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Question #41145

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \mathbb{R} \setminus \{ (2k+1)\pi : k \in \mathbb{Z} \}$ . A binary operation $*$ on $S$ is defined as $x * y = 2 \tan^{-1} \left( \tan \frac{x}{2} + \tan \frac{y}{2} \right)$ Consider the following three statements: $S_1$ : The operation $*$ admits an identity element in $S$ . $S_2$ : The operation $*$ is associative on $S$ . $S_3$ : The function $f: S \to (-\pi, \pi)$ defined by $f(x) = x * 0$ is surjective but not injective. Which of the following is correct?

Options

A.
\text{Only } S_1 \text{ and } S_2 \text{ are true}
B.
\text{Only } S_2 \text{ and } S_3 \text{ are true}
C.
\text{Only } S_2 \text{ is true}
D.
\text{All statements are true}

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

Identity Element Uniqueness|Composite Injectivity/Surjectivity|Tangent Half-Angle Substitution|Inverse Function Periodicity

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous definition of an identity element alongside the properties of inverse trigonometric functions.

The primary trap lies in $S_1$ : students algebraically solve $\tan(x/2) + \tan(e/2) = \tan(x/2)$ to find $e=0$ , incorrectly concluding that $0$ is the identity element.

They fail to verify if $x * 0 = x$ holds for the entire domain $S$ .

Because the operation involves $2 \tan^{-1}(t)$ , its output is strictly bounded to $(-\pi, \pi)$ .

Consequently, elements outside this principal range (like $2\pi$ ) cannot be mapped back to themselves, meaning the algebraic identity fails globally due to the restricted range of the inverse tangent function. $S_3$ further reinforces this by exploring the periodicity and many-to-one nature of the composite function $f(x) = 2 \tan^{-1}(\tan(x/2))$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Identity Element Uniqueness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free