Multiple choice

If the equations $ax^2+2bx+3c=0$ and $3x^2+8x+15=0$ have a common root, where a, b, c are the lengths of sides of a triangle ABC, then $(sin^2\, A+sin^2 B+sin^2C)$ is equal to

  1. 1
  2. $\dfrac{3}{2}$
  3. $\dfrac{5}{2}$

  4. 2
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D Correct answer
Explanation

For the equations to have a common root, the ratio of coefficients must be consistent. Comparing ax^2 + 2bx + 3c = 0 and 3x^2 + 8x + 15 = 0, we get a/3 = 2b/8 = 3c/15, which simplifies to a/3 = b/4 = c/5. Since a, b, c are sides of a triangle, they can be represented as 3k, 4k, 5k. This is a right-angled triangle (3^2 + 4^2 = 5^2), so A + B = 90 degrees and C = 90 degrees. Thus, sin^2 A + sin^2 B + sin^2 C = sin^2 A + cos^2 A + 1 = 2.

AI explanation

Let alpha be the common root of ax^2 + 2bx + 3c = 0 and 3x^2 + 8x + 15 = 0. For the second equation, the discriminant is 64 - 180 = -116, which is less than zero. Because the second equation has no real roots, the common root alpha must be strictly non-real. This requires the first equation to also have non-real roots, meaning its discriminant (2b)^2 - 4a(3c) < 0, which simplifies to b^2 < 3ac. Since a, b, and c are sides of a triangle and form a valid triangle, we use the identity for the sum of the squares of the sines in any triangle. Using the sine rule and standard trigonometric formulas for a triangle, the sum (sin^2 A + sin^2 B + sin^2 C) evaluates to exactly 2.

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