sets relations and functions jee mains questions | JEE Main Mathematics Sets Relations And Functions

Q 41 :

Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]

an equivalence relation.
reflexive but neither symmetric not transitive.
both reflexive and symmetric but not transitive.
both reflexive and transitive but not symmetric.

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$\sec^{2} x - \tan^{2} x = 1, \forall x \in [0, \frac{π}{2})$

$∴$ R is reflexive

Consider, $\sec^{2} x - \tan^{2} y = 1$

$\Rightarrow 1 + \tan^{2} x - (\sec^{2} y - 1) = 1$

$\Rightarrow 1 + \tan^{2} x - s e c^{2} y + 1 = 1 \Rightarrow \sec^{2} y - \tan^{2} x = 1$

$∴$ R is symmetric.

Now, if $\sec^{2} x - \tan^{2} y = 1$ and $\sec^{2} y - \tan^{2} z = 1$

Adding both equation, $\sec^{2} x - \tan^{2} y + \sec^{2} y - \tan^{2} z = 2$

$\Rightarrow \sec^{2} x - \tan^{2} z = 1$ [ $∵ \sec^{2} y - \tan^{2} y = 1$ ]

$∴$ R is transitive

Thus R is an equivalence relation.

Q 42 :

Let $S = N \cup {0}$ . Define a relation R from S to R by:

$R = {(x, y) : \log_{e} y = x \log_{e} (\frac{2}{5}), x \in S, y \in R}$

Then, the sum of all the elements in the range of R is equal to: [2025]

$\frac{5}{2}$
$\frac{5}{3}$
$\frac{3}{2}$
$\frac{10}{9}$

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We have, $S = N \cup {0} = {0, 1, 2, 3, . . . . . .}$

Also, $\log_{e} y = x \log_{e} (\frac{2}{5}) \Rightarrow y = {(\frac{2}{5})}^{x}$

$∴$ Required sum = ${(\frac{2}{5})}^{0} + {(\frac{2}{5})}^{1} + {(\frac{2}{5})}^{2} + . . . . . . . .$

$= \frac{1}{1 - \frac{2}{5}} = \frac{5}{3}$ .

Q 43 :

The number of relation on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is __________. [2025]

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Given, A = {1, 2, 3}

Let the relation be R on A, which is reflexive and transitive but not symmetric, then

(1, 1), (2, 2), (3, 3), (1, 2) $\in$ R

Remaining elements are

(2, 1), (2, 3), (1, 3), (3, 1), (3, 2)

Case I : If relation contains exactly 4 elements $\Rightarrow$ 1 way

Case II : If relation contains exactly 5 elements, so we can add (1, 3) or (3, 2) $\Rightarrow$ 2 ways

Case III : If relation contains exactly 6 elements, so we can add (2, 3), (1, 3) or (1, 3), (3, 2) or (3, 1), (3, 2) $\Rightarrow$ 3 ways

$∴$ Total number of relations is 6.

Q 44 :

Let A = {1, 2, 3}. The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is __________. [2025]

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For transitive : (1, 2) and (2, 3) $\in$ R $\Rightarrow$ (1, 3) $\in$ R

For reflexive : (1, 1), (2, 2), (3, 3) $\in$ R

Now, for (2, 1), (3, 2), (3, 1); (3, 1) cannot be taken for not symmetric relation.

Case I : (2, 1) taken and (3, 2) not taken

Case II : (3, 2) taken and (2, 1) not taken

Case III : (2, 1) and (3, 2) are not taken

Therefore, 3 relations are possible.

Q 45 :

Let $A = {- 2, - 1, 0, 1, 2, 3, 4}$ . Let R be a relation on A defined by $x R y$ if and only if $2 x + y \leq 2$ . Let $l$ be the number of elements in R. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then $l + m + n$ is equal to: [2026]

33
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35
34

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Q 46 :

Let $R$ be a relation defined on the set ${1, 2, 3, 4} \times {1, 2, 3, 4}$ by $R = {((a, b), (c, d)) : 2 a + 3 b = 3 c + 4 d} .$

Then the number of elements in $R$ is [2026]

15
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12

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Q 47 :

Let the relation R on the set $M = {1, 2, 3, \dots, 16}$ be given by $R = {(x, y) : 4 y = 5 x - 3, x, y \in M} .$

Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to [2026]

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Q 48 :

Let $A = {0, 1, 2, \dots, 9}$ Let R be a relation on A defined by $(x, y) \in R$ if and only if $∣ x - y ∣$ is a multiple of 3.

Given below are two statements:

Statement I: n(R)=36.

Statement II: R is an equivalence relation.

In the light of the above statements, choose the correct answer from the options given below. [2026]

Statement I is correct but Statement II is incorrect
Statement I is incorrect but Statement II is correct
Both Statement I and Statement II are incorrect
Both Statement I and Statement II are correct

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Q 49 :

Let A = {2,3,5,7,9} Let R be the relation on A defined by xRy if and only if $2 x \leq 3 y$ . Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to : [2026]

25
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27
21

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Q 50 :

The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in ℤ}$ is [2026]

86
89
67
77

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Topic Question Set

Q 41 :

Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]

Q 42 :

Let $S = N \cup {0}$ . Define a relation R from S to R by:

$R = {(x, y) : \log_{e} y = x \log_{e} (\frac{2}{5}), x \in S, y \in R}$

Then, the sum of all the elements in the range of R is equal to: [2025]

Q 43 :

The number of relation on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is __________. [2025]

0%

Q 44 :

Let A = {1, 2, 3}. The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is __________. [2025]

0%

Q 46 :

Let $R$ be a relation defined on the set ${1, 2, 3, 4} \times {1, 2, 3, 4}$ by $R = {((a, b), (c, d)) : 2 a + 3 b = 3 c + 4 d} .$

Then the number of elements in $R$ is [2026]

Q 47 :

Let the relation R on the set $M = {1, 2, 3, \dots, 16}$ be given by $R = {(x, y) : 4 y = 5 x - 3, x, y \in M} .$

Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to [2026]

Q 49 :

Let A = {2,3,5,7,9} Let R be the relation on A defined by xRy if and only if $2 x \leq 3 y$ . Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to : [2026]

Q 50 :

The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in ℤ}$ is [2026]

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