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GATE 2026 CS (CS2) – Question 27

Engineering Mathematics · Calculus · 1 mark · Multiple select

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

  1. The value of $I(a)$ is independent of the value of $a$.
  2. The value of $I(a)$ can vary with the value of $a$.
  3. There exists $a\in(-\infty,+\infty)$ such that $I(a)$ is a positive real number.
  4. There exists $a\in(-\infty,+\infty)$ such that $I(a)$ is a negative real number.

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Show answer and explanation

Correct answer: (A) The value of $I(a)$ is independent of the value of $a$.; (C) There exists $a\in(-\infty,+\infty)$ such that $I(a)$ is a positive real number.

Explanation

Compute the integral directly: $$I(a)=\int_{-1}^{1}(3x^2-ax+1)\,dx=\int_{-1}^{1}3x^2\,dx-a\int_{-1}^{1}x\,dx+\int_{-1}^{1}1\,dx.$$ We get $$\int_{-1}^{1}3x^2\,dx=2, \qquad \int_{-1}^{1}x\,dx=0, \qquad \int_{-1}^{1}1\,dx=2.$$ Hence $I(a)=4$, independent of $a$. Since 4 is positive for every real $a$, statements (A) and (C) are true, while (B) and (D) are false.