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Question #41900

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = (-1, 1)$ . A binary operation $*$ is defined on $S$ by $x * y = f^{-1}(f(x) + f(y))$ , where $f(x) = \int_0^x \frac{a - |t|}{1 - t^2} dt$ and $a$ is a real parameter. The set of all values of $a$ for which $*$ is a well-defined binary operation on $S$ is:

Options

A.
$(-\infty, 0] \cup (1, \infty)$
B.
$(-\infty, 0] \cup [1, \infty)$
C.
$(-\infty, -1] \cup [1, \infty)$
D.
$(1, \infty)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Invertibility in Binary Operations|Newton-Leibniz Formula with variable limits|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

The problem defines a binary operation via an integral function, requiring students to deduce that closure implies the function's range must be $\mathbb{R}$ .

Applying the Newton-Leibniz formula gives the derivative, where the modulus function forces a piecewise monotonicity analysis, yielding $a \le 0$ or $a \ge 1$ .

The 'lethal collision' occurs when students assume strict monotonicity is sufficient for invertibility on $\mathbb{R}$ , failing to check the improper integral at the boundary $a=1$ .

At $a=1$ , the singularity cancels, restricting the range to $(-\ln 2, \ln 2)$ and destroying the binary operation's closure.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Invertibility in Binary Operations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free