Correct Answer – Option 3 : 12 and 13

Trigonometry identity used:

Sin²θ + cos²θ = 1

Calculation:

12 sin²θ + 13 cos²θ

= 12 sin²θ + 12 cos²θ + cos²θ

= 12 (sin²θ + cos²θ) + cos²θ

= 12 + cos²θ

For minimum value,

Minimum value of cosθ = –1

But cos²θ  ≥ 0, when θ  = 90°

So, cos0° = 1,

Then, required minimum value

= 12 + 0

= 12.

For the maximum value,

Maximum value of cosθ = 1

And cos²θ =1

Then, required maximum value,

= 12 + 1 = 13

∴The minimum and maximum values of 12 sin²θ +13 cos²θ are 12 and 13.