This doesn't really feel like a GMAT question to me -- it isn't unfair, per se, but the use of 'magnitude' is suspect: this might mean absolute value, in which the case the GMAT would say so, or it might (and generally does) mean the 'relative size' of a complex number (the 'distance' between that number and the origin of the complex plane), in which case the GMAT wouldn't use it at all.

That said, to solve this we can use the formulas for the sum of the roots and the product of the roots (also somewhat unusual for the GMAT) of a quadratic. If our quadratic is of the form ax^2 + bx + c = 0, the sum of the roots is (-b)/a and the product is (c/a). So our quadratic gives

Sum of roots = (-q/p)
Product of roots = (p/p) = 1

So if one of the roots is r, the other root must be 1/r, as r * (1/r) = 1. So the roots are reciprocal.

You could also solve this using the quadratic formula. The roots are (-q +/- √(q^2 - 4p^2))/(2p). The product of these roots is 1, so the roots are reciprocal.

kanika123 wrote:What can you say about the roots of px²+qx+p=0?

An alternate approach is to plug in quadratics that satisfy the answer choices.
When the correct answer choice is satisfied, the coefficient of x² will be equal to the last term.

A. One root is double the other.
(x+2)(x+1) = 0 --> roots are x=-2 and x=-1.
(x+2)(x+1) = x² + 3x + 2.
Since the coefficient of x² is not equal to the last term, eliminate A.

B. One root is the square of the other.
(x-2)(x-4) = 0 --> roots are x=2 and x=4.
(x-2)(x-4) = x² - 6x + 8.
Since the coefficient of x² is not equal to the last term, eliminate B.

C. One root is the reciprocal of the other.
(x+2)(x + 1/2) = 0 --> roots are x=-2 and x = -1/2.
(x+2)(x + 1/2) = x² + (1/2)x + 2x + 1.
Success!
The coefficient of x² is equal to the last term.

To confirm, test one more random case:
(3x-1)(x-3) --> roots are x = 1/3 and x = 3.
(3x-1)(x-3) = 3x² - 9x - x + 3.
Success again!
The coefficient of x² is equal to the last term.

The cases above illustrate that answer choice C must be true.

The correct answer is C.