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Question #66374

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_t$ be the line passing through the points $A(t) = (1+3t, 2+4t, t^2-6t-2)$ and $B(t) = (1-t, 2, -t^2+2t-2)$ for $t \neq 0$ . Let $R(t)$ be the unique point of intersection of the line $L_t$ and the plane $\Pi: x + y + z = 1$ . Let $P_0(\alpha, \beta, \gamma) = \lim_{t \to 0} R(t)$ . If $L = \lim_{x \to 0} \frac{1}{x^2} \int_{0}^{x} \left( e^{\alpha t} - \cos(\beta t) + \gamma t \right) dt$ find the value of $12L$ .

Options

A.
4
B.
6
C.
8
D.
12

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

Limit Existence at Discontinuity|Line-Plane intersection logic|Ratio of Division (k:1)

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the resolution of a geometric singularity where a line becomes parallel to, and eventually lies within, a plane.

Students often incorrectly assume that because the limiting line $L_0$ lies entirely in $\Pi$ (yielding an indeterminate $0/0$ intersection state), the limit of the intersection point $R(t)$ diverges or does not exist.

The trap is failing to use the section formula analytically to resolve the $0/0$ form, which reveals that $R(t)$ converges to a specific, well-defined point on the limiting line.

The recursive integral limit further tests L'Hôpital's rule and prevents back-solving from the options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Limit Existence at Discontinuity
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free