Before You Begin

Question #40765

Difficulty: 8/10 | 🟢 Free

The Question

Let $L(t)$ be the variable line given by the equation $x - y = \Delta(t)$ , where $\Delta(t) = \begin{vmatrix} \sec t & \ln(t+1) & \sqrt{t^2+1} \\ \cos t & \sin t & \tan t \\ e^{2t} & t+1 & \cos t \end{vmatrix}$ Let $P$ be the point $(5, 1)$ . If $P'(t)$ is the reflection of $P$ across the line $L(t)$ , and $D(t)$ is the distance of $P'(t)$ from the origin, find the value of $\frac{d}{dt} [D^2(t)]$ at $t = 0$ .

Options

A.
4
B.
-4
C.
8
D.
0

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

Reflection Principle|Symmetry Utilization|Differentiation of Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the Reflection Principle with the Differentiation of Determinants.

The primary trap lies in the algebraic complexity of the determinant \Delta(t).

Students often attempt to fully expand the determinant before differentiating, which leads to a massive, error-prone algebraic collapse.

The 'pro' approach recognizes that the rate of change of the reflected point's distance only requires the determinant and its derivative evaluated at a specific point (t=0).

By applying Jacobi's formula (row-wise differentiation of determinants) directly at t=0, the computation simplifies drastically.

Furthermore, utilizing the symmetry of the reflection across a line with slope 1 allows for a quick parametric representation of the reflected coordinates without relying on cumbersome distance formulas.

Quick Facts About This Question

Question Information

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