Before You Begin

Question #41626

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(14, 14, \sqrt{2})$ be a point in $\mathbb{R}^3$ and $\Pi$ be the plane $x + y + \sqrt{2}z = 4$ . Let $P'$ be the image of $P$ in the plane $\Pi$ . If $Q(6, 0, 0)$ is another point and $V$ is the volume of the tetrahedron $OPP'Q$ (where $O$ is the origin), then the value of $[V]$ is (where $[ \cdot ]$ denotes the greatest integer function):

Options

A.
237
B.
238
C.
239
D.
240

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

Image of a point in a plane|Scalar Triple Product Volume Logic|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This problem tests spatial vector operations combined with a lethal precision trap involving the Greatest Integer Function.

The volume evaluates to $169\sqrt{2}$ .

If a student prematurely approximates $\sqrt{2} \approx 1.414$ or even $1.4142$ , they will calculate $V \approx 238.966$ or $238.9998$ , leading to an incorrect answer of 238.

By squaring the exact radical form to get $V^2 = 57122$ , and recognizing that $239^2 = 57121$ , it becomes rigorously clear that $V$ is marginally greater than 239.

This penalizes floating-point approximations and enforces exact algebraic manipulation until the final bounding step.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Image of a point in a plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free