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Question #76384

Difficulty: 8/10 | 🟢 Free

The Question

A moving line passing through the origin has direction cosines $l(t), m(t), n(t)$ at time $t > 0$ . It is given that $l(t) = \frac{\cos t}{\sqrt{2}}$ , $m(t) = \frac{\sin t}{\sqrt{2} t}$ , and $n(t) > 0$ . If $\alpha = \lim_{t \to 0^+} n'(t)$ , evaluate $\lim_{x \to \alpha} \frac{\sin(\pi x^2)}{x^2 - \alpha^2}$

Options

A.
$-\frac{\pi}{2}$
B.
$\frac{\pi}{2}$
C.
$-\frac{\pi \sqrt{3}}{2}$
D.
$\frac{\pi \sqrt{3}}{2}$

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

Direction Cosines constraint|Rate of Change Dependency|Indeterminate Forms

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 dependency of rates of change under the fundamental 3D geometry constraint $l^2+m^2+n^2=1$ .

The primary trap is an Indeterminate Form ( $0/0$ ) that arises if one differentiates the constraint implicitly to get $n' = -(ll'+mm')/n$ .

At $t \to 0$ , both the numerator and denominator approach zero, rendering direct substitution useless.

Students must recognize that the rate of change of the direction cosines is not independent of the instantaneous position and must employ Taylor series expansions to resolve the singularity before evaluating the secondary limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Direction Cosines constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free