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Question #40996

Difficulty: 8/10 | 🟢 Free

The Question

A line $L$ with direction cosines $l, m, n$ passes through the point $A(1, -1, 2)$ . If $L$ intersects the plane $2x + 2y + 2z = 9$ at a point $B$ such that the distance $AB = \frac{5}{2}$ , then the standard deviation of the data set $\{l, m, n\}$ is:

Options

A.
$\frac{\sqrt{2}}{3}$
B.
$\frac{2}{9}$
C.
$\frac{1}{3}$
D.
$\frac{2}{3}$

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

Standard Deviation and Coefficient of Variation|Direction Cosines constraint|Line-Plane intersection logic

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D Geometry and Statistics.

The primary trap lies in recognizing that the variables l, m, n are not just arbitrary parameters of a line, but direction cosines constrained by the fundamental identity l² + m² + n² = 1.

Students often attempt to solve for individual values of l, m, n using the plane intersection, which is impossible since there is only one linear equation derived (l + m + n = ±1).

The 'lethal collision' occurs if they fail to use the variance formula in conjunction with the geometric identity of direction cosines.

Another common mistake is calculating the variance (2/9) and forgetting to take the square root for the standard deviation, leading them to select option B.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Standard Deviation and Coefficient of Variation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free