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B2HRGK7dEczQF7C94Njgy2xukf0qgrp2 | maths | sets-and-relations | number-of-sets-and-relations | Consider the two sets :
<br/>A = {m $$ \in $$ R : both the roots of<br/> x<sup>2</sup>
– (m + 1)x + m + 4 = 0 are real}
<br/>and B = [–3, 5).
<br/>Which of the following is not true? | [{"identifier": "A", "content": "A $$ \\cap $$ B = {\u20133}"}, {"identifier": "B", "content": "B \u2013 A = (\u20133, 5)"}, {"identifier": "C", "content": "A $$ \\cup $$ B = R"}, {"identifier": "D", "content": "A - B = ($$ - $$$$ \\propto $$, $$ - $$3) $$ \\cup $$ (5, $$ \\propto $$)"}] | ["D"] | null | As roots are real so, $$D \ge 0$$<br><br>$${(m + 1)^2} - 4(m + 4) \ge 0$$<br><br>$$ \Rightarrow {m^2} - 2m - 15 \ge 0$$<br><br>$$ \Rightarrow $$ $$(m - 5)(m + 3) \ge 0$$<br><br>$$m\, \in \,$$($$ - $$$$ \propto $$, $$ - $$3] $$ \cup $$ [5, $$ \propto $$)<br><br>$$A= ( - $$$$ \propto $$, $$ - $$3] $$ \cup $$ [5, $$ \prop... | mcq | jee-main-2020-online-3rd-september-morning-slot | 8,182 |
1p4U7IKhG25B9jE6nSjgy2xukfakf6tf | maths | sets-and-relations | number-of-sets-and-relations | Let $$\mathop \cup \limits_{i = 1}^{50} {X_i} = \mathop \cup \limits_{i = 1}^n {Y_i} = T$$ where each X<sub>i</sub> contains 10 elements and each Y<sub>i</sub> contains 5 elements. If each element of the set T is an element of exactly 20 of sets X<sub>i</sub>’s and exactly 6 of sets Y<sub>i</sub>’s, then n is equal ... | [{"identifier": "A", "content": "30"}, {"identifier": "B", "content": "50"}, {"identifier": "C", "content": "15"}, {"identifier": "D", "content": "45"}] | ["A"] | null | $$\mathop \cup \limits_{i = 1}^{50} {X_i} = $$ X<sub>1</sub>, X<sub>2</sub>,....., X<sub>50</sub> = 50 sets. Given each sets having 10 elements.
<br><br>So total elements = 50 $$ \times $$ 10
<br><br>$$\mathop \cup \limits_{i = 1}^n {Y_i} =$$ $$ Y<sub>1</sub>, Y<sub>2</sub>,....., Y<sub>n</sub> = n sets. Given each s... | mcq | jee-main-2020-online-4th-september-evening-slot | 8,183 |
3iz5nbJtSww5lL4Sg9jgy2xukfxgtuou | maths | sets-and-relations | number-of-sets-and-relations | 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 ______. | [] | null | 28 | Number of subsets of A = 2<sup>m</sup>
<br><br>Number of subsets of B = 2<sup>n</sup>
<br><br>Given = 2<sup>m</sup> – 2<sup>n</sup>
= 112
<br><br>$$ \therefore $$ m = 7, n = 4 (2<sup>7</sup> – 2<sup>4</sup>
= 112)
<br><br>$$ \therefore $$ m $$ \times $$ n = 7 $$ \times $$ 4 = 28 | integer | jee-main-2020-online-6th-september-morning-slot | 8,184 |
ElJhwaKkT1JdIO9sfl1kmhw02h6 | maths | sets-and-relations | number-of-sets-and-relations | The number of elements in the set {x $$\in$$ R : (|x| $$-$$ 3) |x + 4| = 6} is equal to : | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "1"}] | ["B"] | null | <b>Case 1 :</b><br><br>x $$ \le $$ $$-$$4<br><br>($$-$$x $$-$$ 3)($$-$$x $$-$$ 4) = 6<br><br>$$ \Rightarrow $$ (x + 3)(x + 4) = 6<br><br>$$ \Rightarrow $$ x<sup>2</sup> + 7x + 6 = 0<br><br>$$ \Rightarrow $$ x = $$-$$1 or $$-$$6<br><br>but x $$ \le $$ $$-$$4<br><br>x = $$-$$6<br><br><b>Case 2 :</b><br><br>x $$\in$$ ($$-... | mcq | jee-main-2021-online-16th-march-morning-shift | 8,186 |
Cy64jNuK5iXgalKmT51kmiwrnp6 | maths | sets-and-relations | number-of-sets-and-relations | Let A = {2, 3, 4, 5, ....., 30} and '$$ \simeq $$' be an equivalence relation on A $$\times$$ A, defined by (a, b) $$ \simeq $$ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to : | [{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "7"}] | ["D"] | null | ad = bc<br><br>(a, b) R (4, 3) $$ \Rightarrow $$ 3a = 4b<br><br>a = $${4 \over 3}$$b<br><br>b must be multiple of 3<br><br>b = {3, 6, 9 ..... 30}<br><br>(a, b) = {(4, 3), (8, 16), (12, 9), (16, 12), (20, 15), (24, 18), (28, 21)}<br><br>$$ \Rightarrow $$ 7 ordered pair | mcq | jee-main-2021-online-16th-march-evening-shift | 8,187 |
1l567giit | maths | sets-and-relations | number-of-sets-and-relations | <p>Let R<sub>1</sub> and R<sub>2</sub> be relations on the set {1, 2, ......., 50} such that</p>
<p>R<sub>1</sub> = {(p, p<sup>n</sup>) : p is a prime and n $$\ge$$ 0 is an integer} and</p>
<p>R<sub>2</sub> = {(p, p<sup>n</sup>) : p is a prime and n = 0 or 1}.</p>
<p>Then, the number of elements in R<sub>1</sub> $$-$$ ... | [] | null | 8 | Given, ${R}_1=\left\{\left(p, p^n\right): p\right.$ is a Prime and $n \geq 0$ is an integer $\}$
<br/><br/>and, set $A=\{1,2,3 \ldots \ldots .50\}$
<br/><br/>$p$ is a Prime number which can take 15 values $2,3,5,7,11,13,17,19,23,29,31,37,41,43$ and 47
<br/><br/>$\therefore$ We can calculate no. of elements in $\mathrm{... | integer | jee-main-2022-online-28th-june-morning-shift | 8,190 |
1l58apr54 | maths | sets-and-relations | number-of-sets-and-relations | <p>Let A = {n $$\in$$ N : H.C.F. (n, 45) = 1} and</p>
<p>Let B = {2k : k $$\in$$ {1, 2, ......., 100}}. Then the sum of all the elements of A $$\cap$$ B is ____________.</p> | [] | null | 5264 | <p>Sum of all elements of A $$\cap$$ B = 2 [Sum of natural numbers upto 100 which are neither divisible by 3 nor by 5]</p>
<p>$$ = 2\left[ {{{100 \times 101} \over 2} - 3\left( {{{33 \times 34} \over 2}} \right) - 5\left( {{{20 \times 21} \over 2}} \right) + 15\left( {{{6 \times 7} \over 2}} \right)} \right]$$</p>
<p>$... | integer | jee-main-2022-online-26th-june-morning-shift | 8,191 |
1l58aqy4b | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $$ and $$B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $$. Then A + B is equal to _____________.</p> | [] | null | 1100 | <p>$$\sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\{ i,\,j\} } } $$</p>
<p>= {1, 1} {1, 2} {1, 3} ..... {1, 10}</p>
<p>{2, 1} {2, 2} {2, 3} ..... {2, 10}</p>
<p>{3, 1} {3, 2} {3, 3} ..... {3, 10}</p>
<p>$$ \vdots $$</p>
<p>{10, 1} {10, 2} {10, 3} ..... {10, 10}</p>
<p>Now, $$A = \sum\limits_{i = 1}^{10} {\sum\li... | integer | jee-main-2022-online-26th-june-morning-shift | 8,192 |
1l5bb3057 | maths | sets-and-relations | number-of-sets-and-relations | <p>The sum of all the elements of the set $$\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\} $$ is __________.</p> | [] | null | 1633 | <p>The numbers upto 24 which gives g.c.d. with 24 equals to 1 are 1, 5, 7, 11, 13, 17, 19 and 23.</p>
<p>Sum of these numbers = 96</p>
<p>There are four such blocks and a number 97 is there upto 100.</p>
<p>$$\therefore$$ Complete sum</p>
<p>= 96 + (24 $$\times$$ 8 + 96) + (48 $$\times$$ 8 + 96) + (72 $$\times$$ 8 + 96... | integer | jee-main-2022-online-24th-june-evening-shift | 8,193 |
1l6hzials | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$A=\{1,2,3,4,5,6,7\}$$ and $$B=\{3,6,7,9\}$$. Then the number of elements in the set $$\{C \subseteq A: C \cap B \neq \phi\}$$ is ___________.</p> | [] | null | 112 | <p>As C $$\cap$$ B $$\ne$$ $$\phi$$, c must be not be formed by {1, 2, 4, 5}</p>
<p>$$\therefore$$ Number of subsets of A = 2<sup>7</sup> = 128</p>
<p>and number of subsets formed by {1, 2, 4, 5} = 16</p>
<p>$$\therefore$$ Required no. of subsets = 2<sup>7</sup> $$-$$ 2<sup>4</sup> = 128 $$-$$ 16 = 112</p> | integer | jee-main-2022-online-26th-july-evening-shift | 8,195 |
1l6p0j53w | maths | sets-and-relations | number-of-sets-and-relations | <p>Let R be a relation from the set $$\{1,2,3, \ldots, 60\}$$ to itself such that $$R=\{(a, b): b=p q$$, where $$p, q \geqslant 3$$ are prime numbers}. Then, the number of elements in R is :</p> | [{"identifier": "A", "content": "600"}, {"identifier": "B", "content": "660"}, {"identifier": "C", "content": "540"}, {"identifier": "D", "content": "720"}] | ["B"] | null | <p>We have a set S = {1, 2, 3, ..., 60}, and a relation R defined on the set S. An element (a, b) belongs to the relation R if and only if b can be expressed as the product of two prime numbers p and q, where both p and q are greater than or equal to 3.</p>
<p>In terms of number theory, prime numbers are integers great... | mcq | jee-main-2022-online-29th-july-morning-shift | 8,196 |
1l6p3rlat | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$S=\{4,6,9\}$$ and $$T=\{9,10,11, \ldots, 1000\}$$. If $$A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.$$ $$\epsilon S\}$$, then the sum of all the elements in the set $$T-A$$ is equal to __________.</p> | [] | null | 11 | <p>Here $$S = \{ 4,6,9\} $$</p>
<p>And $$T = \{ 9,10,11,\,\,......,\,\,1000\} $$.</p>
<p>We have to find all numbers in the form of $$4x + 6y + 9z$$, where $$x,y,z \in \{ 0,1,2,\,......\} $$.</p>
<p>If a and b are coprime number then the least number from which all the number more than or equal to it can be express as ... | integer | jee-main-2022-online-29th-july-morning-shift | 8,197 |
lgnzb2hr | maths | sets-and-relations | number-of-sets-and-relations | The number of elements in the set <br/><br/>$\left\{n \in \mathbb{N}: 10 \leq n \leq 100\right.$ and $3^{n}-3$ is a multiple of 7$\}$ is ___________. | [] | null | 15 | To determine the number of elements in the given set, we need to find how many natural numbers $n$ between $10$ and $100$ (inclusive) satisfy the condition that $3^n - 3$ is a multiple of $7$.
