Let the range of the function be . Then the distance of the point from the line 3x + 4y + 12 = 0 is: [2025]

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Solution

Q.
Let the range of the function $f (x) = 6 + 16 \cos x \cdot \cos (\frac{π}{3} - x) \cdot \cos (\frac{π}{3} + x) \cdot \sin 3 x \cdot \cos 6 x, x \in R$ be $[α, β]$ . Then the distance of the point $(α, β)$ from the line 3x + 4y + 12 = 0 is: [2025]

1 8

2 11

3 9

4 10

Ans.
(2)

$f (x) = 6 + 16 (\frac{1}{4} \cos 3 x) \sin 3 x \cdot \cos 6 x$

$[∵ \cos θ \cos (\frac{π}{3} - θ) \cos (\frac{π}{3} + θ) = \frac{1}{4} \cos 3 θ]$

$= 6 + 4 \cos 3 x \sin 3 x \cos 6 x = 6 + \sin 12 x$

$∵$ Range of sin x = [–1, 1]

$∴$ Range of f(x) is [5, 7]

$\Rightarrow (α, β) \equiv (5, 7)$

$∴$ Distance of point (5, 7) from the line 3x + 4y + 12 = 0

$=$ $| \frac{15 + 28 + 12}{5} |$ $= 11$

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