If the roots of the quadratic equation \( \Large x^{2}+px+q=0 \) are \( \Large \tan 30 ^{\circ} and\ \tan 15 ^{\circ} \) respectively, then the value of\( \Large 2+q-p \) is

A) 3

B) 0

C) 1

D) 2

Correct answer:
A) 3

Description for Correct answer:
Since, \( \Large \tan 30 ^{\circ} and\ \tan 15 ^{\circ} \) are roots of equation

\( \Large x^{2}+px+q=0 \)

\( \Large \tan 30 ^{\circ} + \tan 15 ^{\circ} = -P \)

and \( \Large \tan 30 ^{\circ} \tan 15 ^{\circ} = -q \)

\( \Large \therefore 2+q-P=2+\tan 30 ^{\circ} + \tan 15 ^{\circ} + \left(\tan 30 ^{\circ} + \tan 15 ^{\circ} \right) \)

=\( \Large 2+\tan 30 ^{\circ} \tan 15 ^{\circ} +1- \tan 30 ^{\circ} \tan 15 ^{\circ} \)

\( \Large \left(\because \tan 45 ^{\circ} =\frac{\tan 30 ^{\circ} + \tan 15 ^{\circ} }{1- \tan 30 ^{\circ} \tan 15 ^{\circ} }\right)=2+q-P=3 \)


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