Multiple choice

Directions: The following question has two statements, (1) and (2). Answer the question using the following options: What is the value of k, if y 2 - 12y + k = 0? (1) One root of the equation is 3. (2) One root of the equation is 6 more than the other.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  4. EACH statement ALONE is sufficient.

  5. Statements (1) and (2) TOGETHER are NOT sufficient.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For y^2 - 12y + k = 0, sum of roots = 12. (1) One root is 3, so other is 9. k = 3*9 = 27. Sufficient. (2) Roots are r and r+6. 2r+6 = 12, so r=3. Roots are 3 and 9. k = 27. Sufficient.

AI explanation

Using statement 1, if one root of y^2 - 12y + k = 0 is 3, then substituting it gives 9 - 36 + k = 0, which solves to k = 27, making it sufficient. Using the sum of roots property from Vieta's formulas, statement 2 provides the relationship y1 = y2 + 6, and knowing the sum of the roots is 12 gives the system y1 + y2 = 12 and y1 - y2 = 6. This system solves to y1 = 9 and y2 = 3, and their product gives k = 27, showing each statement alone is sufficient to determine the value.

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