Question:
Assertion (A) : A line can have direction cosines \( \)
Reason (R) : \( \cos \theta = 1 \) is possible for \( \theta = 0 \).
Assertion (A) : A line can have direction cosines \( \)
Reason (R) : \( \cos \theta = 1 \) is possible for \( \theta = 0 \).
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Direction cosines always lie between -1 and 1, inclusive, but their squared sum must be exactly 1.
If components are \( \), they are direction ratios (DRs), and the corresponding DCs would be \( \).
If components are \( \), they are direction ratios (DRs), and the corresponding DCs would be \( \).
Updated On: Sep 10, 2026
- Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
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The Correct Option is D
Solution and Explanation
Concept:
• Direction Cosines \( (L, M, N) \) must satisfy the fundamental constraint: \( L^2 + M^2 + N^2 = 1 \).
• Direction cosines are the values \( \cos \alpha, \cos \beta, \cos \gamma \), where \( \alpha, \beta, \gamma \) are the angles with the axes.
Step 1: Evaluate Assertion (A)
For a line to have direction cosines \( \), the sum of their squares must equal 1.
Calculation:
\( L^2 + M^2 + N^2 = 1^2 + 1^2 + 1^2 = 1 + 1 + 1 = 3 \).
Since \( 3 \neq 1 \), a line cannot have these direction cosines.
Assertion (A) is false.
Step 2: Evaluate Reason (R)
The statement is \( \cos \theta = 1 \) is possible for \( \theta = 0 \).
By trigonometric table values, \( \cos 0^\circ = 1 \).
So, Reason (R) is true.
Step 3: Conclusion
Assertion (A) is false and Reason (R) is true.
• Direction Cosines \( (L, M, N) \) must satisfy the fundamental constraint: \( L^2 + M^2 + N^2 = 1 \).
• Direction cosines are the values \( \cos \alpha, \cos \beta, \cos \gamma \), where \( \alpha, \beta, \gamma \) are the angles with the axes.
Step 1: Evaluate Assertion (A)
For a line to have direction cosines \( \), the sum of their squares must equal 1.
Calculation:
\( L^2 + M^2 + N^2 = 1^2 + 1^2 + 1^2 = 1 + 1 + 1 = 3 \).
Since \( 3 \neq 1 \), a line cannot have these direction cosines.
Assertion (A) is false.
Step 2: Evaluate Reason (R)
The statement is \( \cos \theta = 1 \) is possible for \( \theta = 0 \).
By trigonometric table values, \( \cos 0^\circ = 1 \).
So, Reason (R) is true.
Step 3: Conclusion
Assertion (A) is false and Reason (R) is true.
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