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Question #68598

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_t$ be a family of curves satisfying the differential equation $\frac{dy}{dx} = (x - t)e^x$ , where $t \in \mathbb{R}$ is a parameter. For each $t$ , the curve $C_t$ passes through the origin. Let $P_t$ be the point of local extremum of the curve $C_t$ . As $t$ varies, $P_t$ traces a curve $L_1$ . A new curve $L_2$ is defined as the locus of a point $Q$ , which is obtained by applying the transformation matrix $M = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$ to the coordinates of $P_t$ (treating $P_t$ as a column vector). Let $A$ be the area of the region bounded by $L_2$ , the normal to $L_2$ at its point of local maximum, and the coordinate axes. Find the value of $A$ .

Options

A.
$4\ln 2 - \frac{5}{2}$
B.
$\frac{3}{2} - 2\ln 2$
C.
$4\ln 2 - \frac{9}{2}$
D.
$2\ln 2 - \frac{3}{2}$

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

Geometric Interpretation|Integration by Parts (LIATE Rule)|Locus Problems via Transformation|Locus of a Moving Point

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 intersection of differential equations, locus transformations, and definite integration.

A primary trap lies in the integration by parts of the ODE, where students often fail to define the integration constant as a function of the parameter $t$ , leading to an incorrect locus $L_1$ .

Furthermore, the geometric constraint requires identifying the normal at the local maximum of $L_2$ .

Students frequently confuse the normal with the tangent ( $y = 2\ln 2 - 1$ ), which completely alters the bounded region and leads to the distractor option $\frac{3}{2} - 2\ln 2$ .

The transformation matrix acts as a cognitive pivot, forcing a frame-of-reference shift before the final integration.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Geometric Interpretation
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free