First slide

Functions (XII)

Question

If $g : [- 2, 2] \to R, where f (x) = x^{3} + \tan x + [\frac{x^{2} + 1}{P}]$ is an odd function, then the value of parametric P, where [.] denotes the greatest integer function, is

Moderate

A

$- 5 < P < 5$
B

$P < 5$
C

$P > 5$
D

none of these

Solution

$g (x) = x^{3} + \tan x + [\frac{x^{2} + 1}{P}]$
or $g (- x) = x^{3} + \tan (- x) + [\frac{{(- x)}^{2} + 1}{P}]$
$= - x^{3} - \tan x + [\frac{x^{2} + 1}{P}]$
Since g(x) is odd,
$g (x) + g (- x) = 0$
$∴ (x^{3} + \tan x + [\frac{x^{2} + 1}{P}]) + (\begin{array}{l} - x^{3} - \tan x + [\frac{x^{2} + 1}{P}] \end{array}) = 0$
or $2 [\frac{(x^{2} + 1)}{P}] = 0 or 0 \leq \frac{x^{2} + 1}{P} < 1$
Now, $x \in [- 2, 2]$
$\begin{array}{l} ∴ x^{2} + 1 \in [1, 5] \\ ∴ P > 5 \end{array}$

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