Multiple choice

If the roots of the quadratic equation mx2 + nx + r = 0 are sinθ and cosθ, which of the following is the correct relation of r in terms of m and n?

  1. (m2 - n2)/2

  2. (n2 - m2)/2

  3. (n2 - m2)/2m

  4. (m2 - n2)/2n
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C Correct answer
Explanation

Roots are sin(theta) and cos(theta). Sum of roots = sin(theta) + cos(theta) = -n/m. Product of roots = sin(theta) * cos(theta) = r/m. Square the sum: sin^2 + cos^2 + 2sin*cos = n^2/m^2. 1 + 2(r/m) = n^2/m^2. 2r/m = n^2/m^2 - 1 = (n^2 - m^2)/m^2. r = (n^2 - m^2) / 2m.

AI explanation

Using the relationships between roots and coefficients for the equation mx^2 + nx + r = 0, the sum of the roots is sin + cos = -n/m and the product is sin cos = r/m. Squaring the sum gives (sin + cos)^2 = sin^2 + cos^2 + 2sin cos, which becomes n^2/m^2 = 1 + 2r/m using the identity sin^2 + cos^2 = 1. Solving this for r yields r = (n^2 - m^2)/2m.

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