<br/><br/>Recall that for any integers $a$ and $b$, $a$ is a multiple of $b$ if there exists an integer $k$ such that $a = bk$... | integer | jee-main-2023-online-15th-april-morning-shift | 8,199 |
lgnzfh40 | maths | sets-and-relations | number-of-sets-and-relations | Let $A=\{1,2,3,4\}$ and $\mathrm{R}$ be a relation on the set $A \times A$ defined by <br/><br/>$R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}$. Then the number of elements in $\mathrm{R}$ is ____________. | [] | null | 6 | $$
2a + 3b = 4c + 5d
$$
<br/><br/>
Given A = {1, 2, 3, 4}, the maximum value of $2a + 3b$ is 20, when (a, b) = (4, 4), and the minimum value of $4c + 5d$ is 9, when (c, d) = (1, 1). Therefore, the possible values for $2a + 3b = 4c + 5d$ are 9, 13, 14, 17, 18, and 19.
<br/><br/>
Now, let's find the combinations of (a, b... | integer | jee-main-2023-online-15th-april-morning-shift | 8,200 |
1lgsumb17 | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$\mathrm{A}=\{1,3,4,6,9\}$$ and $$\mathrm{B}=\{2,4,5,8,10\}$$. Let $$\mathrm{R}$$ be a relation defined on $$\mathrm{A} \times \mathrm{B}$$ such that $$\mathrm{R}=\left\{\left(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\right): a_{1} \leq b_{2}\right.$$ and $$\left.b_{1} \leq a_{2}\right\}$$. Then the n... | [{"identifier": "A", "content": "180"}, {"identifier": "B", "content": "26"}, {"identifier": "C", "content": "52"}, {"identifier": "D", "content": "160"}] | ["D"] | null | Given that the sets are $A = \{1, 3, 4, 6, 9\}$ and $B = \{2, 4, 5, 8, 10\}$, for the relation $\mathrm{R}$ on the set $A \times B$, we need to find the combinations of pairs that satisfy the conditions $a_1 \leq b_2$ and $b_1 \leq a_2$.
<br/><br/>We find the number of combinations by considering the possible values ... | mcq | jee-main-2023-online-11th-april-evening-shift | 8,201 |
1lgvqdfo3 | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$\mathrm{A}=\{2,3,4\}$$ and $$\mathrm{B}=\{8,9,12\}$$. Then the number of elements in the relation
$$\mathrm{R}=\left\{\left(\left(a_{1}, \mathrm{~b}_{1}\right),\left(a_{2}, \mathrm{~b}_{2}\right)\right) \in(A \times B, A \times B): a_{1}\right.$$ divides $$\mathrm{b}_{2}$$ and $$\mathrm{a}_{2}$$ divides $$\le... | [{"identifier": "A", "content": "18"}, {"identifier": "B", "content": "24"}, {"identifier": "C", "content": "36"}, {"identifier": "D", "content": "12"}] | ["C"] | null |
<p>Given sets :
<br/>$ A = {2,3,4} $
<br/>$ B = {8,9,12} $</p>
<p>We want to find the number of elements of the form $( (a_1, b_1), (a_2, b_2) )$ such that :</p>
<ol>
<li>$ a_1 $ divides $ b_2 $</li>
<li>$ a_2 $ divides $ b_1 $</li>
</ol>
<p>For the first condition :
<br/>$ a_1 $ divides $ b_2 $
<br/>Given $ a_1 \in A... | mcq | jee-main-2023-online-10th-april-evening-shift | 8,202 |
1lgxwachw | maths | sets-and-relations | number-of-sets-and-relations | <p>The number of elements in the set $$\{ n \in Z:|{n^2} - 10n + 19| < 6\} $$ is _________.</p> | [] | null | 6 | Given, $\left|n^2-10 n+19\right|<6$
<br/><br/>$\Rightarrow-6 < n^2-10 n+19 < 6$
<br/><br/>Take, $-6 < n^2-10 n+19$ and $n^2-10 n+19 < 6$
<br/><br/>$$
\begin{array}{ll}
\Rightarrow n^2-10 n+25 > 0 & \text { and }\quad n^2-10 n+13 < 0 \\\\
\Rightarrow(n-5)^2 > 0 & \text { and } n=\frac{10 \pm \sqrt{100-52}}{2}<0
\end{a... | integer | jee-main-2023-online-10th-april-morning-shift | 8,203 |
1lh23u6jh | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$\mathrm{A}=\{1,2,3,4, \ldots ., 10\}$$ and $$\mathrm{B}=\{0,1,2,3,4\}$$. The number of elements in the relation $$R=\left\{(a, b) \in A \times A: 2(a-b)^{2}+3(a-b) \in B\right\}$$ is ___________.</p> | [] | null | 18 | <p>Given sets :
<br/><br/>A={1,2,3,4, ............,10}
<br/><br/> B={0,1,2,3,4}
<br/><br/>We are looking for pairs $(a,b) \in A \times A$ such that :
<br/><br/>$ 2(a-b)^2 + 3(a-b) \in B $</p>
<p>Let's break down the relation :</p>
<p><strong>Case 1 :</strong> $ a-b = 0 $
<br/><br/>$ 2(a-b)^2 + 3(a-b) = 0 $
<br/><br... | integer | jee-main-2023-online-6th-april-morning-shift | 8,204 |
jaoe38c1lscn4dk2 | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$A$$ and $$B$$ be two finite sets with $$m$$ and $$n$$ elements respectively. The total number of subsets of the set $$A$$ is 56 more than the total number of subsets of $$B$$. Then the distance of the point $$P(m, n)$$ from the point $$Q(-2,-3)$$ is :</p> | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "10"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "6"}] | ["B"] | null | <p>$$\begin{aligned}
& 2^{\mathrm{m}}-2^{\mathrm{n}}=56 \\
& 2^{\mathrm{n}}\left(2^{\mathrm{m}-\mathrm{n}}-1\right)=2^3 \times 7 \\
& 2^{\mathrm{n}}=2^3 \text { and } 2^{\mathrm{m}-\mathrm{n}}-1=7 \\
& \Rightarrow \mathrm{n}=3 \text { and } 2^{\mathrm{m}-\mathrm{n}}=8 \\
& \Rightarrow \mathrm{n}=3 \text { and } \mathrm... | mcq | jee-main-2024-online-27th-january-evening-shift | 8,206 |
jaoe38c1lsd55mch | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$$. Let $$R$$ be a relation on $$\mathrm{A}$$ defined by $$(x, y) \in R$$ if and only if $$2 x=3 y$$. Let $$R_1$$ be a symmetric relation on $$A$$ such that $$R \subset R_1$$ and the number of elements in $$R_1$$ is $$\mathrm{n}$$. Then, the minimum value of $$\math... | [] | null | 66 | <p>$$\begin{aligned}
& \mathrm{R}=\{(3,2),(6,4),(9,6),(12,8), \ldots \ldots \ldots .(99,66)\} \\
& \mathrm{n}(\mathrm{R})=33 \\
& \therefore 66
\end{aligned}$$</p> | integer | jee-main-2024-online-31st-january-evening-shift | 8,207 |
jaoe38c1lse5zdng | maths | sets-and-relations | number-of-sets-and-relations | <p>Let $$A=\{1,2,3,4\}$$ and $$R=\{(1,2),(2,3),(1,4)\}$$ be a relation on $$\mathrm{A}$$. Let $$\mathrm{S}$$ be the equivalence relation on $$\mathrm{A}$$ such that $$R \subset S$$ and the number of elements in $$\mathrm{S}$$ is $$\mathrm{n}$$. Then, the minimum value of $$n$$ is __________.</p> | [] | null | 16 | $$
\begin{aligned}
& A=\{1,2,3,4\} \\\\
& R=\{(1,2),(2,3),(1,4)\}
\end{aligned}
$$
<br/><br/>$S$ is equivalence
for $R < S$ and reflexive
<br/><br/>$$
\{(1,1),(2,2),(3,3),(4,4)\}
$$
<br/><br/>for symmetric
<br/><br/>$$
\{(2,1),(4,1),(3,2)\}
$$
<br/><br/>for transitive
<br/><br/>$$
\{(1,3),(3,1),(4,2),(2,4)\}
$$
<br/><... | integer | jee-main-2024-online-31st-january-morning-shift | 8,208 |
ljafxwvp | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is : | [{"identifier": "A", "content": "a function"}, {"identifier": "B", "content": "transitive"}, {"identifier": "C", "content": "not symmetric"}, {"identifier": "D", "content": "reflexive"}] | ["C"] | null | <p>Let's evaluate each of the properties for the relation $R$.</p>
<p><b>Relation R :</b> $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$</p>
<ul>
<li><p>A relation is a function if each element in the domain is related to exactly one element in the codomain. In this case, for example, 2 is related to both 4 and 3, so $R$ is... | mcq | aieee-2004 | 8,212 |
ljaeehms | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is : | [{"identifier": "A", "content": "reflexive and symmetric only"}, {"identifier": "B", "content": "an equivalence relation"}, {"identifier": "C", "content": "reflexive only"}, {"identifier": "D", "content": "reflexive and transitive only"}] | ["D"] | null | <p>We have to examine whether the relation $R$ satisfies the properties of reflexivity, symmetry, and transitivity.</p>
<p><b>Relation R :</b> $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ on set $A=\{3,6,9,12\}$.</p>
<p>We will evaluate each of the three properties :</p>
<ul>
<li><p>Reflexivity : A re... | mcq | aieee-2005 | 8,213 |
ljadq9q3 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let $W$ denote the words in the English dictionary. Define the relation $R$ by
<br/><br/>$R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is | [{"identifier": "A", "content": "reflexive, symmetric and not transitive"}, {"identifier": "B", "content": "reflexive, symmetric and transitive"}, {"identifier": "C", "content": "reflexive, not symmetric and transitive"}, {"identifier": "D", "content": "not reflexive, symmetric and transitive"}] | ["A"] | null | <p>Let's evaluate the relation $R$ for the properties of reflexivity, symmetry, and transitivity.</p>
<p><b>Relation R :</b> $R={(x, y) \in W \times W \mid}$ the words $x$ and $y$ have at least one letter in common}.</p>
<ul>
<li>Reflexivity : Each word in English obviously has at least one letter in common with it... | mcq | aieee-2006 | 8,214 |
jRE7pdboFFzdk15T | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let R be the real line. Consider the following subsets of the plane $$R \times R$$ :
<br/> $$S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right\}$$
<br/> $$T = \left\{ {(x,y): x - y\,\,\,is\,\,an\,\,{\mathop{\rm int}} eger\,} \right\}$$,
<p> Which one of the following is true ? </p> | [{"identifier": "A", "content": "Neither S nor T is an equivalence relation on R"}, {"identifier": "B", "content": "Both S and T are equivalence relation on R"}, {"identifier": "C", "content": "S is an equivalence relation on R but T is not"}, {"identifier": "D", "content": "T is an equivalence relation on R but S is n... | ["D"] | null | Given $$S = \left\{ {\left( {x,y} \right):y = x + 1\,\,} \right.\,$$
<br><br>and $$\,\,\,\left. {0 < x < 2} \right\}$$
<br><br>As $$\,\,\,\,x \ne x + 1\,\,\,$$
<br><br>for any $$\,\,\,x \in \left( {0,2} \right) \Rightarrow \left( {x,x} \right) \notin S$$
<br><br>$$\therefore$$ $$S$$ is not reflexive.
<br><br>... | mcq | aieee-2008 | 8,215 |
ljad974k | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Consider the following relations
<br/><br/>$R=\{(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w\}$;
<br/><br/>$S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $q m=p m\}$. Then | [{"identifier": "A", "content": "$R$ is an equivalence relation but $S$ is not an equivalence relation\n"}, {"identifier": "B", "content": "Neither $R$ nor $S$ is an equivalence relation"}, {"identifier": "C", "content": "$S$ is an equivalence relation but $R$ is not an equivalence relation"}, {"identifier": "D", "cont... | ["C"] | null | <p>Let's evaluate each relation for the properties of an equivalence relation: reflexivity, symmetry, and transitivity.</p>
<p><b>Relation R :</b> $R=(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w$.</p>
<ul>
<li>Reflexivity : For all $x$ in $R$, $x = 1x$. Since 1 is a rational number, ev... | mcq | aieee-2010 | 8,216 |
ljaclkwr | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let $R$ be the set of real numbers.
<br/><br/><b>Statement I :</b> $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$.
