If α, β are the roots of the equation x² - 3x + A = 0; γ and δ are the roots of x² – 12x + B = 0, and the sequence α, β, γ, δ is known to be an increasing G.P., then

Let the terms of the increasing geometric progression be alpha, alpha r, alpha r squared, and alpha r cubed. For the first equation, the sum of the roots alpha plus beta equals three, so alpha plus alpha r equals three, meaning alpha times the quantity one plus r equals three. For the second equation, the sum gamma plus delta equals twelve, giving alpha r squared plus alpha r cubed equals twelve, which factors to alpha r squared times the quantity one plus r equals twelve. Dividing the second equation by the first gives r squared equals four, and since the progression is increasing, the common ratio r is two. Substituting r equals two into the first equation gives alpha equals one, making the sequence one, two, four, and eight. The product of the roots for the first equation is A, which equals one times two, giving two. The product of the roots for the second equation is B, which equals four times eight, giving thirty two. Because the calculated values are A equals two and B equals thirty two, the correct choice is none of these.

• Thanks
• Wrong explanation