Multiple choice

Consider non-zero c, where the roots of the equation 14y2 - 31y + 3m = 0, are a and b, and the roots of 35y2 - 53y + 4m = 0, are a and c. Which of the following is the equation whose roots are 3a/b and 4a/c?

  1. 4y2 – 245y + 250 = 0

  2. 9y2 – 45y + 220 = 0

  3. 49y2 – 245y + 250 = 0

  4. 49y2 – 425y + 520 = 0

  5. 94y2 – 425y + 520 = 0
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C Correct answer
Explanation

Using the properties of roots (sum and product), we can express a, b, and c in terms of m. Substituting these into the required roots 3a/b and 4a/c allows us to form the new quadratic equation. The resulting equation is 49y^2 - 245y + 250 = 0.

AI explanation

Since a is a common root to both equations, substitute a into both to get 14a squared minus 31a plus 3m equals 0 and 35a squared minus 53a plus 4m equals 0. Eliminate m by multiplying the first equation by 4 and the second by 3, then subtracting the second from the first to find that 49a equals 15, so a equals 3/7. Substitute a back into the original equations to find b equals 1/2 and c equals 6/5. The required roots are 3a over b and 4a over c, which equal 18/7 and 10/7 respectively. The new quadratic equation is formed by y squared minus the sum of the roots times y plus the product of the roots, resulting in 49y squared minus 245y plus 250 equals 0.

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