Class CBSE Class 12 Mathematics Vector Algebra Q #560
KNOWLEDGE BASED
APPLY
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
Let \(\theta\) be the angle between two unit vectors \(\hat{a}\) and \(\hat{b}\) such that \(\sin\theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to:
(A) \(\pm\frac{3}{5}\)
(B) \(\pm\frac{3}{4}\)
(C) \(\pm\frac{4}{5}\)
(D) \(\pm\frac{4}{3}\)
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AI Tutor Explanation

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Step-by-Step Solution

  1. We know that \(\hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos\theta\).

  2. Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, \(|\hat{a}| = 1\) and \(|\hat{b}| = 1\). Therefore, \(\hat{a} \cdot \hat{b} = \cos\theta\).

  3. Given that \(\sin\theta = \frac{3}{5}\), we can find \(\cos\theta\) using the identity \(\sin^2\theta + \cos^2\theta = 1\).

  4. So, \(\cos^2\theta = 1 - \sin^2\theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}\).

  5. Therefore, \(\cos\theta = \pm\sqrt{\frac{16}{25}} = \pm\frac{4}{5}\).

  6. Thus, \(\hat{a} \cdot \hat{b} = \cos\theta = \pm\frac{4}{5}\).

Correct Answer: \(\pm\frac{4}{5}\)

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of dot product and trigonometric identities to find the value of the dot product.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the relationship between the dot product of two vectors and the angle between them, as well as the use of trigonometric identities.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of vector algebra and trigonometric relationships, which are core concepts covered in the textbook.

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