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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NTA Abhyas 2020 · Mathematics · Inverse Trigonometric Functions

If x, y, z are in arithmetic progression and ^ -1 x, ^ -1 y and ^...

If x,y,zx, y, z are in arithmetic progression and tan⁡−1x,tan⁡−1y\tan^{-1} x, \tan^{-1} y and tan⁡−1z\tan^{-1} z are also in arithmetic progression, then

Options

  1. Ax=y=zx = y = z
  2. Bx=y=−zx = y = - z
  3. Cx=1,y=2,z=3x = 1, y = 2, z = 3
  4. Dx=2,y=4,z=6x = 2, y = 4, z = 6

Correct answer

A. x=y=zx = y = z

Step-by-step solution

Since, x,y,zx, y, z are in AP

∴y=x+z2∴ y = \dfrac{x + z}{2}....(i)

And tan⁡−1x,tan⁡−1y\tan^{-1} x, \tan^{-1} y and tan⁡−1z\tan^{-1} z are also in AP.

∴2tan⁡−1y=tan⁡−1x+tan⁡−1z∴ 2\tan^{-1} y = \tan^{-1} x + \tan^{-1} z

⇒tan⁡−1(2y1−y2)=tan⁡−1⇒ \tan^{-1} \left(\dfrac{2y}{1 - y^{2}}\right) = \tan^{-1} \left(\dfrac{x + z}{1 - xz}\right)<br/><br/>br /><br />⇒ \dfrac{2y}{1 - y^{2}} = \dfrac{2y}{1 - xz} [from equation. (i)]

⇒y2=xz⇒ y^{2} = xz

⇒x,y,z⇒ x, y, z are in GP.

∴ x = y = z
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