $) has non integral real solution, then all possible values of $ a $ lie in the interval :

If $f(x)=\log\dfrac{1-x}{1+x}$, $|x|, find \(f\!\left(\dfrac{2x}{1+x^2}\right)$.

Let $P(A)$ be defined as power set of set $A=\{1,2,3,\ldots,2025\}$. Let $R$ be a relation defined on $P(A)$ as $(B,C)\in R$ if $B$ is superset of $C$, then $R$ is

The domain of the function $\sqrt{x-5}$ is $[z, \infty)$, where $z$ is __________.

Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let $z = 4x + 6y$ be the objective function. The minimum value of z occurs at

Let $f: (-1, 1) \to B$, be a function defined by $f(x) = \tan^{-1}\dfrac{2x}{1-x^2}$, then $f$ is both one-one and onto when $B$ is the interval

Given $ f(x) = a^x $ ($ a > 0 $) and $ 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

In a linear programming problem, the objective function is $z = 4x + 3y$. The corner points of the feasible region are (0, 8), (2, 5), (4, 3), and (9, 0). Find the minimum value of $z$.

$(\sim(p \vee q)) \vee (\sim p \wedge q$ is logically equivalent to

Since $P$ is true, $Q$ is false and $R$ is true, the true statement among the following is:

Consider the following two binary relations on the set $A = \{a, b, c\}$:$R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$ and$R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}$.Then:

Which of the following is NOT a tautology?

The range of the function $f(x) = 7 - {}^{x}P_{x-3}$ is

Given $ f(x) = \dfrac{x^2}{1-x^2} $. The set A should be chosen so that f is a function from A to $[0, \infty)$. Which of the following is the correct set A?

Let $P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\}$ and $Q = \{\theta : \sin\theta + \cos\theta = \sqrt{2}\sin\theta\}$ be two sets. Then

Let a function $f : (0, \infty) \to (0, \infty)$ be defined by $f(x) = \left|1 - \dfrac{1}{x}\right|$. Then f is:

Here, n(A) = 5, n(B) = 7. What is the minimum number of elements in A ∪ B?

The numbers of subsets that can be formed from the set $A = \{4, 5, 6\}$ are

Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement $p \Rightarrow (q \vee r)$ is

Sets A and B have 5 and 7 elements respectively. What can be the minimum number of elements in $A \cup B$ is __________.

The negation of the statement "If I become a teacher, then I will open a school" is

In a group of 140 students, 70 opted Mathematics, 46 opted Physics and 28 opted Chemistry. 23 opted both Mathematics and Physics, 9 opted both Physics and Chemistry, 14 opted both Mathematics and Chemistry, and 4 opted all three subjects. The number of students who did not opt for any of the three courses is:

Let S = {x ∈ ℝ : x ≥ 0 and $2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0$}. Then, S

Let $R=\{(a,b): a/b$ is a prime number$\}$ on first twenty natural numbers. Consider: (I) $R$ is reflexive and symmetric but not transitive; (II) Range of $R^{-1}$ has 20 elements; (III) Domain of $R^{-1}$ has 10 elements. Which statements are true?

Given $P(n) = x^2 - n + 41$ is prime. Which of the following options is correct regarding $P(3)$ and $P(5)$?

Relations & Functions Questions (810)