If the roots of the quadratic equation x^2 + px + q = 0 are 30^ and 15^ , respectively, then the value of 2 + q - p is: - Mathematics MCQ AIEEE 2006 2026 | ExamTest.live

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If the roots of the quadratic equation x^{2} + p x + q = 0 are tan 3 0^{\circ} and tan 1 5^{\circ}, respectively, then the value of 2 + q - p is:

Solution & Step-by-step Explanation

For the equation x^{2} + p x + q = 0 :Sum of roots: tan 3 0^{\circ} + tan 1 5^{\circ} = - p Product of roots: tan 3 0^{\circ} \cdot tan 1 5^{\circ} = q We know that: tan (3 0^{\circ} + 1 5^{\circ}) = tan 4 5^{\circ} = 1 Using the identity tan (A + B) = \frac{t a n A + t a n B}{1 - t a n A t a n B} : 1 = \frac{tan 3 0 ^{\circ} + tan 1 5 ^{\circ}}{1 - tan 3 0 ^{\circ} tan 1 5 ^{\circ}} Substituting p and q : 1 = \frac{- p}{1 - q} ⟹ 1 - q = - p ⟹ q - p = 1 Therefore, 2 + q - p = 2 + 1 = 3 .

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If the roots of the quadratic equation x^{2} + p x + q = 0 are tan 3 0^{\circ} and tan 1 5^{\circ}, respectively, then the value of 2 + q - p is:

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