<br/><br/><b>Statement II :</b> $ B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equivalence relation on $R$. | [{"identifier": "A", "content": "Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I."}, {"identifier": "B", "content": "Statement I is true, Statement II is false."}, {"identifier": "C", "content": "Statement I is false, Statement II is true."}, {"identifier": "D", "con... | ["B"] | null | <p>An equivalence relation on a set must satisfy three properties: reflexivity (every element is related to itself), symmetry (if an element is related to a second, the second is related to the first), and transitivity (if a first element is related to a second, and the second is related to a third, then the first is r... | mcq | aieee-2011 | 8,217 |
DOhFcGHeiIM3tbQSkae3L | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let <b>N</b> denote the set of all natural numbers. Define two binary relations on <b>N</b> as <sub>R</sub> = {(x, y) $$ \in $$ <b>N $$ \times $$ N</b> : 2x + y = 10} and R<sub>2</sub> = {(x, y) $$ \in $$ <b>N $$ \times $$ N</b> : x + 2y = 10}. Then : | [{"identifier": "A", "content": "Range of R<sub>1</sub> is {2, 4, 8)."}, {"identifier": "B", "content": "Range of R<sub>2</sub> is {1, 2, 3, 4}."}, {"identifier": "C", "content": "Both R<sub>1</sub> and R<sub>2</sub> are symmetric relations."}, {"identifier": "D", "content": "Both R<sub>1</sub> and R<sub>2</sub> are tr... | ["B"] | null | For R<sub>1</sub>; 2x + y = 10 and x, y $$ \in $$ N possible values for x and y are :
<br><br>x = 1, y = 8 i.e. (1, 8);
<br><br>x = 2, y = 6 i.e (2, 6);
<br><br>x = 3, y = 4 i.e (3, 4);
<br><br>x = 4, y = 2 i.e (4, 2... | mcq | jee-main-2018-online-16th-april-morning-slot | 8,219 |
Ped5VSnU29ThaR3uOOjgy2xukf3yias0 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let R<sub>1</sub>
and R<sub>2</sub>
be two relation defined as
follows :
<br/>R<sub>1</sub>
= {(a, b) $$ \in $$ R<sup>2</sup>
: a<sup>2</sup>
+ b<sup>2</sup> $$ \in $$ Q} and
<br/>R<sub>2</sub>
= {(a, b) $$ \in $$ R<sup>2</sup>
: a<sup>2</sup>
+ b<sup>2</sup> $$ \notin $$ Q},
<br/>where Q is the
set of all rat... | [{"identifier": "A", "content": "Neither R<sub>1</sub>\n nor R<sub>2</sub>\n is transitive."}, {"identifier": "B", "content": "R<sub>2</sub>\n is transitive but R<sub>1</sub>\n is not transitive."}, {"identifier": "C", "content": "R<sub>1</sub>\n and R<sub>2</sub>\n are both transitive."}, {"identifier": "D", "content"... | ["A"] | null | For R<sub>1</sub> :<br><br>Let a = 1 + $$\sqrt 2 $$, b = 1 $$-$$ $$\sqrt 2 $$, c = $${8^{{1 \over 4}}}$$<br><br>aR<sub>1</sub>b : a<sup>2</sup> + b<sup>2</sup> = 6 $$ \in $$ Q<br><br>bR<sub>1</sub>c : b<sup>2</sup> + c<sup>2</sup> = 3 $$-$$ 2$$\sqrt 2 $$ + 2$$\sqrt 2 $$ = 3 $$ \in $$ Q<br><br>aR<sub>1</sub>c : a<sup>2<... | mcq | jee-main-2020-online-3rd-september-evening-slot | 8,220 |
cS4oUt1tfL1cKkH1tC1klughovj | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, $$-$$1) is the set : | [{"identifier": "A", "content": "$$S = \\{ (x,y)|{x^2} + {y^2} = \\sqrt 2 \\} $$"}, {"identifier": "B", "content": "$$S = \\{ (x,y)|{x^2} + {y^2} = 2\\} $$"}, {"identifier": "C", "content": "$$S = \\{ (x,y)|{x^2} + {y^2} = 1\\} $$"}, {"identifier": "D", "content": "$$S = \\{ (x,y)|{x^2} + {y^2} = 4\\} $$"}] | ["B"] | null | Given R = {(P, Q) | P and Q are at the same distance from the origin}.<br><br>Then equivalence class of (1, $$-$$1) will contain al such points which lies on circumference of the circle of centre at origin and passing through point (1, $$-$$1).<br><br>i.e., radius of circle = $$\sqrt {{1^2} + {1^2}} = \sqrt 2 $$<br><b... | mcq | jee-main-2021-online-26th-february-morning-slot | 8,221 |
UWZpD3f8zFp3ctRl2Z1kmm3f6jr | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Define a relation R over a class of n $$\times$$ n real matrices A and B as <br/><br/>"ARB iff there exists a non-singular matrix P such that PAP<sup>$$-$$1</sup> = B". <br/><br/>Then which of the following is true? | [{"identifier": "A", "content": "R is reflexive, transitive but not symmetric"}, {"identifier": "B", "content": "R is symmetric, transitive but not reflexive."}, {"identifier": "C", "content": "R is reflexive, symmetric but not transitive"}, {"identifier": "D", "content": "R is an equivalence relation"}] | ["D"] | null | For reflexive relation,<br/><br/> $\forall(A, A) \in R$ for matrix $P$.<br/><br/>
$\Rightarrow A=P A P^{-1}$ is true for $P=1$<br/><br/>
So, $R$ is reflexive relation.<br/><br/>
For symmetric relation,<br/><br/>
Let $(A, B) \in R$ for matrix $P$.<br/><br/>
$$
\Rightarrow \quad A=P B P^{-1}
$$<br/><br/>
After pre-multip... | mcq | jee-main-2021-online-18th-march-evening-shift | 8,222 |
1krxllilh | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let N be the set of natural numbers and a relation R on N be defined by $$R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0\} $$. Then the relation R is : | [{"identifier": "A", "content": "symmetric but neither reflexive nor transitive"}, {"identifier": "B", "content": "reflexive but neither symmetric nor transitive"}, {"identifier": "C", "content": "reflexive and symmetric, but not transitive"}, {"identifier": "D", "content": "an equivalence relation"}] | ["B"] | null | $${x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0$$<br><br>$$ \Rightarrow x({x^2} - {y^2}) - 3y({x^2} - {y^2}) = 0$$<br><br>$$ \Rightarrow (x - 3y)(x - y)(x + y) = 0$$<br><br>Now, x = y $$\forall$$(x, y) $$\in$$N $$\times$$ N so reflexive but not symmetric & transitive.<br><br>See, (3, 1) satisfies but (1, 3) does not. Also ... | mcq | jee-main-2021-online-27th-july-evening-shift | 8,223 |
1ktipm2vd | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Which of the following is not correct for relation R on the set of real numbers ? | [{"identifier": "A", "content": "(x, y) $$\\in$$ R $$ \\Leftrightarrow $$ 0 < |x| $$-$$ |y| $$\\le$$ 1 is neither transitive nor symmetric."}, {"identifier": "B", "content": "(x, y) $$\\in$$ R $$ \\Leftrightarrow $$ 0 < |x $$-$$ y| $$\\le$$ 1 is symmetric and transitive."}, {"identifier": "C", "content": "(x, y) ... | ["B"] | null | Note that (a, b) and (b, c) satisfy 0 < |x $$-$$ y| $$\le$$ 1 but (a, c) does not satisfy it so 0 $$\le$$ |x $$-$$ y| $$\le$$ 1 is symmetric but not transitive.
<br><br>For example,
<br><br>x = 0.2, y = 0.9, z = 1.5
<br><br>0 ≤ |x – y| = 0.7 ≤ 1
<br><br>0 ≤ |y – z| = 0.6 ≤ 1
<br><br>But |x – z| = 1.3 > 1
<br><br... | mcq | jee-main-2021-online-31st-august-morning-shift | 8,224 |
1l54578k2 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let a set A = A<sub>1</sub> $$\cup$$ A<sub>2</sub> $$\cup$$ ..... $$\cup$$ A<sub>k</sub>, where A<sub>i</sub> $$\cap$$ A<sub>j</sub> = $$\phi$$ for i $$\ne$$ j, 1 $$\le$$ j, j $$\le$$ k. Define the relation R from A to A by R = {(x, y) : y $$\in$$ A<sub>i</sub> if and only if x $$\in$$ A<sub>i</sub>, 1 $$\le$$ i $$\... | [{"identifier": "A", "content": "reflexive, symmetric but not transitive."}, {"identifier": "B", "content": "reflexive, transitive but not symmetric."}, {"identifier": "C", "content": "reflexive but not symmetric and transitive."}, {"identifier": "D", "content": "an equivalence relation."}] | ["D"] | null | <p>$$R = \{ (x,y):y \in {A_i},\,iff\,x \in {A_i}\,1 \le i \ge k\} $$</p>
<p>(1) Reflexive</p>
<p>(a, a) $$\Rightarrow$$ $$a \in {A_i}$$ iff $$a \in {A_i}$$</p>
<p>(2) Symmetric</p>
<p>(a, b) $$\Rightarrow$$ $$a \in {A_i}$$ iff $$b \in {A_i}$$</p>
<p>(b, a) $$\in$$R as $$b \in {A_i}$$ iff $$a \in {A_i}$$</p>
<p>(3) Tran... | mcq | jee-main-2022-online-29th-june-morning-shift | 8,225 |
1l55h1x3k | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let R<sub>1</sub> = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\le$$ 13} and</p>
<p>R<sub>2</sub> = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\ne$$ 13}. Then on N :</p> | [{"identifier": "A", "content": "Both R<sub>1</sub> and R<sub>2</sub> are equivalence relations"}, {"identifier": "B", "content": "Neither R<sub>1</sub> nor R<sub>2</sub> is an equivalence relation"}, {"identifier": "C", "content": "R<sub>1</sub> is an equivalence relation but R<sub>2</sub> is not"}, {"identifier": "D"... | ["B"] | null | $R_{1}=\{(a, b) \in N \times N:|a-b| \leq 13\}$ and
<br/><br/>
$R_{2}=\{(a, b) \in N \times N:|a-b| \neq 13\}$
<br/><br/>
In $R_{1}: \because|2-11|=9 \leq 13$
<br/><br/>
$\therefore \quad(2,11) \in R_{1}$ and $(11,19) \in R_{1}$ but $(2,19) \notin R_{1}$
<br/><br/>
$\therefore \quad R_{1}$ is not transitive
<br/><br/>
... | mcq | jee-main-2022-online-28th-june-evening-shift | 8,226 |
1l6jaxbfq | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$R_{1}$$ and $$R_{2}$$ be two relations defined on $$\mathbb{R}$$ by</p>
<p>$$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrightarrow a \geq b$$</p>
<p>Then,</p> | [{"identifier": "A", "content": "$$R_{1}$$ is an equivalence relation but not $$R_{2}$$"}, {"identifier": "B", "content": "$$R_{2}$$ is an equivalence relation but not $$R_{1}$$"}, {"identifier": "C", "content": "both $$R_{1}$$ and $$R_{2}$$ are equivalence relations"}, {"identifier": "D", "content": "neither $$R_{1}$$... | ["D"] | null | <p>$$a\,{R_1}\,b \Leftrightarrow ab \ge 0$$</p>
<p>So, definitely $$(a,a) \in {R_1}$$ as $${a^2} \ge 0$$</p>
<p>If $$(a,b) \in {R_1} \Rightarrow (b,a) \in {R_1}$$</p>
