Question

Set $A$ has $m$elements and set $B$ has $n$elements. If the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$, then the value of $m·n$ is___.

JEE Main 2020 (06 Sep Shift 1)

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Question

Let $X=\left\{n\in N:1\le n\le 50\right\}$. If $A=\left\{n\in X: \mathrm{n }\mathrm{i}\mathrm{s }\mathrm{a }\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e }\mathrm{o}\mathrm{f} 2\right\}$ and $\mathrm{B}=\left\{n\in X: \mathrm{n }\mathrm{i}\mathrm{s }\mathrm{a }\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e }\mathrm{o}\mathrm{f }7\right\}$, then the number of elements in the smallest subset of $X$, containing both $\mathrm{A}$ and $\mathrm{B}$, is.

JEE Main 2020 (07 Jan Shift 2)

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Question

Let $A=\left\{2,3,4,5,\dots .,30\right\}$ and $\text{'}\simeq \text{'}$ be an equivalence relation on $A\times A,$ defined by $\left(a,b\right)\simeq \left(c,d\right),$ if and only if $ad=bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $\left(4,3\right)$ is equal to :

JEE Main 2021 (16 Mar Shift 2)

Options

• A: $5$
• B: $6$
• C: $8$
• D: $7$

Question

Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as "$ARB$ iff there exists a non-singular matrix $P$ such that $PA{P}^{-1}=B$". Then which of the following is true ?

JEE Main 2021 (18 Mar Shift 2)

Options

• A: $R$ is symmetric, transitive but not reflexive
• B: $R$ is reflexive, symmetric but not transitive
• C: $R$ is an equivalence relation
• D: $R$ is reflexive, transitive but not symmetric

Question

Let $Z$ be the set of all integers,
$A=\left\{(x,y)\in Z\times Z:(x-2{)}^2+y^2\le 4\right\}$

$B=\left\{(x,y)\in Z\times Z:x^2+y^2\le 4\right\}\text{ and }$

$C=\left\{(x,y)\in Z\times Z:(x-2{)}^2+(y-2{)}^2\le 4\right\}$
If the total number of relations from $A\cap B$ to $A\cap C$ is $2^p$, then the value of $p$ is:

JEE Main 2021 (27 Aug Shift 2)

Options

• A: $25$
• B: $9$
• C: $16$
• D: $49$

Question

Let a set $A=A_1\cup A_2\cup \dots \cup A_k$, where $A_i\cap A_j=\varphi$ for $i\ne j; 1\le i,j\le k.$ Define the relation $R$ from $A$ to $A$ by $R=${$\left(x,y\right):y\in A_i$ if and only if $x\in A_i,1\le i\le k$}. Then, $R$ is:

JEE Main 2022 (29 Jun Shift 1)

Options

• A: reflexive, symmetric but not transitive
• B: reflexive, transitive but not symmetric
• C: reflexive but not symmetric and transitive
• D: an equivalence relation

Question

Let $A=\left\{1,2,3,4,5,6,7\right\}$. Define $B=${$T\subseteq A$: either $1\notin T$ or $2\in T$} and $C=${$T\subseteq A:T$ the sum of all the elements of $T$ is a prime number.} Then the number of elements in the set $B\cup C$ is _______.

JEE Main 2022 (25 Jul Shift 2)

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Question

Let $A=\left\{1,2,3,4,5,6,7\right\}$ and $B=\left\{3,6,7,9\right\}$. Then the number of elements in the set $\left\{C\subseteq A:C\cap B\ne \varphi \right\}$ is ______

JEE Main 2022 (26 Jul Shift 2)

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Question

Let $A=\left\{1,a_1,a_2\dots \dots a_{18},77\right\}$ be a set of integers with $1<a_1<a_2<\dots ..<a_{18}<77$. Let the set $A+A=\left\{x+y:x,y\in A\right\}$ contain exactly $39$ elements. Then, the value of $a_1+a_2+\dots ..+a_{18}$ is equal to ______.

JEE Main 2022 (28 Jun Shift 1)

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Question

Let $P\left(\mathrm{S}\right)$ denote the power set of $S=\left\{1,2,3,\dots ,10\right\}$. Define the relations $R_1$ and $R_2$ on $P\left(S\right)$ as $A{R}_{1}B$ if $\left(A\cap B^c\right)\cup \left(B\cap A^c\right)=\varphi$ and $A{R}_2 B$ if $A\cup B^c=B\cup A^c,\forall A,B\in P\left(S\right)$ . Then :

JEE Main 2023 (01 Feb Shift 2)

Options

• A: both $R_1$ and $R_2$ are equivalence relations
• B: only $R_1$ is an equivalence relation
• C: only $R_2$ is an equivalence relation
• D: both $R_1$ and $R_2$ are not equivalence relations

Question

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

JEE Main 2023 (29 Jan Shift 2)

Options

• A: not reflexive
• B: transitive but not symmetric
• C: symmetric but not transitive
• D: an equivalence relation

Question

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

JEE Main 2023 (31 Jan Shift 1)

Options

• A: symmetric but neither reflexive nor transitive
• B: transitive but neither reflexive nor symmetric
• C: reflexive and symmetric but not transitive
• D: symmetric and transitive but not reflexive

Question

In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25\left({\beta }^2{x}^2+{\alpha }^2{y}^2\right)={\alpha }^2{\beta }^2$ is

JEE Main 2023 (06 Apr Shift 2)

Options

• A: $\frac{\sqrt{119}}{12}$
• B: $\frac{\sqrt{117}}{12}$
• C: $\frac{3\sqrt{15}}{12}$
• D: $\frac{\sqrt{129}}{12}$

Question

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

JEE Main 2023 (15 Apr Shift 1)

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Question

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 ______

JEE Main 2024 (04 Apr Shift 1)

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Question

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

JEE Main 2024 (04 Apr Shift 2)

Options

• A: Both (I) and (II) are correct.
• B: Only (II) is correct.
• C: Neither (I) nor (II) is correct.
• D: Only (I) is correct.

Question

Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3, \ldots, 20\}$ be given by $R_1=\{(x, y): 2 x-3 y=2\}$ and $R_2=\{(x, y):-5 x+4 y=0\}$. 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

JEE Main 2024 (06 Apr Shift 1)

Options

• A: 12
• B: 16
• C: 8
• D: 10

Question

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

JEE Main 2024 (09 Apr Shift 1)

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Question

The number of elements in the set $S=\left\{\left(x,y,z\right):x,y,z\in Z,x+2y+3z=42,x,y,z\ge 0\right\}$ equals ________

JEE Main 2024 (01 Feb Shift 1)

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