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.

1

2

3

4

(1)

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

$∴$ R is reflexive

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

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

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

$∴$ 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]

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

1

2

3

4

(2)

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]

0%

5

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]

0%

3

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

For reflexive : (1, 1), (2, 2), (2, 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.

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%

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Topic Question Set

Q 41 : Define a relation R on the interval [0,π2) by xRy if and only if sec2x–tan2y=1. Then R is : [2025]

Q 42 : Let S=N∪{0}. Define a relation R from S to R by: R={(x,y):logey=xloge(25), x∈S, y∈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%

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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]

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]