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Question #75794

Difficulty: 8/10 | 🟢 Free

The Question

A variable plane $P$ has equal intercepts $I(t)$ on the absolute axes $Ox, Oy, Oz$ . An observer moves along the line $x=y=z$ , such that their position vector is $\vec{r}_0 = t\hat{i} + t\hat{j} + t\hat{k}$ . In the observer's frame, the plane has equal intercepts $R(t)$ on its local axes (which are parallel to the absolute axes). It is given that the rate of change of the absolute intercept $I(t)$ satisfies the differential equation $\frac{dI}{dt} = \frac{R(t)}{k}$ , where $k$ is the term independent of $x$ in the binomial expansion of $\left(x^2 + \frac{1}{x}\right)^9$ . If the initial absolute intercept is $I(0) = k$ , find the value of $I(k)$ .

Options

A.
$168(3-e)$
B.
$84e$
C.
$168$
D.
$84(3-e)$

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

Position Vector Origin Dependence|Variable plane intercepts|Term Independent of x|Implicit to Explicit Conversion

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 'Implicit to Explicit Conversion' trap within the context of shifting coordinate frames.

Students must first solve a binomial expansion to find the parameter $k$ .

The core trap lies in relating the absolute intercepts $I(t)$ to the local intercepts $R(t)$ .

Students often incorrectly assume $R(t) = I(t)$ (ignoring the origin shift) or $R(t) = I(t) - t$ (failing to account for the 3D nature of the plane equation $x+y+z=I(t)$ where each coordinate shifts by $t$ , resulting in a $3t$ shift).

This leads to incorrect differential equations.

Solving the correct linear differential equation requires careful integration by parts, rewarding students who rigorously track implicit dependencies during coordinate transformations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Position Vector Origin Dependence
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free