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Question #41120

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \int_0^x \frac{t^2 - at + 1}{(t^2+1)^2} \, dt$ Which of the following represents the set of all real values of $a$ for which $f$ is bijective?

Options

A.
$[-2, 2]$
B.
$(-2, 2)$
C.
$\mathbb{R}$
D.
$\emptyset$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Injective Mapping (One-One)|Newton-Leibniz Formula with variable limits|Bounded vs Unbounded Regions

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between injectivity and bijectivity for functions defined by integrals.

The primary trap is that students will correctly apply the Newton-Leibniz formula to find the condition for monotonicity ( $f'(x) \ge 0 \implies a \in [-2, 2]$ ) and immediately select the corresponding option, assuming strict monotonicity implies bijectivity.

However, they must also verify surjectivity.

Because the integrand is $O(1/t^2)$ , the integral converges as $x \to \pm \infty$ , meaning the function is bounded and fails to cover the codomain $\mathbb{R}$ .

Recognizing this asymptotic behavior is a classic JEE Advanced conceptual hurdle.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Injective Mapping (One-One)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free