Let x=m/n ( m, n are co-prime natural numbers ) be a solution of the equation cos (2 sin -1 x )=1/9 and let α, β(α β) be the roots of the equation m x 2 -n x- m + n =0 . Then the point (α, β) lies on the line

Mathematics · Inverse Trigonometric Functions

JEE Main 2024 — 29 January, Shift 2 — Question 15

Let $x=\frac{m}{n}$ ( $m, n$ are co-prime natural numbers $)$ be a solution of the equation $\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}$ and let $\alpha, \beta(\alpha>\beta)$ be the roots of the equation $m x^{2}-n x-$ $\mathrm{m}+\mathrm{n}=0$ . Then the point $(\alpha, \beta)$ lies on the line

Option A:
$3 x+2 y=2$
Option B:
$5 x-8 y=-9$
Option C:
$3 x-2 y=-2$
Option D:
$5 x+8 y=9$
Correct

Answer: D

Step-by-step solution

Assume $\sin ^{-1} \mathrm{x}=\theta$

\begin{aligned}& \cos (2 \theta)=\frac{1}{9} & \sin \theta= \pm \frac{2}{3}\end{aligned}

as m and n are co-prime natural numbers,

\mathrm{x}=\frac{2}{3}

i.e. $m=2, n=3$

So, the quadratic equation becomes $2 \mathrm{x}^{2}-3 \mathrm{x}+1=$ 0

whose roots are $\alpha=1, \beta=\frac{1}{2}$

$\left(1, \frac{1}{2}\right)$ lies on $5 x+8 y=9$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 29 January, Shift 2
Subject: Mathematics
Chapter: Inverse Trigonometric Functions
Topic: Properties related to Inverse Trigonometric Functions