<p>But if $$(a,b) \in {R_1},(b,c) \in {R_1}$$</p>
<p>$$\Rightarrow$$ Then $$(a,c)$$ may or may not belong to R<sub>1</sub></p>
<p>{Consider $$a = - 5,b ... | mcq | jee-main-2022-online-27th-july-morning-shift | 8,227 |
1l6m5n24e | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>For $$\alpha \in \mathbf{N}$$, consider a relation $$\mathrm{R}$$ on $$\mathbf{N}$$ given by $$\mathrm{R}=\{(x, y): 3 x+\alpha y$$ is a multiple of 7$$\}$$. The relation $$R$$ is an equivalence relation if and only if :</p> | [{"identifier": "A", "content": "$$\\alpha=14$$"}, {"identifier": "B", "content": "$$\\alpha$$ is a multiple of 4"}, {"identifier": "C", "content": "4 is the remainder when $$\\alpha$$ is divided by 10"}, {"identifier": "D", "content": "4 is the remainder when $$\\alpha$$ is divided by 7"}] | ["D"] | null | <p>$$R = \{ (x,y):3x + \alpha y$$ is multiple of 7$$\} $$, now R to be an equivalence relation</p>
<p>(1) R should be reflexive : $$(a,a) \in R\,\forall \,a \in N$$</p>
<p>$$\therefore$$ $$3a + a\alpha = 7k$$</p>
<p>$$\therefore$$ $$(3 + \alpha )a = 7k$$</p>
<p>$$\therefore$$ $$3 + \alpha = 7{k_1} \Rightarrow \alpha ... | mcq | jee-main-2022-online-28th-july-morning-shift | 8,228 |
1ldo4r397 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$P(S)$$ denote the power set of $$S=\{1,2,3, \ldots ., 10\}$$. Define the relations $$R_{1}$$ and $$R_{2}$$ on $$P(S)$$ as $$\mathrm{AR}_{1} \mathrm{~B}$$ if $$\left(\mathrm{A} \cap \mathrm{B}^{\mathrm{c}}\right) \cup\left(\mathrm{B} \cap \mathrm{A}^{\mathrm{c}}\right)=\emptyset$$ and $$\mathrm{AR}_{2} \mathrm{... | [{"identifier": "A", "content": "only $$R_{2}$$ is an equivalence relation"}, {"identifier": "B", "content": "both $$R_{1}$$ and $$R_{2}$$ are not equivalence relations"}, {"identifier": "C", "content": "both $$R_{1}$$ and $$R_{2}$$ are equivalence relations"}, {"identifier": "D", "content": "only $$R_{1}$$ is an equiv... | ["C"] | null | $\begin{aligned} & \mathrm{S}=\{1,2,3, \ldots \ldots 10\} \\\\ & \mathrm{P}(\mathrm{S})=\text { power set of } \mathrm{S} \\\\ & \mathrm{AR}_1 \mathrm{B} \Rightarrow(\mathrm{A} \cap \overline{\mathrm{B}}) \cup(\overline{\mathrm{A}} \cap \mathrm{B})=\phi \\\\ & \mathrm{R}_1 \text { is reflexive, symmetri... | mcq | jee-main-2023-online-1st-february-evening-shift | 8,229 |
ldo96adc | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Among the relations
<br/><br/>$\mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}-\{0\}, 2+\frac{\mathrm{a}}{\mathrm{b}}>0\right\}$
<br/><br/> and $\mathrm{T}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}, \mathrm{a}^{2}-\mathrm{b}^{2} \in \mathbb{Z}\right\}$, | [{"identifier": "A", "content": "$\\mathrm{S}$ is transitive but $\\mathrm{T}$ is not\n"}, {"identifier": "B", "content": "both $\\mathrm{S}$ and $\\mathrm{T}$ are symmetric"}, {"identifier": "C", "content": "neither $S$ nor $T$ is transitive"}, {"identifier": "D", "content": "$T$ is symmetric but $S$ is not"}] | ["D"] | null | For relation $\mathrm{T}=\mathrm{a}^{2}-\mathrm{b}^{2}=-\mathrm{I}$
<br/><br/>Then, $(\mathrm{b}, \mathrm{a})$ on relation $\mathrm{R}$
<br/><br/>$\Rightarrow \mathrm{b}^{2}-\mathrm{a}^{2}=-\mathrm{I}$
<br/><br/>$\therefore \mathrm{T}$ is symmetric
<br/><br/>$\mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, ... | mcq | jee-main-2023-online-31st-january-evening-shift | 8,230 |
1ldomayox | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$R$$ be a relation on $$\mathbb{R}$$, given by $$R=\{(a, b): 3 a-3 b+\sqrt{7}$$ is an irrational number $$\}$$. Then $$R$$ is</p> | [{"identifier": "A", "content": "an equivalence relation"}, {"identifier": "B", "content": "reflexive and symmetric but not transitive"}, {"identifier": "C", "content": "reflexive and transitive but not symmetric"}, {"identifier": "D", "content": "reflexive but neither symmetric nor transitive"}] | ["D"] | null | <b>For reflexive :</b>
<br/><br/>$3 a-3 a+\sqrt{7}$ is an irrational number $\forall a \in R R$ is reflexive
<br/><br/><b>For symmetric :</b>
<br/><br/>Let $3 a-3 b+\sqrt{7}$ is an irrational number
<br/><br/>$\Rightarrow 3 b-3 a+\sqrt{7}$ is an irrational number
<br/><br/>For example, Let $3 a-3 b=\sqrt{7}$
<br/... | mcq | jee-main-2023-online-1st-february-morning-shift | 8,231 |
1ldprx4f6 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$\mathrm{R}$$ be a relation on $$\mathrm{N} \times \mathbb{N}$$ defined by $$(a, b) ~\mathrm{R}~(c, d)$$ if and only if $$a d(b-c)=b c(a-d)$$. Then $$\mathrm{R}$$ is</p> | [{"identifier": "A", "content": "symmetric and transitive but not reflexive"}, {"identifier": "B", "content": "reflexive and symmetric but not transitive"}, {"identifier": "C", "content": "transitive but neither reflexive nor symmetric"}, {"identifier": "D", "content": "symmetric but neither reflexive nor transitive"}] | ["D"] | null | Given, $(a, b) R(c, d) \Rightarrow a d(b-c)=b c(a-d)$
<br/><br/><b>Symmetric :</b>
<br/><br/>(c, d) $R(a, b) \Rightarrow \operatorname{cb}(\mathrm{d}-\mathrm{a})=\mathrm{da}(\mathrm{c}-\mathrm{b}) $
<br/><br/>$\Rightarrow$ Symmetric.
<br/><br/><b>Reflexive :</b>
<br/><br/>(a, b) R (a, b) $\Rightarrow a b(b-a) \neq... | mcq | jee-main-2023-online-31st-january-morning-shift | 8,232 |
1ldselj5c | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let R be a relation defined on $$\mathbb{N}$$ as $$a\mathrm{R}b$$ if $$2a+3b$$ is a multiple of $$5,a,b\in \mathbb{N}$$. Then R is</p> | [{"identifier": "A", "content": "an equivalence relation"}, {"identifier": "B", "content": "non reflexive"}, {"identifier": "C", "content": "symmetric but not transitive"}, {"identifier": "D", "content": "transitive but not symmetric"}] | ["A"] | null | <p>a R b if 2a + 3b = 5m, m $$\in$$ $$l$$</p>
<p>(1) $$(a,a) \in R$$ as $$2a + 3a = 5a,a \in N$$</p>
<p>Hence, R is reflexive</p>
<p>(2) If $$(a,b) \in R$$ then $$2a + 3 = 5m$$</p>
<p>Now, $$5(a + b) = 5n$$</p>
<p>$$3a + 2b + 2a + 3b = 5n$$</p>
<p>$$\therefore$$ $$3a + 2b = 5(n - m)$$</p>
<p>$$\therefore$$ $$(b,a) \in ... | mcq | jee-main-2023-online-29th-january-evening-shift | 8,234 |
1ldwxc9aj | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation, is __________.</p> | [] | null | 13 | $R=\{(a, b)(b, c)(b, d)\}$
<br/><br/>
$S:\{a, b, c, d\}$
<br/><br/>
Adding $(a, a),(b, b),(c, c),(d, d)$ make reflexive.
<br/><br/>
Adding $(b, a),(c, b),(d, b)$ make Symmetric
<br/><br/>
And adding $(a, d),(a, c)$ to make transitive
<br/><br/>
Further $(d, a) \&(c, a)$ to be added to make Symmetricity.
<br/><br/>
Furt... | integer | jee-main-2023-online-24th-january-evening-shift | 8,235 |
1lgoxyjgf | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$$ and $$\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$$ or $$\left.b^{2}=a+1\right\}$$ be a relation on $$\mathrm{A}$$. Then the minimum number of elements, that must be added to the relation $$\mathrm{R}$$ so that it becomes reflexive and symmetric, is... | [] | null | 7 | $$
\begin{aligned}
A & =\{-4,-3,-2,0,1,3,4\} \\\\
R= & \{(-4,4),(-3,3),(0,0),(1,1) \\
& (3,3),(4,4),(0,1),(3,-2)\}
\end{aligned}
$$
<br/><br/>Relation to be reflexive $(a, a) \in R \forall a \in A$
<br/><br/>$\Rightarrow (-4,-4),(-3,-3),(-2,-2)$ also should be added in $R$.
<br/><br/>Relation to be symmetric if $(a, b... | integer | jee-main-2023-online-13th-april-evening-shift | 8,237 |
1lgrgm8hz | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>The number of relations, on the set $$\{1,2,3\}$$ containing $$(1,2)$$ and $$(2,3)$$, which are reflexive and transitive but not symmetric, is __________.</p> | [] | null | 3 | <p>To find the number of such relations, let's first understand what it means for a relation to be reflexive, transitive, and not symmetric.</p>
<p>A relation $$R$$ on a set $$S$$ is <strong>reflexive</strong> if every element is related to itself. That is, $$(a, a) \in R$$ for all $$a \in S$$.</p>
<p>A relation is <... | integer | jee-main-2023-online-12th-april-morning-shift | 8,238 |
1lgyld2f2 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$\mathrm{A}=\{1,2,3,4,5,6,7\}$$. Then the relation $$\mathrm{R}=\{(x, y) \in \mathrm{A} \times \mathrm{A}: x+y=7\}$$ is :</p> | [{"identifier": "A", "content": "reflexive but neither symmetric nor transitive"}, {"identifier": "B", "content": "transitive but neither symmetric nor reflexive"}, {"identifier": "C", "content": "symmetric but neither reflexive nor transitive"}, {"identifier": "D", "content": "an equivalence relation"}] | ["C"] | null | Here, $A=\{1,2,3,4,5,6,7\}$
<br/><br/>Since, $x+y=7 \Rightarrow y=7-x$
<br/><br/>So, $\mathrm{R}=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$
<br/><br/>$\because(a, b) \in \mathrm{R} \Rightarrow(b, a) \in \mathrm{R}$
<br/><br/>$\therefore \mathrm{R}$ is symmetric only. | mcq | jee-main-2023-online-8th-april-evening-shift | 8,239 |
1lh00d5h8 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$A=\{0,3,4,6,7,8,9,10\}$$ and $$R$$ be the relation defined on $$A$$ such that $$R=\{(x, y) \in A \times A: x-y$$ is odd positive integer or $$x-y=2\}$$. The minimum number of elements that must be added to the relation $$R$$, so that it is a symmetric relation, is equal to ____________.</p> | [] | null | 19 | We have, $A=\{0,3,4,6,7,8,9,10\}$
<br/><br/>Case I : $x-y$ is odd, if one is odd and one is even and $x>y$.
