GATE CS 2025 Set 2 — Question 64
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Question
A quadratic polynomial over complex numbers is said to be square invariant if . 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)
Correct answer
0.5 to 0.5
Solution
A quadratic polynomial is square invariant if the set of its roots is identical to the set of the squares of its roots . This implies that the squaring operation must be a permutation of the set of roots.There are two possible cases for this permutation:
The possible distinct sets of roots are , , and .
Substituting into the first equation: .
The solutions for are , where is a cube root of unity.
The polynomials with equal roots are and .
Number of favorable outcomes .Probability .
1.Case 1: Identity Permutation (Each root is its own square)
The possible distinct sets of roots are , , and .
2.Case 2: Transposition Permutation (Roots are swapped by squaring)
and Substituting into the first equation: .
The solutions for are , where is a cube root of unity.
- If , then (already counted in Case 1).
- If , then (already counted in Case 1).
- If , then . The set of roots is .
- If , then (same set as above).
- (roots: )
- (roots: )
- (roots: )
- (roots: )
The polynomials with equal roots are and .
Number of favorable outcomes .Probability .
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