SA

Let $R$ be a relation defined over $N$, where $N$ is set of natural numbers, defined as "$m R n$ if and only if $m$ is a multiple of $n$, where $m, n \in N$." Find whether $R$ is reflexive, symmetric and transitive or not. [Delhi 2025]

LA

Show that a function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined as $f(x)=\frac{5x-3}{4}$ is both one-one and onto.

LA

If $N$ denotes the set of all natural numbers and $R$ is the relation on $N \times N$ defined by $(a, b) R (c, d)$ if $ad(b + c) = bc(a + d)$. Show that $R$ is an equivalence relation. [Delhi 2015, All India 2023]

VSA

Let $X = \{-1, 0, 1\}, Y = \{0, 2\}$ and a function $f: X \to Y$ defined by $y = 2x^4$, then check whether $f$ is one-one or onto.

SA

$f(x) = \frac{2 \tan x}{1 - \tan^2 x}$. Find the range of $f(x)$ for $x \in R$. Show your steps. [Competency]

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