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Question #41645

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(\alpha, \beta, \gamma)$ be a point in 3D space whose coordinates are given by: $\alpha = 35 \int_{-\pi/2}^{\pi/2} \text{sgn}(x) \sin^3 x \cos^4 x \, dx$ $\beta = \frac{32}{\pi} \int_{-\pi/2}^{\pi/2} \text{sgn}(x) \cos^6 x \, dx$ $\gamma = \frac{32}{\pi} \int_{-\pi/2}^{\pi/2} (1 + \text{sgn}(x)) \sin^6 x \, dx$ where $\text{sgn}(x)$ denotes the signum function. The perpendicular distance from the point $P$ to the plane $x + 2y + 2z - 6 = 0$ is:

Options

A.
$6$
B.
$\frac{34}{3}$
C.
$18$
D.
$\frac{26}{3}$

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

Distance from point to plane|Wallis' Formula|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the integration of piecewise functions over symmetric intervals combined with Wallis' formula and 3D geometry.

The primary trap lies in the signum function.

Students often overlook the parity changes induced by $\text{sgn}(x)$ .

For instance, $\sin^3 x \cos^4 x$ is odd, but multiplying by the odd $\text{sgn}(x)$ makes it even, doubling the integral instead of evaluating to zero.

Conversely, $\cos^6 x$ becomes odd, evaluating to zero.

Failing to account for these parity shifts leads to incorrect coordinates (e.g., $(0, 10, 10)$ ) and subsequently a wrong distance calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Distance from point to plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free