Multiple choice

Let p and q be the roots of the equation ${x^2} - 2x + A = 0,$ and let r and s be the roots of the equation ${x^2} - 18x + B = 0.$ If p < q < r < s are in arithmetic progression. The value of (A+B) equals

  1. $80$
  2. $77$
  3. $75$
  4. $74$
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D Correct answer
Explanation

Roots p, q are for x^2 - 2x + A = 0 (p+q=2, pq=A). Roots r, s are for x^2 - 18x + B = 0 (r+s=18, rs=B). AP: p, q, r, s. Let common difference be d. q=p+d, r=p+2d, s=p+3d. (p+q)=2p+d=2. (r+s)=2p+5d=18. Subtracting: 4d=16, d=4. 2p+4=2, p=-1. Roots are -1, 3, 7, 11. A = pq = -3. B = rs = 77. A+B = 74.

AI explanation

Since the four roots form an arithmetic progression, let them be a minus 3d, a minus d, a plus d, and a plus 3d. The sum of the roots for the first equation is p plus q equals 2a, and using Vieta's formulas we have p plus q equals 2, so a equals 1. The sum of all four roots is p plus q plus r plus s equals 2 plus 18 equals 20, which also equals 4a, confirming a is 1. The product of the roots p and q is (1 minus 3d)(1 minus d) equals A, and the product of r and s is (1 plus d)(1 plus 3d) equals B. Adding these products yields A plus B equals (1 minus 3d)(1 minus d) plus (1 plus d)(1 plus 3d) equals 2 plus 6d^2 plus 6d^2 equals 74.

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