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Question #40336

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be the set of all points $(x, y)$ in the $xy$ -plane satisfying the equations $x^2 + y^2 = 25$ and $xy = 12$ . Let $B$ be the range of the function $f(t) = \left[ \frac{12}{t^2 - 2t + 4} \right]$ for $t \in \mathbb{R}$ , where $[ \cdot ]$ denotes the greatest integer function. Find the number of injective mappings from $A$ to $B$ .

Options

A.
24
B.
120
C.
256
D.
625

Check Your Answer

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

Function Definition (The 'One-to-One' Constraint)|Geometry-based Combinations|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This problem tests the evaluation of discrete codomains generated by step functions and mapping constraints.

The primary trap lies in finding the range of the inner function $h(t)$ .

Students often correctly find the maximum value $4$ but incorrectly assume the minimum GIF value is $1$ , missing that $h(t) \to 0$ as $t \to \infty$ (which yields a GIF of $0$ ).

Missing $0$ leads to $|B|=4$ , triggering the trap answers $24$ (injective) or $256$ (total mappings).

The secondary trap is ignoring the 'injective' constraint entirely, leading to the total mappings trap of $5^4 = 625$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Function Definition (The 'One-to-One' Constraint)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free