Vectors in 2D Practice Questions - NDA Mathematics Question Bank

Question 1 of 2 NDA MCQ

Detailed Explanation

Solution

Option (a) is correct.

Explanation:

The unit vector \(\hat{a}\) making an angle of \(30^\circ\) with the positive x-axis can be expressed in components as:

\[ \hat{a} = \cos(30^\circ) \hat{i} + \sin(30^\circ) \hat{j} \]

Substituting the values of \(\cos(30^\circ)\) and \(\sin(30^\circ)\): \(\cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(30^\circ) = \frac{1}{2}\) Thus, we have:

\[ \hat{a} = \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j} \]

Thus, the correct option is:

\[ \hat{a} = \frac{\sqrt{3} \hat{i} + \hat{j}}{2} \]

Hints & Visual Guide

📌 Hints / Properties Used:

Unit vector components: \(\hat{a} = \cos(\theta) \hat{i} + \sin(\theta) \hat{j}\)
Angle trigonometric values at special angles: \(30^\circ\)

📊 Visual Diagram Suggestion:

A coordinate system diagram showing the x and y axes, the angle \(30^\circ\), and the vector \(\hat{a}\) represented as an arrow originating from the origin.

📖 Factual Verification & Reference:

Verified against NCERT Class XI Mathematics, Chapter 4 (Vectors)

Question 2 of 2 NDA MCQ

Detailed Explanation

Vectors in 2D

Solution

Hints & Visual Guide

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