If x, y, z are in A.P. and tan^{-1}x, tan^{-1 | vedclass

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

Question

(A) Given that $x, y, z$ are in $A.P.$,we have $2y = x + z$. Since $	an^{-1}x, 	an^{-1}y, 	an^{-1}z$ are in $A.P.$,we have $2	an^{-1}y = 	an^{-1}

Vedclass · Accepted Answer

$x = y = z$

Vedclass · Answer

(A) Given that $x, y, z$ are in $A.P.$,we have $2y = x + z$. Since $	an^{-1}x, 	an^{-1}y, 	an^{-1}z$ are in $A.P.$,we have $2	an^{-1}y = 	an^{-1}x + 	an^{-1}z$. Using the formula $	an^{-1}A + 	an^{-1}B = 	an^{-1}\left(\frac{A+B}{1-AB}ight)$,we get $	an^{-1}\left(\frac{2y}{1-y^2}ight) = 	an^{-1}\left(\frac{x+z}{1-xz}ight)$. This implies $\frac{2y}{1-y^2} = \frac{x+z}{1-xz}$. Substituting $x+z = 2y$,we get $\frac{2y}{1-y^2} = \frac{2y}{1-xz}$. This equation holds if $2y = 0$ or $1-y^2 = 1-xz$. If $2y = 0$,then $y=0$,which implies $x+z=0$ or $x=-z$. If $1-y^2 = 1-xz$,then $y^2 = xz$. Since $x, y, z$ are in $A.P.$ ($y^2 = xz$ implies $G.P.$),the only way for numbers to be in both $A.P.$ and $G.P.$ is if $x = y = z$.

If $x, y, z$ are in $A.P.$ and $\tan^{-1}x, \tan^{-1}y, \tan^{-1}z$ are also in $A.P.$,then

Similar Questions

Let $g(x) = f(x) + f(1-x)$ and $f''(x) > 0$ for $x \in (0, 1)$. If $g$ is decreasing in the interval $(0, \alpha)$ and increasing in the interval $(\alpha, 1)$,then $\tan^{-1}(2\alpha) + \tan^{-1}\left(\frac{1}{\alpha}\right) + \tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)$ is equal to:

The number of solutions of the equation $\sin \left[2 \cos^{-1} \left\{\cot \left(2 \tan^{-1} x\right)\right\}\right] = 0$ is

Considering only the principal values of the inverse trigonometric functions,the set $\{x \geq 0 : \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$

Which pair$(s)$ of function$(s)$ is/are equal? (where ${x}$ and $[x]$ denote the fractional part and integral part functions respectively.)

Let $f(x) = \cot \left( \sin^{-1} \sqrt{\frac{2}{3 + \cos 2x}} \right)$. Then,the value of $f'\left( \frac{2\pi}{3} \right)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module