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Question #40822

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_t$ be a family of conics given by the equation $\frac{x^2}{f(t)} + \frac{y^2}{g(t)} = 1$ , where $t \in (0, 1)$ , $f(t) = 2 + t \left[ \frac{3}{t} \right] + t^2 \sin\left(\frac{1}{t}\right)$ , and $g(t) = 8 - t \left\{ \frac{3}{t} \right\} + 4t^2 \sin\left(\frac{1}{t}\right)$ . Here $[ \cdot ]$ and $\{ \cdot \}$ denote the greatest integer and fractional part functions respectively. Let $D_t$ be the diameter of $C_t$ bisecting a system of parallel chords of slope $m = \frac{1}{t}$ . If $M(t)$ is the slope of $D_t$ , find the value of $\lim_{t \to 0^+} \frac{M(t) + 2t}{t}$ .

Options

A.
$\frac{2}{5}$
B.
-2
C.
1
D.
$\frac{18}{5}$

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

Diameter of Conics|Squeeze Theorem (Sandwich Theorem)|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of coordinate geometry (diameters of conics) and real analysis (limits involving jump discontinuities).

The primary trap lies in evaluating the limit of $t [3/t]$ as $t \to 0^+$ .

Students frequently make the error of assuming $\lim_{t \to 0} t [3/t] = 0$ by treating the GIF as a bounded quantity or incorrectly applying $0 \times \infty$ logic.

By forcing the conversion to the fractional part and rigorously applying the Squeeze Theorem, the problem ensures that only students who understand the piecewise bounding of discontinuous parameters will arrive at the correct conic parameters in the singular state.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Diameter of Conics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free