12(√3 - √2)
12(√2π)
12(√3π)
12(√3 + 1)
The given quadratic equation is
18x² - πx + π²/2 = 0
⇒ x = (π ± √(π² - 4(18)(π²/2)))/(2(18))
⇒ x = (π ± √(π² - 36π²))/(36)
⇒ x = (π ± √(-35π²))/(36)
⇒ x = (π ± i3√35π)/36
The roots are complex, and there is an error in the question.
Let's assume the quadratic equation is 18x² - πx + π²/2 = 0.
Then, using the quadratic formula,
x = (π ± √(π² - 4 * 18 * π²/2))/(2 * 18)
x = (π ± √(-35π²))/36
The roots are complex, so there's likely a mistake in the problem statement. Let's assume the quadratic equation was intended to be 18x² - 6πx + π² = 0 instead.
18x² - 6πx + π² = 0
(6x - π)² = 0
x = π/6
This gives only one root. Let's assume the equation is 18x² - 9πx + π² = 0
(6x - π)(3x - π) = 0
x = π/6, π/3
Let α = π/6 and β = π/3
Now, y = g(f(x)) = cos(f(x)²) = cos(x)
Area under the curve = ∫(π/6)^(π/3) cos(x) dx = sin(x)|(π/6)^(π/3) = sin(π/3) - sin(π/6) = √3/2 - 1/2 = (√3 - 1)/2
This doesn't match any of the options. Let's re-examine the equation.
Let's assume the quadratic equation was 18x² - 6πx + π² = 0. Then x = π/6 (repeated root).
If we take α = π/6 and β = π/3 as in the previous incorrect solution, then
Area = ∫(π/6)^(π/3) cos(x) dx = sin(x)|(π/6)^(π/3) = (√3 - 1)/2
This still does not match the given options.
There appears to be an error in either the question or the provided solution.