For a real number a, let I(a) = ∈t_-1^1 (3x^2 - ax + 1)\, dx. Which of the following statements is/are true?

The value of I(a) is independent of the value of a

There exists a ∈ (-∈fty, +∈fty) such that I(a) is a positive real number

The value of I(a) is independent of the value of a

The value of I(a) can vary with the value of a

There exists a ∈ (-∈fty, +∈fty) such that I(a) is a positive real number

There exists a ∈ (-∈fty, +∈fty) such that I(a) is a negative real number

GATE CSE 2026 Session 2 Q27: Definite integral I(a) of 3x² − ax + 1 over [−1, 1]

For a real number $a$ , let $I(a) = \displaystyle\int_{-1}^{1} (3x^2 - ax + 1)\, dx$ . Which of the following statements is/are true?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A, C (official GATE 2026 final answer key)

Concept used: The integral of an odd function over a symmetric interval is zero.

First step: The term $-ax$ is odd, so it integrates to 0 over $[-1, 1]$ ; integrate $3x^2 + 1$ .

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Source: official GATE 2026 Session 2 CS question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key