Question

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

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

Hint:

Direction cosines $l, m, n$ represent components of a unit vector, so their vector length $\sqrt{l^2+m^2+n^2}$ must always be exactly 1.

Answer 1

The Correct Option is D

Concept: Let $(l, m, n)$ be the direction cosines of any line in 3D space. A fundamental identity governing direction cosines is that the sum of their squares must always equal exactly 1: \[ l^2 + m^2 + n^2 = 1 \]

Step 1: Evaluate the Assertion statement.
The assertion states that a line can have direction cosines $l = 1, m = 1, n = 1$. Let us check if these satisfy the fundamental identity: \[ l^2 + m^2 + n^2 = 1^2 + 1^2 + 1^2 = 1 + 1 + 1 = 3 \neq 1 \] Since the sum of squares equals 3 instead of 1, these values cannot be the direction cosines of any line. Thus, the Assertion is false.

Step 2: Evaluate the Reason statement.
The reason states that $\cos\theta = 1$ is possible for $\theta = 0$. We know from standard trigonometry that $\cos(0) = 1$. Therefore, this mathematical statement is true. Since the Assertion is false and the Reason is true, the correct choice is option (D).