Before You Begin

Question #41117

Difficulty: 8/10 | 🟢 Free

The Question

A curve is defined parametrically by $x(t) = t - a \sin t$ and $y(t) = t - b \sin t$ , where $t \in \mathbb{R}$ and $a, b \in \mathbb{R}$ . The set of all pairs $(a, b)$ for which $y$ can be expressed as a single-valued injective function of $x$ on the entire real line $\mathbb{R}$ is:

Options

A.
$\{(a, b) : |a| \le 1 \text{ and } |b| \le 1\} \cup \{(a, b) : a = b\}$
B.
$\{(a, b) : |a| \le 1 \text{ and } |b| \le 1\}$
C.
$\{(a, b) : |a| < 1 \text{ and } |b| < 1\} \cup \{(a, b) : a = b\}$
D.
$\{(a, b) : |a| \ge 1 \text{ and } |b| \ge 1\}$

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)|Rolle's Theorem Root Existence|Parametric Derivative Endpoint Trap|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

The problem tests the rigorous definition of a function and injectivity via parametric derivatives.

The primary trap is the 'Parametric Derivative Endpoint Trap': students often assume that $f'(x)$ must be strictly positive and finite for strict monotonicity, incorrectly excluding the boundary cases $|a|=1$ and $|b|=1$ where the derivative becomes zero or undefined at discrete points.

Furthermore, students frequently overlook the case where $x(t)$ is non-monotonic ( $|a| > 1$ ) but $a=b$ , which perfectly collapses the parametric curve into the identity line $y=x$ , a trivially injective function.

This dual-layered trap heavily penalizes blind application of the first derivative test without analyzing discrete singularities and the fundamental definition of a mapping.

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