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Question #40743

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be two families of straight lines defined by: $L_1: (\sin \theta) x + (\cos \theta) y = \sin 2\theta$ $L_2: (\tan \theta) x + y = 2\sin \theta + |\cos 2\theta|$ where $\theta \in \left(0, \frac{\pi}{2}\right)$ . Let $d(\theta)$ be the perpendicular distance between $L_1$ and $L_2$ . Let $S$ be the number of points in $\left(0, \frac{\pi}{2}\right)$ where $d(\theta)$ is non-differentiable, $M$ be the number of local extrema of $d(\theta)$ in $\left(0, \frac{\pi}{2}\right)$ , and $L = \lim_{\theta \to \frac{\pi}{2}^-} d(\theta)$ . Find the value of $S + M + L$ .

Options

A.
1
B.
2
C.
3
D.
4

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

Distance Between Parallel Lines|Family of Lines|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of coordinate geometry and differential calculus.

The primary trap lies in the absolute value generated by the distance formula, which creates a point of non-differentiability at $\theta = \pi/4$ .

Students often drop the absolute value, incorrectly assuming $d(\theta) = \cos \theta \cos 2\theta$ , which leads to missing both the non-differentiability ( $S=0$ ) and the local minimum ( $M=1$ ).

The secondary trap involves the limit of the parameter space as $\theta \to \pi/2^-$ .

At this limit, the lines become concurrent (coincident on the y-axis), and direct substitution into $L_2$ yields an undefined $\tan(\pi/2)$ .

Students must recognize that the distance function remains well-behaved and approaches 0, testing their understanding of continuity at domain boundaries.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Distance Between Parallel Lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free