<br/><br/>$\therefore$ Possibilites are $\{(3,0),(4,3),(6,3),(7,6),(7,4)$, $(7,0),(8,7),(8,3),(9,8),(9,6),(9,4),(9,0),(10,9),(10$, $7),(10,3)\}$
<br/><br/>No. of cases $=15$
<br/><br/>Case II : $x-y=2$
<br/><... | integer | jee-main-2023-online-8th-april-morning-shift | 8,240 |
lsam6kw5 | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Consider the relations $R_1$ and $R_2$ defined as $a R_1 b \Leftrightarrow a^2+b^2=1$ for all $a, b \in \mathbf{R}$ and $(a, b) R_2(c, d) \Leftrightarrow$ $a+d=b+c$ for all $(a, b),(c, d) \in \mathbf{N} \times \mathbf{N}$. Then : | [{"identifier": "A", "content": "$R_1$ and $R_2$ both are equivalence relations"}, {"identifier": "B", "content": "Only $R_1$ is an equivalence relation"}, {"identifier": "C", "content": "Only $R_2$ is an equivalence relation"}, {"identifier": "D", "content": "Neither $R_1$ nor $R_2$ is an equivalence relation"}] | ["C"] | null | <p>To determine if the given relations $R_1$ and $R_2$ are equivalence relations, we need to check whether each of them satisfies the three defining properties of an equivalence relation: reflexivity, symmetry, and transitivity.</p>
<p>Let's start by analysing $R_1$:</p>
<p>Reflexivity: A relation $R$ on a set $S$ is... | mcq | jee-main-2024-online-1st-february-evening-shift | 8,241 |
lsbkh6se | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation <br/><br/>$\mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{A} \cap \mathrm{B} \neq \phi ; \mathrm{A}, \mathrm{B} \in \mathrm{M}\}$ is : | [{"identifier": "A", "content": "symmetric only"}, {"identifier": "B", "content": "reflexive only"}, {"identifier": "C", "content": "symmetric and reflexive only"}, {"identifier": "D", "content": "symmetric and transitive only"}] | ["A"] | null | <p>Let $$S=\{1,2,3, \ldots, 10\}$$</p>
<p>$$R=\{(A, B): A \cap B \neq \phi ; A, B \in M\}$$</p>
<p>For Reflexive,</p>
<p>$$M$$ is subset of '$$S$$'</p>
<p>So $$\phi \in \mathrm{M}$$</p>
<p>for $$\phi \cap \phi=\phi$$</p>
<p>$$\Rightarrow$$ but relation is $$\mathrm{A} \cap \mathrm{B} \neq \phi$$</p>
<p>So it is not ref... | mcq | jee-main-2024-online-27th-january-morning-shift | 8,242 |
jaoe38c1lseymk2g | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$R$$ be a relation on $$Z \times Z$$ defined by $$(a, b) R(c, d)$$ if and only if $$a d-b c$$ is divisible by 5. Then $$R$$ is</p> | [{"identifier": "A", "content": "Reflexive and transitive but not symmetric\n"}, {"identifier": "B", "content": "Reflexive and symmetric but not transitive\n"}, {"identifier": "C", "content": "Reflexive but neither symmetric nor transitive\n"}, {"identifier": "D", "content": "Reflexive, symmetric and transitive"}] | ["B"] | null | <p>$$(a, b) R(a, b)$$ as $$a b-a b=0$$</p>
<p>Therefore reflexive</p>
<p>Let $$(a, b) R(c, d) \Rightarrow a d-b c$$ is divisible by 5</p>
<p>$$\Rightarrow \mathrm{bc}-\mathrm{ad}$$ is divisible by $$5 \Rightarrow(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b})$$</p>
<p>Therefore symmetric</p>
<p>Relation not... | mcq | jee-main-2024-online-29th-january-morning-shift | 8,243 |
lv2er45n | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let a relation $$\mathrm{R}$$ on $$\mathrm{N} \times \mathbb{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$$.
Consider the two statements:</p>
<p>(I) $$\mathrm{R}$$ is reflexive but not symmetric.</p>
<p>(II) $$\mathrm{R}$$ is transiti... | [{"identifier": "A", "content": "Only (II) is correct.\n"}, {"identifier": "B", "content": "Both (I) and (II) are correct.\n"}, {"identifier": "C", "content": "Neither (I) nor (II) is correct.\n"}, {"identifier": "D", "content": "Only (I) is correct."}] | ["D"] | null | <p>$$\begin{aligned}
& \left(x_1, y_1\right) R\left(x_2, y_2\right) \\
& \text { If } x_1 \leq x_2 \text { or } y_1 \leq y_2
\end{aligned}$$</p>
<p>For reflexive;</p>
<p>$$\begin{aligned}
& \left(x_1, y_1\right) R\left(x_1, y_1\right) \\
& \Rightarrow x_1 \leq x_1 \text { or } y_1 \leq y_1
\end{aligned}$$</p>
<p>So, $$... | mcq | jee-main-2024-online-4th-april-evening-shift | 8,244 |
lv3vefnm | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$A=\{2,3,6,8,9,11\}$$ and $$B=\{1,4,5,10,15\}$$. Let $$R$$ be a relation on $$A \times B$$ defined by
$$(a, b) R(c, d)$$ if and only if $$3 a d-7 b c$$ is an even integer. Then the relation $$R$$ is</p> | [{"identifier": "A", "content": "reflexive but not symmetric.\n"}, {"identifier": "B", "content": "an equivalence relation.\n"}, {"identifier": "C", "content": "reflexive and symmetric but not transitive.\n"}, {"identifier": "D", "content": "transitive but not symmetric."}] | ["C"] | null | <p>$$(a, b) R(c, d) \Rightarrow 3 a d-7 b c \in$$ even</p>
<p>For reflexive</p>
<p>$$(a, b) R(a, b) \Rightarrow 3 a b-7 b a=-4 a b \in$$ even</p>
<p>For symmetric</p>
<p>$$(a, b) R(c, d)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw4jkysz/b1310f77-4268-428b-aeb5-13316c958a9a/413519a0-10ec... | mcq | jee-main-2024-online-8th-april-evening-shift | 8,245 |
lvb294aa | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let $$\mathrm{A}=\{1,2,3,4,5\}$$. Let $$\mathrm{R}$$ be a relation on $$\mathrm{A}$$ defined by $$x \mathrm{R} y$$ if and only if $$4 x \leq 5 \mathrm{y}$$. Let $$\mathrm{m}$$ be the number of elements in $$\mathrm{R}$$ and $$\mathrm{n}$$ be the minimum number of elements from $$\mathrm{A} \times \mathrm{A}$$ that a... | [{"identifier": "A", "content": "23"}, {"identifier": "B", "content": "26"}, {"identifier": "C", "content": "25"}, {"identifier": "D", "content": "24"}] | ["C"] | null | <p>$$\begin{aligned}
& A=\{1,2,3,4,5\} \\
& x R y \Leftrightarrow 4 x \leq 5 y \\
& 4 x \leq 5 y \quad \Rightarrow \quad \frac{x}{y} \leq \frac{5}{4} \quad \Rightarrow \frac{x}{y} \leq 1.25
\end{aligned}$$</p>
<p>$$\begin{aligned}
& R=\{(1,2),(1,3),(1,4),(1,5),(1,1),(2,2),(2,3),(2,4), \\
& (2,5),(3,3),(3,4),(3,5),(4,4)... | mcq | jee-main-2024-online-6th-april-evening-shift | 8,246 |
lvc57b2n | maths | sets-and-relations | symmetric,-transitive-and-reflexive-properties | <p>Let the relations $$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+... | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "12"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "10"}] | ["D"] | null | <p>$$\begin{aligned}
& R_1=\{(x, y): 2 x-3 y=2\} \\
& R_2=\{(x, y):-5 x+4 y=0\} \\
& 2 x-3 y=2
\end{aligned}$$</p>
<p>So $$2 x$$ and $$3 y$$ both has to be even or odd simultaneously and $$2 x$$ can't be odd so $$2 x$$ and $$3 y$$ both will be even</p>
<p>$$R_1=\{(4,2),(7,4),(10,6),(13,8),(16,10),(19,12)\}$$</p>
<p>For... | mcq | jee-main-2024-online-6th-april-morning-shift | 8,247 |
ljalzt2e | maths | sets-and-relations | venn-diagram | If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then : | [{"identifier": "A", "content": "$A=C$"}, {"identifier": "B", "content": "$B=C$"}, {"identifier": "C", "content": "$A \\cap B=\\phi$"}, {"identifier": "D", "content": "$A=B$"}] | ["B"] | null | <p>From the given conditions, we have :</p>
<ol>
<li><p>A ∩ B = A ∩ C : The intersection of set A with set B is the same as the intersection of set A with set C. This indicates that all elements common to A and B are also common to A and C, and vice versa.</p>
</li>
<br/><li><p>A ∪ B = A ∪ C : The union of set A with s... | mcq | aieee-2009 | 8,248 |
l7sKsDq2kn1zBOJ3jUvQF | maths | sets-and-relations | venn-diagram | In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is | [{"identifier": "A", "content": "42"}, {"identifier": "B", "content": "102"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "38"}] | ["D"] | null | We're given that there are 140 students numbered from 1 to 140.
<br/><br/>1. Define the set $A$ to be the set of even numbered students. The cardinality of $A$ (the number of elements in $A$), denoted as $n(A)$, can be computed as the greatest integer less than or equal to $140/2$. Hence, $n(A) = \left[\frac{140}{2}\... | mcq | jee-main-2019-online-10th-january-morning-slot | 8,249 |
bm95iHHu7AMPxaoEUQ18hoxe66ijvwvu5os | maths | sets-and-relations | venn-diagram | Two newspapers A and B are published in a city.
It is known that 25% of the city populations reads
A and 20% reads B while 8% reads both A and
B. Further, 30% of those who read A but not B
look into advertisements and 40% of those who
read B but not A also look into advertisements,
while 50% of those who read both A an... | [{"identifier": "A", "content": "13.5"}, {"identifier": "B", "content": "13"}, {"identifier": "C", "content": "12.8"}, {"identifier": "D", "content": "13.9"}] | ["D"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265966/exam_images/s3wpew3c8wnvsbkrzigh.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263345/exam_images/cerp3qr3wsysa2fxukmv.webp"><img src="https://res.c... | mcq | jee-main-2019-online-9th-april-evening-slot | 8,250 |
O7qOj5q59aigrfqtMA3rsa0w2w9jxae4toq | maths | sets-and-relations | venn-diagram | Let A, B and C be sets such that $$\phi $$ $$ \ne $$ A $$ \cap $$ B $$ \subseteq $$ C. Then which of the following statements is not true ? | [{"identifier": "A", "content": "If (A \u2013 B) $$ \\subseteq $$ C, then A $$ \\subseteq $$ C"}, {"identifier": "B", "content": "B $$ \\cap $$ C $$ \\ne $$ $$\\phi $$"}, {"identifier": "C", "content": "(C $$ \\cup $$ A) $$ \\cap $$ (C $$ \\cup $$ B) = C"}, {"identifier": "D", "content": "If (A \u2013 C) $$ \\subseteq ... | ["D"] | null | According to the question, we have the following Venn diagram.
