1
JEE Main 2025 (Online) 2nd April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
Let $A=\{1,2,3, \ldots ., 100\}$ and $R$ be a relation on $A$ such that $R=\{(a, b): a=2 b+1\}$. Let $\left(a_1\right.$, $\left.a_2\right),\left(a_2, a_3\right),\left(a_3, a_4\right), \ldots .,\left(a_k, a_{k+1}\right)$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :
A
6
B
8
C
7
D
5
2
JEE Main 2025 (Online) 2nd April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let A be the set of all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is :

A
Symmetric and transitive but not reflective
B
Symmetric but neither reflective nor transitive
C
Transitive but neither reflexive nor symmetric
D
Reflexive but neither symmetric nor transitive
3
JEE Main 2025 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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

$$ \mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\} . $$

Then, the sum of all the elements in the range of $R$ is equal to :

A
$\frac{3}{2}$
B
$\frac{10}{9}$
C
$\frac{5}{2}$
D
$\frac{5}{3}$
4
JEE Main 2025 (Online) 29th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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

A

both reflexive and symmetric but not transitive

B

both reflexive and transitive but not symmetric

C

reflexive but neither symmetric not transitive

D

an equivalence relation

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