Multiple choice

Two positive roots of the equation y² - Sy + 45 = 0 are m and n, with m - n = 4. Quantity I: Value of 2S Quantity II: Value of n² + m + 1

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I ≤ Quantity II

  5. Quantity I = Quantity II or no relation
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B Correct answer
Explanation

From the quadratic equation y^2 - Sy + 45 = 0, the roots m and n satisfy m+n = S and mn = 45. Given m-n = 4, we solve the system: (m+n)^2 - (m-n)^2 = 4mn, so S^2 - 16 = 180, S^2 = 196, S = 14. Then m+n=14 and m-n=4 gives m=9, n=5. Quantity I = 2S = 28. Quantity II = n^2 + m + 1 = 25 + 9 + 1 = 35. Since 28 < 35, Quantity I < Quantity II.

AI explanation

Using the sum and product of roots gives m + n = S and mn = 45. Solving the system with m - n = 4 yields the roots 9 and 5, so S = 14. Quantity I is 2S, which is 28, while Quantity II is n^2 + m + 1, which equals 25 + 9 + 1 = 35. Therefore, Quantity I is less than Quantity II.

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