Question

The equation $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3=k{\pi }^3$ has no solution for

• A. $k>\frac{1}{32}$
• B. $k=\frac{1}{32}$
• C. $k<\frac{1}{32}$
• D. $k=1$

Answer 1

Answer: (C)

$k<\frac{1}{32}$

$(si{n}^{-1}x+co{s}^{-1}x)$
$\left(\right(si{n}^{-1}x{)}^2+(co{s}^{-1}x{)}^2-si{n}^{-1}xco{s}^{-1}x)=k{\pi }^3$
or $(si{n}^{-1}x+co{s}^{-1}x{)}^2-3si{n}^{-1}xco{s}^{-1}x$
$=k{\pi }^3.\frac{2}{\pi }=2k{\pi }^3$
or $\frac{{\pi }^2}{4}-3si{n}^{-1}x(\frac{\pi }{2}-si{n}^{-1}x)=2k{\pi }^2$
or $3(si{n}^{-1}x{)}^2-\frac{3\pi }{2}si{n}^{-1}x+{\pi }^2(\frac{1}{4}-2k)=0$
Above is a quadratic equation and will have solution if
$\Delta \ge 0$ or $B^2-4AC\ge 0$
or $\frac{9{\pi }^2}{4}-\mathrm{4.3.}{\pi }^2(\frac{1}{4}-2k)\ge 0$
or $3-16(\frac{1}{4}-2k)\le 0$ or $-1+32k\ge 0$
or $k\ge \frac{1}{32}$, In this case it will have a solution.
Therefore if $k<\frac{1}{32}$ it will have no solution.