Let α = max _(x ∈ R) { 8²ˢⁱⁿ³ˣ.4⁴ᶜᵒˢ³ˣ} and β = min _(x ∈ R) { 8²ˢⁱⁿ³ˣ.4⁴ᶜᵒˢ³ˣ}. If 8x² + bx + c = 0 is a quadratic equation whose roots are α 1/5 and β 1/5 , then the value of c - b is equal to :

Solution

Maths sir · 4 steps

Write everything as a power of 2, and the problem becomes the range of 6 sin 3x + 8 cos 3x.

1. Common base. 8²ˢⁱⁿ³ˣ = 2⁶ˢⁱⁿ³ˣ and 4⁴ᶜᵒˢ³ˣ = 2⁸ᶜᵒˢ³ˣ. So the expression is 2⁶ˢⁱⁿ³ˣ⁺⁸ᶜᵒˢ³ˣ. Since 2ᵗ increases with t, its max and min come from the max and min of the exponent.

   -√(6²+8²) ≤ 6 sin 3x + 8 cos 3x ≤ √(6²+8²) ⇒ -10 ≤ 6 sin 3x + 8 cos 3x ≤ 10

2. Find the roots. α = 2¹⁰ and β = 2⁻¹⁰. So α^(1/5) = 2² = 4 and β^(1/5) = 2⁻² = 1/4.
3. Build the quadratic. Sum of roots = 4 + 1/4 = 17/4 and product = 1. With leading coefficient 8, the equation is 8x² - 34x + 8 = 0, so b = -34 and c = 8.
4. So. c - b = 8 - (-34) = 42, which is option (a).

The idea to keep. For a sin θ + b cos θ, the range is [-√(a²+b²), √(a²+b²)]. When it sits in an exponent, first bring everything to one base.

Answer from the NTA final answer key. Solution by PrepWiser.