GATE 2025 CS2 Q.64: solution and answer

GATE 2025 · CS2Q.64 · 2 marks · NAT

A quadratic polynomial $(x - α) (x - β)$ over complex numbers is said to be square invariant if $(x - α) (x - β) = (x - α^{2}) (x - β^{2})$ . 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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Answer and solution

ANSWER

The probability that the chosen polynomial has equal roots is 0.5.

Official answer: 0.5Source: Official answer key, GATE 2025 CS2 (IIT Roorkee)

Two monic quadratics are equal exactly when their root multisets are equal, so square invariance means ${α, β} = {α^{2}, β^{2}}$ as multisets.
Matching 1: $α^{2} = α$ and $β^{2} = β$ . Then $α, β \in {0, 1}$ , giving the polynomials $x^{2}$ , $x (x - 1)$ and $(x - 1)^{2}$ .
Matching 2: $α^{2} = β$ and $β^{2} = α$ . Then $α^{4} = α$ , so $α (α^{3} - 1) = 0$ and $α \in {0, 1, ω, ω^{2}}$ , where $ω$ is a primitive cube root of unity.
For $α = 0$ or $1$ this gives $β = α$ , already counted. For $α = ω$ we get $β = ω^{2}$ , and $α = ω^{2}$ gives $β = ω$ . Both give the same polynomial $(x - ω) (x - ω^{2}) = x^{2} + x + 1$ . Check: the squares of its roots are $ω^{2}$ and $ω^{4} = ω$ , the same pair.
So there are exactly 4 distinct square invariant polynomials: $x^{2}$ , $x^{2} - x$ , $x^{2} - 2 x + 1$ and $x^{2} + x + 1$ .
Equal roots occur for $x^{2}$ (roots $0, 0$ ) and $(x - 1)^{2}$ (roots $1, 1$ ). That is 2 of the 4. The other two have distinct roots ( $0, 1$ and $ω, ω^{2}$ ).
Probability $= 2/4 = 0.5$ .

KEY IDEA

Square invariance means the root multiset is unchanged by squaring, so the roots are either each fixed by squaring (0 or 1) or swapped by squaring (the two primitive cube roots of unity). List the polynomials and count.

WATCH OUT FOR

Counting only the fixed-point case and forgetting that the two roots can be swapped by squaring ( $ω$ and $ω^{2}$ ). That gives 3 polynomials and the wrong probability $2/3$ .

From the GATE 2025 CS2 paper. Every question of this paper · Download the full paper · All 6 Counting questions, with notes

GATE 2025 CS2 Q.64: solution and answer

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