Question

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

• A. 3
• B. 0
• C. 1
• D. 2

Hint:

Pay very close attention to the final expression you need to evaluate. A small difference, like \( q-p \) versus \( p-q \), can change the answer completely. Always derive the relationship between coefficients first using trigonometric identities before substituting.

Answer 1

The Correct Option is C

For a quadratic equation \( x^2 + Px + Q = 0 \), we know that the sum of the roots is \( -P \) and the product of the roots is \( Q \).
For the given equation \( x^2 + px + q = 0 \), the roots are \( \alpha = \tan 30^{\circ} \) and \( \beta = \tan 15^{\circ} \). Therefore, we have:
Sum of roots: \( \tan 30^{\circ} + \tan 15^{\circ} = -p \)
Product of roots: \( \tan 30^{\circ} \cdot \tan 15^{\circ} = q \)
We use the tangent addition formula, \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \). Let \( A=30^{\circ} \) and \( B=15^{\circ} \). This gives \( A+B = 45^{\circ} \).
\[ \tan(45^{\circ}) = \frac{\tan 30^{\circ} + \tan 15^{\circ}}{1 - \tan 30^{\circ} \tan 15^{\circ}} \]
We know that \( \tan(45^{\circ}) = 1 \). Now substitute the sum and product expressions in terms of p and q:
\[ 1 = \frac{-p}{1 - q} \] Cross-multiplying the equation gives: \[ 1 - q = -p \]
Rearranging the terms, we get a relationship between p and q: \[ p - q = -1 \]
The question asks for the value of the expression \( 2 + p - q \). We can substitute the value we found for the term \( (p - q) \):
\[ 2 + (p - q) = 2 + (-1) = 1 \]
The mathematically correct answer is 1, which corresponds to option (C). This also matches the provided OCR answer key. My previous reading of the expression was incorrect.