<br><br>Here, $A \cap B \subseteq C$ and $A \cap B \neq \phi$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lj7pt113/8d56bc6b-c920-4fcb-9e3c-2bea678573a5/22404c90-114c-11ee-b657-a1fe60b76246/file-6y3zli1lj7pt13t.png?format=p... | mcq | jee-main-2019-online-12th-april-evening-slot | 8,251 |
0lH3KaKfqfK9WMpOSp7k9k2k5kheoi8 | maths | sets-and-relations | venn-diagram | If A = {x $$ \in $$ R : |x| < 2} and B = {x $$ \in $$ R : |x – 2| $$ \ge $$ 3};
then : | [{"identifier": "A", "content": "A \u2013 B = [\u20131, 2)"}, {"identifier": "B", "content": "A $$ \\cup $$ B = R \u2013 (2, 5)"}, {"identifier": "C", "content": "A $$ \\cap $$ B = (\u20132, \u20131)"}, {"identifier": "D", "content": "B \u2013 A = R \u2013 (\u20132, 5)"}] | ["D"] | null | A : x $$ \in $$ (–2, 2);
<br><br>B : x $$ \in $$ (–$$\infty $$, –1] $$ \cup $$ [5, $$\infty $$)
<br><br>$$ \Rightarrow $$ B – A = R – (–2, 5)
<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267710/exam_images/ndmd3v807ggdl7pdczek.webp" style="max-width: 100%;height: auto;display: block;margin:... | mcq | jee-main-2020-online-9th-january-evening-slot | 8,252 |
D2PrxSbbi6MwpADcmujgy2xukf8zgpnm | maths | sets-and-relations | venn-diagram | A survey shows that 63% of the people in a city read newspaper A whereas 76% read
newspaper B. If x% of the people read both the newspapers, then a possible value of x can be: | [{"identifier": "A", "content": "37"}, {"identifier": "B", "content": "65"}, {"identifier": "C", "content": "29"}, {"identifier": "D", "content": "55"}] | ["D"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263695/exam_images/le1gtktn6hvqfqyl9vnt.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267027/exam_images/vaqvolhmyxkkppgwspnb.webp"><img src="https://res.c... | mcq | jee-main-2020-online-4th-september-morning-slot | 8,253 |
A6axaqgESreYfYvDBFjgy2xukfg6ejul | maths | sets-and-relations | venn-diagram | A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be : | [{"identifier": "A", "content": "63"}, {"identifier": "B", "content": "36"}, {"identifier": "C", "content": "54"}, {"identifier": "D", "content": "38"}] | ["B"] | null | C $$ \to $$ person like coffee
<br><br>T $$ \to $$ person like Tea
<br><br>n(C) = 73
<br><br>n(T) = 65
<br><br>n(C $$ \cup $$ T) $$ \le $$ 100
<br><br>n(C) + n(T) – n (C $$ \cap $$ T) $$ \le $$ 100
<br><br>73 + 65 – x $$ \le $$ 100
<br><br>x $$ \ge $$ 38
<br><br>73 – x $$ \ge $$ 0 $$ \Rightarrow $$ x $$ \le $$ 73
<br><... | mcq | jee-main-2020-online-5th-september-morning-slot | 8,254 |
XyHNmjyUgZXbsPIFQb1kmjb97ia | maths | sets-and-relations | venn-diagram | In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?<br/><br/><img src="data:image/png;base64,UklGRoQbAABXRUJQVlA4IHgbAABwggCdASoTArYAPm00lkgkIqIhI5KbEIANiWlu/HyXDcW+TUvlCOTKOR8J/0n... | [{"identifier": "A", "content": "Q and R"}, {"identifier": "B", "content": "None of these"}, {"identifier": "C", "content": "P and R"}, {"identifier": "D", "content": "P and Q"}] | ["B"] | null | As none play all three games the intersection of all
three circles must be zero.
<br><br>Hence none of P, Q, R justify the given statement | mcq | jee-main-2021-online-17th-march-morning-shift | 8,255 |
1ktbcom3g | maths | sets-and-relations | venn-diagram | Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set : | [{"identifier": "A", "content": "{80, 83, 86, 89}"}, {"identifier": "B", "content": "{84, 86, 88, 90}"}, {"identifier": "C", "content": "{79, 81, 83, 85}"}, {"identifier": "D", "content": "{84, 87, 90, 93}"}] | ["C"] | null | <p>This solution begins by applying the principle of inclusion and exclusion, which in the context of this problem, is represented by the formula : </p>
<p>n(A ∪ B) ≥ n(A) + n(B) - n(A ∩ B)</p>
<p>Here, n(A ∪ B) represents the total number of patients in the hospital, which is 100%. n(A) represents the proportion of pa... | mcq | jee-main-2021-online-26th-august-morning-shift | 8,256 |
1lguwarzq | maths | sets-and-relations | venn-diagram | <p>An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?</p> | [{"identifier": "A", "content": "10"}, {"identifier": "B", "content": "15"}, {"identifier": "C", "content": "21"}, {"identifier": "D", "content": "9"}] | ["C"] | null | <ol>
<li><p>We are given the number of medals for events A, B, and C which are 48, 25, and 18 respectively. We are also given that the total number of unique medal recipients across all events is 60 and that 5 people received a medal in all three events.</p>
</li>
<br><li><p>Using the Principle of Inclusion and Exclusi... | mcq | jee-main-2023-online-11th-april-morning-shift | 8,257 |
lv0vxdqq | maths | sets-and-relations | venn-diagram | <p>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... | [] | null | 45 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lx36wai5/74d61a18-6cda-4479-99f2-da165f8c36c1/14310dd0-23fa-11ef-9cd5-572f1863d2cb/file-6y3zli1lx36wai6.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lx36wai5/74d61a18-6cda-4479-99f2-da165f8c36c1/14310dd0-23fa-11ef-9c... | integer | jee-main-2024-online-4th-april-morning-shift | 8,258 |
lvc57bbi | maths | sets-and-relations | venn-diagram | <p>Let $$A=\{n \in[100,700] \cap \mathrm{N}: n$$ is neither a multiple of 3 nor a multiple of 4$$\}$$. Then the number of elements in $$A$$ is</p> | [{"identifier": "A", "content": "300"}, {"identifier": "B", "content": "310"}, {"identifier": "C", "content": "290"}, {"identifier": "D", "content": "280"}] | ["A"] | null | <p>$$n \in[100,700]$$</p>
<p>$$n(A)=$$ Total $$-$$ (multiple of $$3$$ + multiple of 4) + (multiple of 12)</p>
<p>Total $$=601$$</p>
<p>Multiple of $$3=102,105, \ldots, 699$$</p>
<p>$$\begin{aligned}
& n=699=102+(n-1) 3 \\
& \Rightarrow n=200
\end{aligned}$$</p>
<p>Multiple of $$4=100,104 \ldots ., 700$$</p>
<p>$$\begin... | mcq | jee-main-2024-online-6th-april-morning-shift | 8,259 |
S2CpfA5xzaUED7bD | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of
the set is increased by 2, then the median of the new set : | [{"identifier": "A", "content": "is increased by 2"}, {"identifier": "B", "content": "is decreased by 2"}, {"identifier": "C", "content": "is two times the original median"}, {"identifier": "D", "content": "remains the same as that of the original set "}] | ["D"] | null | Here total no of observation is 9 which is a odd number. As we know for odd number 9 the median will be the 5<sup>th</sup> term.
<br><br>Now question says, you increase largest 4 number by 2 which does not affect the 5<sup>th</sup> term so the new median will be the same. | mcq | aieee-2003 | 8,261 |
IEaBRIby7TZbChYp | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | If in a frequency distribution, the mean and median are 21 and 22 respectively, then
its mode is approximately : | [{"identifier": "A", "content": "20.5"}, {"identifier": "B", "content": "22.0"}, {"identifier": "C", "content": "24.0"}, {"identifier": "D", "content": "25.5"}] | ["C"] | null | Given that,
<br><br>Mean = 21 and median = 22
<br><br>We know,
<br><br>Mode + 2 Mean = 3 Median
<br><br>$$\therefore$$ Mode = 3 $$ \times $$ 22 $$-$$ 2 $$ \times $$ 21
<br><br>= 66 $$-$$ 42
<br><br>= 24 | mcq | aieee-2005 | 8,262 |
lB1b64UMGJQfxOW0 | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted
and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is : | [{"identifier": "A", "content": "15.8"}, {"identifier": "B", "content": "14.0"}, {"identifier": "C", "content": "16.8"}, {"identifier": "D", "content": "16.0"}] | ["B"] | null | Initially we have $$16$$ observations and among them one is $$16.$$
<br><br>So, we have $$15$$ unknowns. Let those are $${a_1},a{}_2,{a_3}.....{a_{15}}$$
<br><br>$$\therefore\,\,\,$$ Mean of $$16$$ datal set
<br><br>$$ = {{{a_1} + {a_2} + .....{a_{15}} + 16} \over {16}}$$
<br><br>According to the question,
<br><br>... | mcq | jee-main-2015-offline | 8,264 |
yW2sKypZRX7KsEL99JymR | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is : | [{"identifier": "A", "content": "25"}, {"identifier": "B", "content": "30"}, {"identifier": "C", "content": "35"}, {"identifier": "D", "content": "40"}] | ["C"] | null | Mean $$\left( {\overline x } \right)$$ = $${{{x_1} + {x_2}..... + {x_n}} \over n}$$ = $${{\sum x } \over n}$$
<br><br>Here, Mean = 40 of 25 teachers
<br><br>$$\therefore$$ 40 = $${{\sum x } \over {25}}$$
<br><br>$$ \Rightarrow $$ $$\sum x $$ = 40 $$ \times $$ 25 = 1000
<br><br>After retireing of a 60 year old teacher, ... | mcq | jee-main-2017-online-8th-april-morning-slot | 8,265 |
neBwTeBhdFgcTfYjwLPWU | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | The mean of set of 30 observations is 75. If each observation is multiplied by a non-zero number $$\lambda $$ and then each of them is decreased by 25, their mean remains the same. Then $$\lambda $$ is equal to : | [{"identifier": "A", "content": "$${1 \\over 3}$$"}, {"identifier": "B", "content": "$${2 \\over 3}$$"}, {"identifier": "C", "content": "$${4 \\over 3}$$"}, {"identifier": "D", "content": "$${10 \\over 3}$$"}] | ["C"] | null | As mean is a linear operation, so if each observation is multiplied by $$\lambda $$ and decreased by 25 then the mean becomes 75$$\lambda $$$$-$$25.
<br><br>According to the question,
<br><br>75$$\lambda $$ $$-$$ 25 = 75 $$ \Rightarrow $$ $$\lambda $$ = $${4 \over 3}$$. | mcq | jee-main-2018-online-15th-april-morning-slot | 8,266 |
GqaYQGm3JSi2zukbEp18hoxe66ijvww34j1 | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | The mean and the median of the following ten
numbers in increasing order 10, 22, 26, 29, 34, x,
42, 67, 70, y are 42 and 35 respectively, then $${y \over x}$$ is equal to | [{"identifier": "A", "content": "$${7 \\over 2}$$"}, {"identifier": "B", "content": "$${8 \\over 3}$$"}, {"identifier": "C", "content": "$${9 \\over 4}$$"}, {"identifier": "D", "content": "$${7 \\over 3}$$"}] | ["D"] | null | Given ten numbers are 10, 22, 26, 29, 34, x,
42, 67, 70, y.
<br><br>As the numbers are in increasing order so
<br><br>Mediun = $${{34 + x} \over 2}$$ = 35
<br><br>$$ \Rightarrow $$ x = 36
<br><br>Also given mean = 42
<br><br>$$ \Rightarrow $$ $${{10 + 22 + 26 + 29 + 34 + x + 42 + 67 + 70 + y} \over {10}}$$ = 42
<br><b... | mcq | jee-main-2019-online-9th-april-evening-slot | 8,267 |
h2VdOffcBfbNtOpUQ73rsa0w2w9jwy0yzk9 | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | If for some x $$ \in $$ R, the frequency distribution of the marks obtained by 20 students in a test is :<br/><br/>
<style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-brea... | [{"identifier": "A", "content": "3.0"}, {"identifier": "B", "content": "2.8"}, {"identifier": "C", "content": "2.5"}, {"identifier": "D", "content": "3.2"}] | ["B"] | null | Number of students<br><br>
$$ \Rightarrow {\left( {x + 1} \right)^2} + (2x - 5) + \left( {{x^2} - 3x} \right) + x = 20$$<br><br>
$$ \Rightarrow 2{x^2} + 2x - 4 = 20$$<br><br>
$$ \Rightarrow {x^2} + x - 12 = 0$$<br><br>
$$ \Rightarrow (x + 4)(x - 3) = 0$$<br><br>
$$x = 3$$<br><br>
<style type="text/css">
.tg {border-co... | mcq | jee-main-2019-online-10th-april-morning-slot | 8,268 |
KqISKJ8PMSjrFVZan4jgy2xukg4n89i7 | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | Consider the data on x taking the values<br/> 0, 2, 4,
8,....., 2<sup>n</sup> with frequencies<br/> <sup>n</sup>C<sub>0</sub>
,
<sup>n</sup>C<sub>1</sub>
,
<sup>n</sup>C<sub>2</sub>
,....,
<sup>n</sup>C<sub>n</sub>
respectively. If the<br/> mean of this data is $${{728} \over {{2^n}}}$$, then n is equal to _________ .
