GATE 2025 CS (CS2) – Question 64
A quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$. 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)
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Correct answer: 0.5
Explanation
We need $\{\alpha^2,\beta^2\} = \{\alpha,\beta\}$. Either both roots are in $\{0,1\}$, or $\alpha^2=\beta$ and $\beta^2=\alpha$ (the cube roots of unity $\omega,\omega^2$). This gives 4 polynomials: $x^2$, $x(x-1)$, $(x-1)^2$ and $x^2+x+1$. Two of them have equal roots, so the probability is 2/4 = 0.5.