For the least possible value of n in Z,the so | vedclass

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

Question

(A) Let $a = \cos^{-1} x$ and $b^2 = (\sin^{-1} y)^2$. The given equations are $a + b^2 = \frac{n \pi^2}{4}$ and $a \cdot b^2 = \frac{\pi^4}{16}$.Thes

Vedclass · Accepted Answer

$(\cos(\frac{\pi^2}{4}), \pm 1)$

Vedclass · Answer

(A) Let $a = \cos^{-1} x$ and $b^2 = (\sin^{-1} y)^2$. The given equations are $a + b^2 = \frac{n \pi^2}{4}$ and $a \cdot b^2 = \frac{\pi^4}{16}$.These are roots of the quadratic equation $t^2 - (\frac{n \pi^2}{4})t + \frac{\pi^4}{16} = 0$.For real roots,the discriminant $D \ge 0$,so $(\frac{n \pi^2}{4})^2 - 4(\frac{\pi^4}{16}) \ge 0$.$\frac{n^2 \pi^4}{16} - \frac{\pi^4}{4} \ge 0 \Rightarrow n^2 \ge 4 \Rightarrow n \ge 2$ (since $n \in Z$ and $n > 0$ for real $x, y$).For the least value $n = 2$,the quadratic becomes $t^2 - \frac{2 \pi^2}{4}t + \frac{\pi^4}{16} = 0$,which is $(t - \frac{\pi^2}{4})^2 = 0$.Thus,$a = \frac{\pi^2}{4}$ and $b^2 = \frac{\pi^2}{4}$.Since $a = \cos^{-1} x = \frac{\pi^2}{4}$,we have $x = \cos(\frac{\pi^2}{4})$.Since $b^2 = (\sin^{-1} y)^2 = \frac{\pi^2}{4}$,we have $\sin^{-1} y = \pm \frac{\pi}{2}$,so $y = \sin(\pm \frac{\pi}{2}) = \pm 1$.The solution is $(\cos(\frac{\pi^2}{4}), \pm 1)$.

For the least possible value of $n \in Z$,the solution $(x, y)$ of the equations $\cos ^{-1} x + (\sin ^{-1} y)^2 = \frac{n \pi^2}{4}$ and $(\cos ^{-1} x)(\sin ^{-1} y)^2 = \frac{\pi^4}{16}$ is

Similar Questions

The value of $x$,where $x>0$ and $\tan \left(\sec ^{-1}\left(\frac{1}{x}\right)\right)=\sin \left(\tan ^{-1} 2\right)$ is

The range of the real valued function $f(x) = \operatorname{Cos}^{-1}(-x) + \operatorname{Sin}^{-1}(-x) + \operatorname{Cosec}^{-1}(x)$ is

If $x = \sin \left( 2 \tan^{-1} 2 \right)$ and $y = \sin \left( \frac{1}{2} \tan^{-1} \frac{4}{3} \right)$,then -

If $\sum_{k=1}^n \tan^{-1} \left( \frac{1}{k^2+k+1} \right) = \tan^{-1} ( \theta )$,then $\theta =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module