Questions number 19 and 20 are Assertion and Reason based questions.
Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).
Select the correct answer from the codes (A), (B), (C), and (D) as given below.
Assertion (A): For any non-zero unit vector \( \vec{a} \), \( \vec{a} \cdot (-\vec{a}) = (-\vec{a}) \cdot \vec{a} = -1 \).
Reason (R): Angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \).
Assertion (A): For any non-zero unit vector \( \vec{a} \), \( \vec{a} \cdot (-\vec{a}) = (-\vec{a}) \cdot \vec{a} = -1 \).
Reason (R): Angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \).
Show Hint
- Both Assertion (A) and Reason (R) are true and 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.
The Correct Option is C
Solution and Explanation
Step 1: Verify Assertion (A)
The dot product of a vector \( \vec{a} \) and its negative \( -\vec{a} \) is given by: \[ \vec{a} \cdot (-\vec{a}) = |\vec{a}| \cdot |-\vec{a}| \cdot \cos \theta. \] Here, \( |\vec{a}| = 1 \) (since \( \vec{a} \) is a unit vector) and \( |-\vec{a}| = 1 \). The angle \( \theta \) between \( \vec{a} \) and \( -\vec{a} \) is \( \pi \) (180°), so: \[ \cos \pi = -1. \] Thus: \[ \vec{a} \cdot (-\vec{a}) = 1 \cdot 1 \cdot (-1) = -1. \]
Therefore, Assertion (A) is true.
Step 2: Verify Reason (R)
The Reason (R) states that the angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \) (90°). However, the actual angle between \( \vec{a} \) and \( -\vec{a} \) is \( \pi \) (180°), as they are in opposite directions.
Hence, Reason (R) is false.
Conclusion: Assertion (A) is true, but Reason (R) is false.
Assertion (A): Every scalar matrix is a diagonal matrix.
Reason (R): In a diagonal matrix, all the diagonal elements are 0.
Reason (R): In a diagonal matrix, all the diagonal elements are 0.
Show Hint
- Both Assertion (A) and Reason (R) are true and 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.
The Correct Option is C
Solution and Explanation
Step 1: Analyze Assertion (A)
A scalar matrix is a diagonal matrix where all the diagonal elements are equal, and the non-diagonal elements are zero.
Hence, Assertion (A) is true.
Step 2: Analyze Reason (R)
In a diagonal matrix, the diagonal elements can have any value (not necessarily 0).
Therefore, Reason (R) is false.
Step 3: Conclude the result
Assertion (A) is true, but Reason (R) is false.
Hence, the correct option is (C).
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