If tan ⁡ A and tan ⁡ B are the roots of the quadratic equation 3 x 2 - 10 x - 25 = 0 , then the value of 3 sin 2 ⁡ A + B - 10 sin ⁡ A + B cos ⁡ A + B - 25 cos 2 ⁡ A + B is :

Step-by-step solution

Since, tan ⁡ A and tan ⁡ B are roots of the equation 3 x 2 - 10 x - 25 = 0 . So, tan ⁡ A + tan ⁡ B = 10 3 and tan ⁡ A . tan ⁡ B = - 25 3 ∴ tan ⁡ A + B = tan ⁡ A + tan ⁡ B 1 - tan ⁡ A . tan ⁡ B = 10 3 1 + 25 3 = 10 28 = 5 14 So, sin A + B = ± 5 221 &   cos A + B = ± 14 221 (Note that either both are negative or both positive) ∴ 3 sin 2 ⁡ A + B - 10 sin ⁡ A + B . cos ⁡ A + B - 25 cos 2 ⁡ A + B = 3