The following planes have a common point other than the origin. Let b be the sum and c the product of all possible values of a. Which statements are correct?
$a x + y + 5 z = 0, 2 x + 2 y + a z = 0, 5 x + 6 y - z = 0.$

Question

The following planes have a common point other than the origin. Let b be the sum and c the product of all possible values of a. Which statements are correct?

$a x + y + 5 z = 0, 2 x + 2 y + a z = 0, 5 x + 6 y - z = 0.$

TestHub · Accepted Answer

A nonzero common solution requires determinant zero. The roots have opposite signs and one lies between 1 and 2. Rational binomial terms have even powers of the square root. The geometric series sum is negative. Hence A, B and C are correct. $det M = - 3 (2 a^{2} - a - 4) = 0, b = \frac{1}{2}, c = - 2. f (1) = - 3, f (2) = 2. (2^{b} + 3^{c})^{10} = (2 + \frac{1}{9})^{10} . Even powers: 0, 2, 4, 6, 8, 10, so six rational terms. \frac{c}{1 - b} = \frac{- 2}{1 - 1/2} = - 4 \neq = 4.$

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