Let f(x) = Ax^3 - Bx - tan x cdot text{sgn}(x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For $f(x)$ to be an even function,$f(x) = f(-x)$ must hold for all $x$ in the domain.Given $f(x) = Ax^3 - Bx - 	an x \cdot 	ext{sgn}(x)$.Since $

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) For $f(x)$ to be an even function,$f(x) = f(-x)$ must hold for all $x$ in the domain.Given $f(x) = Ax^3 - Bx - 	an x \cdot 	ext{sgn}(x)$.Since $	an(-x) = -	an x$ and $	ext{sgn}(-x) = -	ext{sgn}(x)$,we have $	an(-x) \cdot 	ext{sgn}(-x) = (-	an x) \cdot (-	ext{sgn}(x)) = 	an x \cdot 	ext{sgn}(x)$.Thus,$f(-x) = A(-x)^3 - B(-x) - 	an x \cdot 	ext{sgn}(x) = -Ax^3 + Bx - 	an x \cdot 	ext{sgn}(x)$.For $f(x) = f(-x)$,we require $Ax^3 - Bx - 	an x \cdot 	ext{sgn}(x) = -Ax^3 + Bx - 	an x \cdot 	ext{sgn}(x)$.This simplifies to $2Ax^3 - 2Bx = 0$ for all $x$,which implies $A = 0$ and $B = 0$.$A = \sin^2 \alpha - \sin \alpha + \frac{1}{4} = (\sin \alpha - \frac{1}{2})^2 = 0 \Rightarrow \sin \alpha = \frac{1}{2}$.$B = 	an^2 \alpha + \frac{2}{\sqrt{3}} 	an \alpha + \frac{1}{3} = (	an \alpha + \frac{1}{\sqrt{3}})^2 = 0$ $\Rightarrow 	an \alpha = -\frac{1}{\sqrt{3}}$.We need $\alpha \in [-\frac{3\pi}{2}, 2\pi]$ such that $\sin \alpha = \frac{1}{2}$ and $	an \alpha = -\frac{1}{\sqrt{3}}$.$\sin \alpha = \frac{1}{2} \Rightarrow \alpha = \frac{\pi}{6}, \frac{5\pi}{6}, -\frac{7\pi}{6}, -\frac{11\pi}{6}$.$	an \alpha = -\frac{1}{\sqrt{3}} \Rightarrow \alpha = \frac{5\pi}{6}, -\frac{\pi}{6}, -\frac{7\pi}{6}, \frac{11\pi}{6}$.The common values are $\alpha = \frac{5\pi}{6}$ and $\alpha = -\frac{7\pi}{6}$.Thus,there are $2$ such values.

Similar Questions

If a function $g(x)$ is defined in $[-1, 1]$ and two vertices of an equilateral triangle are $(0, 0)$ and $(x, g(x))$ and its area is $\frac{\sqrt{3}}{4}$,then $g(x)$ equals :-

Let $P(x)$ be a polynomial with real coefficients such that $P(\sin^2 x) = P(\cos^2 x)$ for all $x \in [0, \pi/2)$. Consider the following statements:
$I.$ $P(x)$ is an even function.
$II.$ $P(x)$ can be expressed as a polynomial in $(2x - 1)^2$.
$III.$ $P(x)$ is a polynomial of even degree.
Then,

For some $a, b, c \in N$,let $f(x)=ax-3$ and $g(x)=x^b+c$,$x \in R$. If $(fog)^{-1}(x)=\left(\frac{x-7}{2}\right)^{1/3}$,then $(fog)(ac) + (gof)(b)$ is equal to $..........$

Let $A = \{1, 2, 3, 5, 8, 9\}$. Then the number of possible functions $f : A \rightarrow A$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to $...............$.

Let $f(x) = \frac{\sqrt{x - 2\sqrt{x - 1}}}{\sqrt{x - 1} - 1}$. Then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module