| [] | null | 6 | Mean = $${{\sum {{x_1}.{f_1}} } \over {\sum {{f_1}} }}$$
<br><br>= $${{0.{}^n{C_0} + 2.{}^n{C_1} + {2^2}.{}^n{C_2} + ... + {2^n}.{}^n{C_n}} \over {{}^n{C_0} + {}^n{C_1} + ... + {}^n{C_n}}}$$
<br><br>We know,
<br><br>(1 + x)<sup>n</sup> = $${{}^n{C_0} + {}^n{C_1}x + {}^n{C_2}{x^2} + ... + {}^n{C_n}{x^n}}$$ ...(1)
<br><b... | integer | jee-main-2020-online-6th-september-evening-slot | 8,269 |
pFSajzAuBaaQVZQ3Qj1kmlm1y0e | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is _________. | [] | null | 35 | Mean $$\left( {\overline x } \right)$$ = $${{{x_1} + {x_2}..... + {x_n}} \over n}$$ = $${{\sum x } \over n}$$
<br><br>Here, Mean = 40 of 25 teachers
<br><br>$$\therefore$$ 40 = $${{\sum x } \over {25}}$$
<br><br>$$ \Rightarrow $$ $$\sum x $$ = 40 $$ \times $$ 25 = 1000
<br><br>After retireing of a 60 year old teacher, ... | integer | jee-main-2021-online-18th-march-morning-shift | 8,270 |
1krub3sh2 | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | Consider the following frequency distribution :<br/><br/><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;... | [] | null | 4 | <table class="tg">
<thead>
<tr>
<th class="tg-0lax">Class</th>
<th class="tg-0lax">Frequency</th>
<th class="tg-0lax">$${x_i}$$</th>
<th class="tg-0lax">$${f_i}{x_i}$$</th>
</tr>
</thead>
<tbody>
<tr>
<td class="tg-0lax">0-6</td>
<td class="tg-0lax">a</td>
<td class="tg-0lax">3</td>
... | integer | jee-main-2021-online-22th-july-evening-shift | 8,271 |
1krw28thi | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | Consider the following frequency distribution :<br/><br/><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;... | [] | null | 164 | $$\because$$ Sum of frequencies = 584<br><br>$$\Rightarrow$$ $$\alpha$$ + $$\beta$$ = 390<br><br>Now, median is at $${{584} \over 2}$$ = 292<sup>th</sup><br><br>$$\because$$ Median = 45 (lies in class 40 - 50)<br><br>$$\Rightarrow$$ $$\alpha$$ + 110 + 54 + 15 = 292<br><br>$$\Rightarrow$$ $$\alpha$$ = 113, $$\beta$$ = 2... | integer | jee-main-2021-online-25th-july-morning-shift | 8,272 |
1ktise4jh | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | The mean of 10 numbers 7 $$\times$$ 8, 10 $$\times$$ 10, 13 $$\times$$ 12, 16 $$\times$$ 14, ....... is ____________. | [] | null | 398 | 7 $$\times$$ 8, 10 $$\times$$ 10, 13 $$\times$$ 12, 16 $$\times$$ 14 ........<br><br>T<sub>n</sub> = (3n + 4) (2n + 6) = 2(3n + 4) (n + 3)<br><br>= 2(3n<sup>2</sup> + 13n + 12) = 6n<sup>2</sup> + 26n + 24<br><br>S<sub>10</sub> = $$\sum\limits_{n = 1}^{10} {{T_n}} = 6\sum\limits_{n = 1}^{10} {{n^2}} + 26\sum\limits_{n... | integer | jee-main-2021-online-31st-august-morning-shift | 8,273 |
1lsgack8c | maths | statistics | calculation-of-mean,-median-and-mode-of-grouped-and-ungrouped-data | <p>Let M denote the median of the following frequency distribution</p>
<p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{bo... | [{"identifier": "A", "content": "104"}, {"identifier": "B", "content": "52"}, {"identifier": "C", "content": "208"}, {"identifier": "D", "content": "416"}] | ["C"] | null | <p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial,... | mcq | jee-main-2024-online-30th-january-morning-shift | 8,275 |
Vtsx1SkaZm1Q91Mf | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | In a series of 2n observations, half of them equal $$a$$ and remaining half equal $$–a$$. If the
standard deviation of the observations is 2, then $$|a|$$ equals | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$\\sqrt 2 $$"}, {"identifier": "C", "content": "$${1 \\over n}$$"}, {"identifier": "D", "content": "$${{\\sqrt 2 } \\over n}$$"}] | ["A"] | null | Mean $$\left( A \right) = {{a - a} \over {2n}} = 0$$
<br><br>Given standard deviation (S.D) = 2
<br><br>$$\therefore\,\,\,$$ $$\sqrt {{{\sum {{{\left( {x - A} \right)}^2}} } \over {2n}}} = 2$$
<br><br>$$ \Rightarrow \,\,\,\sqrt {{{{{\left( {a - 0} \right)}^2} + {{\left( {a - 0} \right)}^2} + ..... + {{\left( {0 - a} ... | mcq | aieee-2004 | 8,278 |
KXk4NEHZZHI9u4CB | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | Let x<sub>1</sub>, x<sub>2</sub>,...........,x<sub>n</sub> be n observations such that
<br/><br/>$$\sum {x_i^2} = 400$$ and $$\sum {{x_i}} = 80$$. Then a
possible value of n among the following is | [{"identifier": "A", "content": "18"}, {"identifier": "B", "content": "15"}, {"identifier": "C", "content": "12"}, {"identifier": "D", "content": "9"}] | ["A"] | null | As we know,
<br><br>$${\sigma ^2} \ge 0$$
<br><br>$$\therefore\,\,\,$$ $${{\sum {x_i^2} } \over n} - {\left( {{{\sum {{x_i}} } \over n}} \right)^2} \ge 0$$
<br><br>$$ \Rightarrow \,\,\,{{400} \over n} - {{6400} \over {{n^2}}} \ge 0$$
<br><br>$$ \Rightarrow \,\,\,n \ge 16$$
<br><br>$$\therefore\,\,\,$$ Possible valu... | mcq | aieee-2005 | 8,279 |
jAJXr12O9W6IUO6E | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | Suppose a population A has 100 observations 101, 102,........, 200, and another
population B has 100 observations 151, 152,......., 250. If V<sub>A</sub> and V<sub>B</sub> represent the
variances of the two populations, respectively, then $${{{V_A}} \over {{V_B}}}$$ is | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "$${9 \\over 4}$$"}, {"identifier": "C", "content": "$${4 \\over 9}$$"}, {"identifier": "D", "content": "$${2 \\over 3}$$"}] | ["A"] | null | Series A = 101, 102 ............ 200
<br><br>Series B = 151, 152 ............ 250
<br><br>Here series B can be obtained if we change the origin of A by 50 units.
<br><br>And we know the variance does not change by changing the origin.
<br><br>So, $$\,\,\,\,$$ $${V_A} = {V_B}$$
<br><br>$$ \Rightarrow \,\,\,\,\,{{{V_... | mcq | aieee-2006 | 8,280 |
O8gldTfNXQMBhdGy | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | <b>Statement - 1 :</b> The variance of first n even natural numbers is $${{{n^2} - 1} \over 4}$$
<br/><br/><b>Statement - 2 :</b> The sum of first n natural numbers is $${{n\left( {n + 1} \right)} \over 2}$$ and the sum of squares of first n natural numbers is $${{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 6... | [{"identifier": "A", "content": "Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1"}, {"identifier": "B", "content": "Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1"}, {"identifier": "C", "content": "Statement-1 is true, St... | ["D"] | null | Let first n even natural numbers = 2,4, 6, 8 ...... 2n
<br><br>$$\therefore$$ Sum of those num = 2 + 4 + 6 + ..... 2n
<br><br>= 2 (1 + 2 + ..... n)
<br><br>= $$2.{{n\left( {n + 1} \right)} \over 2}$$
<br><br>= n (n + 1)
<br><br>$$\therefore\,\,\,$$ Mean $$\left( {\overline x } \right) = {{n\left( {n + 1} \right)} \ove... | mcq | aieee-2009 | 8,282 |
ZY7XixgriqKYkqZL | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding
means are given to be 2 and 4, respectively. The variance of the combined data set is | [{"identifier": "A", "content": "$${5 \\over 2}$$"}, {"identifier": "B", "content": "$${11 \\over 2}$$"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "$${13 \\over 2}$$"}] | ["B"] | null | Given that,
<br><br>$${\sigma _1}^2 = 4$$
<br><br>and $${\sigma _2}^2 = 5$$
<br><br>And also given,
<br><br>$$\overline x = 2\,\,$$ and $$\overline y = 4\,\,$$
<br><br>So, $$\,\,\,$$ $${{\sum {{x_i}} } \over 5} = 2$$
<br><br>$$ \Rightarrow \sum {{x_i}} = 10$$ v
<br><br>and $${{\sum {{y_i}} } \over 5} = 4$$
<br><... | mcq | aieee-2010 | 8,284 |
3mIl7ePHUNR0dez9 | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If the mean deviation about the median of the numbers a, 2a,........., 50a is 50, then |a| equals | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["A"] | null | <b>NOTE :</b>
<br><br>If total no of terms are even then median
<br><br>$$ = {1 \over 2}$$ [ $${n \over 2}$$th term $$ + \left( {{n \over 2} + 1} \right)$$ th term]
<br><br>Here total terms $$ = 50,$$ which is even
<br><br>$$\therefore$$ $$\,\,\,$$ Median $$ = {1 \over 2}$$ [ $${{50} \over 2}$$ th term $$ + \left( {{... | mcq | aieee-2011 | 8,285 |
xBO8kMOnfG3Qz0d1 | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | Let x<sub>1</sub>, x<sub>2</sub>,........., x<sub>n</sub> be n observations, and let $$\overline x $$ be their arithematic mean and $${\sigma ^2}$$ be their variance.
<br/><br/><b>Statement 1 :</b> Variance of 2x<sub>1</sub>, 2x<sub>2</sub>,......., 2x<sub>n</sub> is 4$${\sigma ^2}$$.
<br/><b>Statement 2 :</b> : Arithm... | [{"identifier": "A", "content": "Statement 1 is false, statement 2 is true"}, {"identifier": "B", "content": "Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1"}, {"identifier": "C", "content": "Statement 1 is true, statement 2 is true; statement 2 is not a correct explanati... | ["D"] | null | Given that,
<br><br>for $${x_1},{x_2},....{x_n},$$ $$A.M = \overline x $$
<br><br>and variance $$ = {\sigma ^2}$$
<br><br>Now A.M of
<br><br>$$2{x_1},2x{}_2.....2{x_n} = {{2\left( {{x_1} + {x_2} + ....{x_n}} \right)} \over n} = 2\overline x $$
<br><br>But given $$A.M = 4\overline x $$
<br><br>$$\therefore\,\,\,$$... | mcq | aieee-2012 | 8,286 |
yLM7mSC8LrFHWos6 | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10
to each of the students. Which of the following statistical measures will not change even after the grace
marks were given? | [{"identifier": "A", "content": "median"}, {"identifier": "B", "content": "mode"}, {"identifier": "C", "content": "variance"}, {"identifier": "D", "content": "mean "}] | ["C"] | null | As we know variance does not change with the change of origin. So, here even after adding grace marks $$10$$, the variance will be same.
<br><br>Let's see with an example,
<br><br>Assume initial variance $$ = {{\sum {{{\left( {{x_i} - \overline x } \right)}^2}} } \over N}$$
<br><br>After adding grace marks $$10$$ wi... | mcq | jee-main-2013-offline | 8,287 |
weiErB44wHyXOgoj | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | The variance of first 50 even natural numbers is | [{"identifier": "A", "content": "833"}, {"identifier": "B", "content": "437"}, {"identifier": "C", "content": "$${{437} \\over 4}$$"}, {"identifier": "D", "content": "$${{833} \\over 4}$$"}] | ["A"] | null | Here is total $$50$$ numbers, so $$N=50$$
<br><br>Variance $$ = $$ $${{\sum {{x^2}} } \over {50}} - {\left( {{{\sum x } \over {50}}} \right)^2}$$
<br><br>Here $$\sum {{x^2}} = $$ sum of square of first $$50$$ even natural number.
