Class CBSE Class 12 Mathematics Three Dimensional Geometry Q #823
KNOWLEDGE BASED
UNDERSTAND
1 Marks 2023 MCQ SINGLE
Direction cosines of the line \frac{x-1}{2}=\frac{1-y}{3}=\frac{2z-1}{12} are:
(A) \frac{2}{7},\frac{3}{7},\frac{6}{7}
(B) \frac{2}{\sqrt{157}},-\frac{3}{\sqrt{157}},\frac{12}{\sqrt{157}}
(C) \frac{2}{7},-\frac{3}{7},-\frac{6}{7}
(D) \frac{2}{7},-\frac{3}{7},\frac{6}{7}
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Step-by-Step Solution

  1. Rewrite the given equation in the standard form: \frac{x-1}{2}=\frac{1-y}{3}=\frac{2z-1}{12} can be written as \frac{x-1}{2}=\frac{-(y-1)}{3}=\frac{2(z-\frac{1}{2})}{12} \frac{x-1}{2}=\frac{y-1}{-3}=\frac{z-\frac{1}{2}}{6}
  2. The direction ratios are 2, -3, 6.
  3. Calculate the magnitude of the direction ratios: \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7
  4. Divide each direction ratio by the magnitude to get the direction cosines: \frac{2}{7}, \frac{-3}{7}, \frac{6}{7}

Correct Answer: \frac{2}{7},-\frac{3}{7},\frac{6}{7}

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of direction cosines and how to derive them from the given equation of a line. They need to manipulate the equation into a standard form and then apply the formula.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to find the direction cosines: first, standardize the equation of the line, and then apply the formula to calculate the direction cosines.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of a concept taught in the textbook and requires them to apply a standard formula.

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