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GATE CSE 2025 Session 2 Q64: Probability of equal roots in a square-invariant quadratic

Question

A quadratic polynomial (x−α)(x−β)(x - \alpha)(x - \beta) over complex numbers is said to be square invariant if (x−α)(x−β)=(x−α2)(x−β2)(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2). Suppose from the set of all square invariant quadratic polynomials we choose one at random.

The probability that the roots of the chosen polynomial are equal is __________. (rounded off to one decimal place)

Try it: numerical answer: type the value

Numerical answer. No negative marking.
Show answer and explanation

Answer: 0.5 (official GATE 2025 final answer key)

Concept used: Equal polynomials have equal root multisets: {α,β}={α2,β2}\{\alpha, \beta\} = \{\alpha^2, \beta^2\}.

First step: Either α=α2\alpha = \alpha^2 and β=β2\beta = \beta^2 (roots in {0,1}\{0, 1\}), or α=β2\alpha = \beta^2 and β=α2\beta = \alpha^2 (so α4=α\alpha^4 = \alpha); list the distinct polynomials.

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Source: official GATE 2025 Session 2 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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