<br><br>$$ = {2^2} + {4^2} + ..... + {100^2}$$
<br><br>$$ = {2^2}\left[ {{1^2} + {2^2}... | mcq | jee-main-2014-offline | 8,288 |
TvmSJULEl6IoTAWP | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true? | [{"identifier": "A", "content": "3$$a$$<sup>2</sup> - 26$$a$$ + 55 = 0"}, {"identifier": "B", "content": "3$$a$$<sup>2</sup> - 32$$a$$ + 84 = 0"}, {"identifier": "C", "content": "3$$a$$<sup>2</sup> - 34$$a$$ + 91 = 0"}, {"identifier": "D", "content": "3$$a$$<sup>2</sup> - 23$$a$$ + 44 = 0"}] | ["B"] | null | The formula for standard deviation (S.D)
<br><br>$$ = \sqrt {{{\sum {x_i^2} } \over n} - {{\left( {{{\sum {{x_i}} } \over n}} \right)}^2}} $$
<br><br>Where $$\sum {x_i^2 = } $$ Sum of square of the numbers
<br><br>$$ = {2^2} + {3^2} + {a^2} + {11^2}$$
<br><br>$$ = 4 + 9 + {a^2} + 121$$
<br><br>$$ = 134 + {a^2}$$
<b... | mcq | jee-main-2016-offline | 8,289 |
o2CYmg1ei6ZV5R3QOmfZp | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | The mean of 5 observations is 5 and their variance is 124. If three of the observations
are 1, 2 and 6 ; then the mean deviation from the mean of the data is : | [{"identifier": "A", "content": "2.4"}, {"identifier": "B", "content": "2.8"}, {"identifier": "C", "content": "2.5"}, {"identifier": "D", "content": "2.6"}] | ["B"] | null | Let 5 observations are x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, x<sub>4</sub>, x<sub>5</sub>
<br><br>given, x<sub>1</sub> = 1, x<sub>2</sub> = 2, x<sub>3</sub> = 6
<br><br>Mean = 5
<br><br>$$ \therefore $$ Mean$$\left( {\overline x } \right)$$ = $${{{x_1} + {x_2} + {x_3} + {x_4} + {x_... | mcq | jee-main-2016-online-10th-april-morning-slot | 8,290 |
fDAndzgC3lC0xV4A5gqpU | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If the mean deviation of the numbers 1, 1 + d, ..., 1 +100d from their mean is 255, then a value of d is : | [{"identifier": "A", "content": "10.1"}, {"identifier": "B", "content": "20.2"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "5.05"}] | ["A"] | null | Given numbers are,
<br><br>1, 1 + d, 1 + 2d . . . . . 1 + 100d
<br><br>$$ \therefore $$ Total 101 number are present.
<br><br>$$ \therefore $$ n = 101
<br><br>$$ \therefore $$ mean $$\left( {\overline x } \right)$$ = $${{1 + \left( {1 + d} \right) + ......\left( {1 + ... | mcq | jee-main-2016-online-9th-april-morning-slot | 8,291 |
Zd4npDUYRa223mH398w05 | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | The sum of 100 observations and the sum of their squares are 400 and 2475,
respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If
the incorrect observations are omitted, then the variance of the remaining observations
is : | [{"identifier": "A", "content": "8.25 "}, {"identifier": "B", "content": "8.50"}, {"identifier": "C", "content": "8.00"}, {"identifier": "D", "content": "9.00"}] | ["D"] | null | <p>We have</p>
<p>$$\sum\limits_{i = 1}^{100} {{x_i} = 400} $$</p>
<p>$$\sum\limits_{i = 1}^{100} {x_i^2 = 2425} $$</p>
<p>The variance of the remaining observations is</p>
<p>$${\sigma ^2} = {{\sum {x_i^2} } \over N} - {\left( {{{\sum {{x_i}} } \over N}} \right)^2}$$</p>
<p>$$ \Rightarrow {{2425} \over {97}} - {\left(... | mcq | jee-main-2017-online-9th-april-morning-slot | 8,292 |
vx5i8PURp9Ov44S5 | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If $$\sum\limits_{i = 1}^9 {\left( {{x_i} - 5} \right)} = 9$$ and
<br/><br/>$$\sum\limits_{i = 1}^9 {{{\left( {{x_i} - 5} \right)}^2}} = 45$$, then the standard deviation of the 9 items
<br/>$${x_1},{x_2},.......,{x_9}$$ is | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "2"}] | ["D"] | null | <u>IMPORTANT POINT :-</u>
<br><br>When every number is added or subtracted by a fixed number then the standard Deviation remain unchanged.
<br><br>so let $${x_i} - 5 = {y_i}$$
<br><br>So, new equation is $$\sum\limits_{i = 1}^9 {{y_i}} = 9$$
<br><br>and $$\sum\limits_{i = 1}^9 {y_i^2} = 45$$
<br><br>As, we know. St... | mcq | jee-main-2018-offline | 8,293 |
mplrVzKRZMtSkE8492dXI | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If the mean of the data : 7, 8, 9, 7, 8, 7, $$\lambda $$, 8 is 8, then the variance of this data is : | [{"identifier": "A", "content": "$${7 \\over 8}$$"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "$${9 \\over 8}$$"}, {"identifier": "D", "content": "2"}] | ["B"] | null | $$\overline x $$ = $${{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8} \over 8}$$ = 8
<br><br>$$ \Rightarrow $$$$\,\,\,$$ $${{54 + \lambda } \over 8}$$ = 8 $$ \Rightarrow $$ $$\lambda $$ = 10
<br><br>Now variance = $$\sigma $$<sup>2</sup>
<br><br>= $${{{{\left( {7 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2} ... | mcq | jee-main-2018-online-15th-april-evening-slot | 8,294 |
RGk9zKugGjSBF8zj5MoXV | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | The mean and the standard deviation(s.d.) of five observations are9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is : | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "4"}] | ["C"] | null | Here mean = $$\overline x $$ = 9
<br><br>$$ \Rightarrow $$ $$\overline x $$ = $${{\sum {{x_i}} } \over n}$$ = 9
<br><br>$$ \Rightarrow $$ $${\sum {{x_i}} }$$ = 9 $$ \times $$ 5 = 45
<br><br>Now, standard deviation = 0
<br><br>$$\therefore\,\,\,$$ all the five terms are same i.e.; 9
<br><br>Now f... | mcq | jee-main-2018-online-16th-april-morning-slot | 8,295 |
AjX0JjPR9fp5X27YCGJ9H | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | A student scores the following marks in five tests
:
<br/><br/>45, 54, 41, 57, 43.
<br/><br/>His score is not known for the
sixth test. If the mean score is 48 in the six tests,
then the standard deviation of the marks in six tests
is | [{"identifier": "A", "content": "$$100 \\over {\\sqrt 3}$$"}, {"identifier": "B", "content": "$$10 \\over {\\sqrt 3}$$"}, {"identifier": "C", "content": "$$10 \\over3$$"}, {"identifier": "D", "content": "$$100 \\over3$$"}] | ["B"] | null | Let the score in the sixth test = x
<br><br>Given, Mean ($$\overline x $$) = 48
<br><br>$$ \Rightarrow $$ $${{45 + 54 + 41 + 57 + 43 + x} \over 6}$$ = 48
<br><br>$$ \Rightarrow $$ x = 48
<br><br>Standard deviation (SD)
<br><br>= $$\sqrt {{{\sum\limits_{i = 1}^N {{{\left( {{x_i} - \overline x } \right)}^2}} } \over N}}... | mcq | jee-main-2019-online-8th-april-evening-slot | 8,296 |
ia0Ybgi0hXqiDtEOXS3rsa0w2w9jx620ebc | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If the data x<sub>1</sub>, x<sub>2</sub>,......., x<sub>10</sub> is such that the mean of first four of these is 11, the mean of the remaining six is
16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is : | [{"identifier": "A", "content": "$$\\sqrt 2 $$"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "2$$\\sqrt 2 $$"}, {"identifier": "D", "content": "4"}] | ["B"] | null | $${\sigma ^2} = {{\sum {x_i^2} } \over {10}} - {\left( {{{\sum {{x_i}} } \over {10}}} \right)^2} \to (i)$$<br><br>
Now x<sub>1</sub> + x<sub>2</sub> + x<sub>3</sub> + x<sub>4</sub> = 44 & x<sub>5</sub> + x<sub>6</sub> + ......... + x<sub>10</sub> = 96<br><br>
Hence $${\sigma ^2}$$ = $${{2000} \over {10}} - {\l... | mcq | jee-main-2019-online-12th-april-morning-slot | 8,297 |
8FSQCWDSs03z11aXVQ3rsa0w2w9jx2fqton | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If both the mean and the standard deviation of 50 observations x<sub>1</sub>, x<sub>2</sub>,..., x<sub>50</sub> are equal to 16, then the mean of (x<sub>1</sub> – 4)<sup>2</sup>
, (x<sub>2 </sub>– 4)<sup>2</sup>
,....., (x<sub>50</sub> – 4)<sup>2</sup>
is : | [{"identifier": "A", "content": "400"}, {"identifier": "B", "content": "480"}, {"identifier": "C", "content": "380"}, {"identifier": "D", "content": "525"}] | ["A"] | null | $$Mean(\mu ) = {{\sum {{x_i}} } \over {50}} = 16$$<br><br>
$$ \therefore $$ $$\sum {{x_i}} = 16 \times 50$$<br><br>
$$S.D.\left( \sigma \right) = \sqrt {{{\sum {{x_i}^2} } \over {50}} - {{\left( \mu \right)}^2}} = 16$$<br><br>
$$ \Rightarrow {{\sum {{x_i}^2} } \over {50}} = 256 \times 2$$<br><br>
Required mean = $$... | mcq | jee-main-2019-online-10th-april-evening-slot | 8,298 |
Gf3bGbvx41RN1hxaDf3iQ | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | If the standard deviation of the numbers
–1, 0, 1, k is $$\sqrt 5$$ where k > 0, then k is equal to | [{"identifier": "A", "content": "2$$\\sqrt 6 $$"}, {"identifier": "B", "content": "$$\\sqrt 6 $$"}, {"identifier": "C", "content": "$$2\\sqrt {{{5} \\over 6}} $$"}, {"identifier": "D", "content": "$$2\\sqrt {{{10} \\over 3}} $$"}] | ["A"] | null | standard deviation = $$\sqrt 5$$
<br><br>$$ \therefore $$ Variance = $${\left( {\sqrt 5 } \right)^2}$$ = 5
<br><br>Also variance = $${{\sum {x_i^2} } \over N} - {\mu ^2}$$
<br><br>Where $$\mu $$ = Mean = $${{ - 1 + 0 + 1 + k} \over 4}$$ = $${k \over 4}$$
<br><br>$$ \therefore $$ Variance = $${{{{\left( { - 1} \right)}^... | mcq | jee-main-2019-online-9th-april-morning-slot | 8,299 |
u2v4ONloLZkkoEVwLPxuI | maths | statistics | calculation-of-standard-deviation,-variance-and-mean-deviation-of-grouped-and-ungrouped-data | The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are
3, 4 and 4 ; then the absolute value of the difference of the other two observations, is : | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "5"}] | ["B"] | null | mean $$\overline x $$ = 4, $$\sigma $$<sup>2</sup> = 5.2, n = 5, . x<sub>1</sub> = 3 x<sub>2</sub> = 4 = x<sub>3</sub>
<br><br>$$\sum {{x_i}} = 20$$
<br><br>x<sub>4</sub> + x<sub>5</sub> = 9 . . . . . . (i)
<br><br>$${{\sum {x_i^2} } \over x} - {\left( {\overline x } \right)^2} = \sigma \Rightarrow \sum {x_i^2} = 10... | mcq | jee-main-2019-online-12th-january-evening-slot | 8,301 |
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