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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NDA 2021 · Mathematics · Inverse Trigonometric Functions

Let ^ -1 x+ ^ -1 y+ ^ -1 z=(3 π)/(2) for 0 ≤ x, y, z ≤ 1 . What is...

Let sin⁡−1x+sin⁡−1y+sin⁡−1z=3π2\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2} for 0≤x,y,z≤10 \leq x, y, z \leq 1. What is the value of x1000+y1001+z1002x^{1000}+y^{1001}+z^{1002} ?

Options

  1. A0
  2. B1
  3. C3
  4. D6

Correct answer

C. 3

Step-by-step solution

sin⁡−1(x)+sin⁡−1(y)+sin⁡−1(z)=3π2=π2+π2+π2\sin ^{-1}(x)+\sin ^{-1}(y)+\sin ^{-1}(z)=\frac{3 \pi}{2}=\frac{\pi}{2}+\frac{\pi}{2}+\frac{\pi}{2}⇒sin⁡−1(x)=π2,sin⁡−1(y)=π2,sin⁡−1(z)=π2⇒x=y=z=1[ as sin⁡−1(1)=π2]So, x1000+y1001+z1002⇒1+1+1=3\begin{aligned} \Rightarrow & \sin ^{-1}(x)=\frac{\pi}{2}, \sin ^{-1}(y)=\frac{\pi}{2}, \sin ^{-1}(z)=\frac{\pi}{2} \\ \Rightarrow & x=y=z=1\left[\text{ as } \sin ^{-1}(1)=\frac{\pi}{2}\right] \\ & \text{So, } x^{1000}+y^{1001}+z^{1002} \\ & \Rightarrow 1+1+1=3\end{aligned}$$
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