Multiple choice

If the roots of the equation x2 + px + q = 0 are tan 19° and tan 26°, then which one of the following is correct?

  1. q - p = 1

  2. p - q = 1

  3. p + q = 2

  4. p + q = 3
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A Correct answer
Explanation

Sum of roots = tan 19 + tan 26 = -p. Product of roots = tan 19 * tan 26 = q. Using tan(A+B) = (tan A + tan B) / (1 - tan A * tan B), tan 45 = 1 = (-p) / (1 - q). 1 = -p / (1 - q) => 1 - q = -p => p - q = -1 => q - p = 1.

AI explanation

Using the sum and product of roots for x2 + px + q = 0, we have p = -(tan 19° + tan 26°) and q = tan 19° tan 26°. Evaluating the expression q - p gives tan 19° tan 26° + tan 19° + tan 26°. By the tangent addition formula for 45°, since 19° + 26° = 45°, we know tan(19° + 26°) = (tan 19° + tan 26°) / (1 - tan 19° tan 26°) = 1. This simplifies to tan 19° + tan 26° = 1 - tan 19° tan 26°, which rearranges to tan 19° tan 26° + tan 19° + tan 26° = 1. Therefore, q - p = 1.

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