If R is a relation defined by on, then which of the following is correct?

Question

If R is a relation defined by $a R b$ $i f f$ $a \leq b$ on $N \times N$ , then which of the following is correct?

Smart Achievers · Accepted Answer

(1)

Now

$\lim_{x \to 0} \frac{α (1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots) + β (1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \dots) + γ (x - \frac{x^{3}}{3!} + \dots)}{x^{3}}$ $[∵ \sin x \leq x \forall x \in R]$

Constant terms should be zero $\Rightarrow α + β = 0$

Coefficient of $x$ should be zero $\Rightarrow α - β + γ = 0$

Coefficient of $x^{2}$ should be zero $\lim_{x \to 0} \frac{x^{3} (\frac{α}{3!} - \frac{β}{3!} - \frac{γ}{3!}) + x^{4} (\frac{α}{3!} - \frac{β}{3!} - \frac{γ}{3!})}{x^{3}} = \frac{2}{3}$

$\Rightarrow \frac{α}{2} + \frac{β}{2} = 0,$ Now $\Rightarrow \frac{α}{6} - \frac{β}{6} - \frac{γ}{6} = \frac{2}{3}$

$\Rightarrow α = 1, β = - 1, γ = - 2$

Solution

Q.
If R is a relation defined by $a R b$ $i f f$ $a \leq b$ on $N \times N$ , then which of the following is correct

1 R is not a symmetric relation

2 R is a symmetric relation

3 R is not a reflexive relation

4 R is a equivalence relation

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