Multiple choice

Directions: You are given two quantities, Quantity I and Quantity II. Compare the two quantities and choose the correct option accordingly. One of the roots of the equation 78y2 - ay + 78 = 0 is 13/6, and a is a constant. Quantity I: a Quantity II: 169

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I = Quantity II

  4. No relation
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A Correct answer
Explanation

Substitute y = 13/6 into 78y^2 - ay + 78 = 0: 78*(169/36) - a*(13/6) + 78 = 0. Divide by 6: 13*(169/6) - a*(13/6) + 78 = 0. Multiply by 6/13: 169 - a + 36 = 0 => a = 205. Quantity I (205) > Quantity II (169).

AI explanation

Substitute the given root y = 13/6 into the equation 78y^2 - ay + 78 = 0 to find the value of the constant a. This gives the equation 78(13/6)^2 - a(13/6) + 78 = 0, which simplifies to 2197/3 - 13a/6 + 78 = 0. Multiplying the entire equation by 6 clears the fractions, resulting in 4394 - 13a + 468 = 0, or 13a = 4862. Dividing by 13 gives a = 374, and since 374 is greater than 169, Quantity I is greater than Quantity II.

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