Question

Find the maximum and minimum value of 256^sinθ × 4^3cosθ.
1. - 512, 1/512
2. - 1024, 1/1024
3. 512, - (1/512)
4. 1024, - (1/1024)

Answer 1

Correct Answer - Option 4 : 1024, - (1/1024)

Formula used:

a sin θ + b cos θ

Maximum value = √(a² + b²)

Maximum value = - √(a² + b²)

Calculation:

256^sinθ × 4^3cosθ = 2^{8 sinθ} × 2^{6 cosθ}

⇒ 2^{(8 sinθ + 6 cosθ)}

⇒ Maximum value of (8 sinθ + 6 cosθ) = √(64 + 36)

⇒ √100 = 10

⇒ Maximum value of 2^{(8 sinθ + 6 cosθ)}

⇒ 2¹⁰ = 1024

⇒ Minimum value of (8 sinθ + 6 cosθ) = - √(64 + 36)

⇒ - √100 = - 10

⇒ Minimum value of 2^{(8 sinθ + 6 cosθ)}

⇒ 2^-10 = 1/1024

∴ Required answer are 1024, - (1/1024)