Question

Assertion (A): A line can have direction cosines $\langle 1,1,1\rangle$.
Reason (R): $\cos\theta=1$ is possible for $\theta=0^\circ$.

• A. Both (A) and (R) are true and (R) is the correct explanation of (A).
• B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Hint:

Direction cosines must always satisfy the identity \(l^2+m^2+n^2=1\).

Answer 1

The Correct Option is D

Step 1: Recall the property of direction cosines.
If \(l,m,n\) are the direction cosines of a line, then they must satisfy the fundamental identity \[ l^2+m^2+n^2=1 \] This condition is necessary because direction cosines represent the cosines of angles that a line makes with the coordinate axes.
Step 2: Test the assertion.
The assertion claims that the direction cosines of a line can be \[ \langle 1,1,1\rangle \] Check the condition: \[ 1^2+1^2+1^2=3 \] But the required value must be \[ 1 \] Thus the condition for direction cosines is not satisfied.
Therefore the statement that a line can have direction cosines \(\langle1,1,1\rangle\) is false.
Step 3: Check the reason statement.
The reason states \[ \cos\theta=1 \] This occurs when \[ \theta=0^\circ \] This is a standard trigonometric result and is mathematically correct.
Thus the reason statement is true.
Step 4: Final conclusion.
Since the assertion is false but the reason is true, the correct option becomes \[ (D) \] Final Answer: $\boxed{\text{(D) Assertion is false but Reason is true}}$