LA

Show that $f : R \to R$ defined by $f(x) = 2x + \sin x$ for $x \in R$ is a bijection. [Competency]

VSA

A function $f:A\rightarrow B$ defined as $f(x)=2x$ is both one-one and onto. If $A=\{1,2,3,4\}$, then find the set $B$. OR Evaluate : $\sin^{-1}(\sin\frac{3\pi}{4})+\cos^{-1}(\cos\frac{3\pi}{4})+\tan^{-1}(1)$

LA

A relation $R$ is defined on a set of real numbers $\mathbb{R}$ as:$$R = \{(x, y) : x \cdot y \text{ is an irrational number}\}$$Check whether $R$ is reflexive, symmetric, and transitive or not.

VSA

Check whether the mapping $f : N \to N$ given by $f(n) = 1 + n^2, n \in N$ when $N$ is the set of natural numbers, is one-one or onto. [Competency]

LA

34. (a) 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.

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