For any two real numbers a and b, we define a R b if and only if sin²(a) + cos²(b) = 1. The relation R is

• reflexive but not symmetric

• symmetric but not transitive

• transitive but not reflexive

• an equivalence relation

The total number of injections (one-one into mappings) from {a₁, a₂, a₃, a₄} to {b₁, b₂, b₃, b₄, b₅, b_6, b₇} is

• 400

• 420

• 800

• 840

183.

Let IR be the set of real numbers and f : IR ➔ IR be such that for all x, y ∈ IR, . Prove that f is a constant function.

The even function of the following is

• $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{a}}^{\mathrm{x}} + {\mathrm{a}}^{- \mathrm{x}}}{{\mathrm{a}}^{\mathrm{x}} - {\mathrm{a}}^{- \mathrm{x}}}$

• $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{a}}^{\mathrm{x}} + 1}{{\mathrm{a}}^{\mathrm{x}} - 1}$

• $\mathrm{f}\left(\mathrm{x}\right) = \mathrm{x} . \frac{{\mathrm{a}}^{\mathrm{x}} - 1}{{\mathrm{a}}^{\mathrm{x}} + 1}$

• $\mathrm{f}\left(\mathrm{x}\right) = {\log }_2\left(\mathrm{x} + \sqrt{{\mathrm{x}}^2 + 1}\right)$

If f(x + 2y, x - 2y) = xy, then f(x, y) is equal to

• $\frac{1}{4}\mathrm{xy}$

• $\frac{1}{4}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)$

• $\frac{1}{8}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)$

• $\frac{1}{2}\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)$

Let R be the set of real numbers and the mapping f : R $\to$ R and g : R $\to$ R be defined by f(x) = 5 - x² and g(x) =3$\text{λ}$ - 4, then the value of (fog) (- 1) is

• - 44

• - 54

• - 32

• - 64

A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6} are two sets, and function f : Aa B is defined by f(x) = x + 2 $\forall$ x $\in$ A, then the function!

• bijective

• onto

• one-one

• many-one

The function f(x) = $\sec \left[\log \left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^{{}^2}}\right)\right]$

• odd

• even

• neither odd nor even

• constant

The domain of the function f(x) = $\sqrt{{\cos }^{-1}\left(\frac{1 - \left|\mathrm{x}\right|}{2}\right)}$ is

• (- 3, 3)

• [- 3, 3]

• $\left(- \infty , - 3\right) \cup \left(3, \infty \right)$

• $(- \infty , - 3] \cup [3, \infty )$

A mapping from N to N is defined as follows f : N $\to$ N f(n) = (n + 5)², n $\in$ N

(N is the set of natural numbers). Then,

• f is not one to one

• f is onto

• f is both one to one and onto

• f is one to one but not onto