Let the range of the function f(x)=1/2+sin 3 x+cos 3 x, x in IR be [a, b] . If α and β are respectively the A.M. and the G.M. of a and b , then α/β is equal to :

Mathematics · Functions

JEE Main 2024 — 9 April, Shift 2 — Question 9

Let the range of the function $f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \operatorname{IR}$ be $[a, b]$ . If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$ , then $\frac{\alpha}{\beta}$ is equal to :

Option A:
$\sqrt{2}$
Correct
Option B:
$2$
Option C:
$\sqrt{\pi}$
Option D:
$\pi$

Answer: A

Step-by-step solution

$f(x) =\frac{1}{2+\sin 3 x+\cos 3 x}$

$\left[\frac{1}{2+\sqrt{2}}, \frac{1}{2-\sqrt{2}}\right]$

$\frac{\alpha}{\beta}=\frac{a+b}{2 \sqrt{a b}}=\frac{1}{2}\left(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}\right)$

$=\frac{1}{2}\left(\sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}+\sqrt{\frac{2+\sqrt{2}}{2-\sqrt{2}}}\right)$

$=\frac{(2-\sqrt{2})+(2+\sqrt{2})}{2 \times \sqrt{2}}=\sqrt{2}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 9 April, Shift 2
Subject: Mathematics
Chapter: Functions
Topic: Domain & range of functions

Let A=\(x, y): 2 x+3 y=23, x, y in N\ and B=\x:(x, y) in A\ . Then the number of one-one functions from A to B is equal to

Answer: A

Step-by-step solution

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