Multiple choice

lf the equations $ax^{2}+2bx+3c=0$ and $3x^{2}+8x+15=0$ have a common root, where $a,\ b,\ c$ are the length of the sides of a $\Delta \mathrm{A}BC$, then $\sin^{2}\mathrm{A}+\sin^{2}B+\sin^{2}C$ is equal to

  1. $1$
  2. $\displaystyle \frac{3}{2}$
  3. $\sqrt{2}$
  4. $2$
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D Correct answer
Explanation

The equation 3x^2 + 8x + 15 = 0 has complex roots. For the equations to have a common root, the coefficients must be proportional. Comparing ax^2 + 2bx + 3c = 0 and 3x^2 + 8x + 15 = 0, we get a/3 = 2b/8 = 3c/15, so a/3 = b/4 = c/5. Since a, b, c are sides of a triangle, they can be 3, 4, 5. This is a right triangle with hypotenuse 5. Thus, sin^2 A + sin^2 B + sin^2 C = (3/5)^2 + (4/5)^2 + (5/5)^2 = 9/25 + 16/25 + 1 = 1 + 1 = 2.

AI explanation

By comparing the coefficients of the given equations ax^2 + 2bx + 3c = 0 and 3x^2 + 8x + 15 = 0, we get the ratios a/3 = 2b/8 = 3c/15. Simplifying this gives a/3 = b/4 = c/5, meaning the triangle with sides a, b, and c is a right-angled triangle where c is the hypotenuse. For a right-angled triangle, the sum of the squares of the sines of its angles is sin^2 A + sin^2 B + sin^2 C = 1 + 1 = 2.

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