UPSC NDA I 2021 Mathematics Solved Paper

Concept:The acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by $\tan\theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right|$. When one line is vertical ($m = \infty$), the angle can be found directly from the slopes of the other line.Explanation:For the line $x - 2 = 0$, rewrite as $x = 2$. This is a vertical line, so its slope $m_1$ is undefined. The angle this line makes with the positive $x$-axis is $\theta_1 = 90^{\circ}$. For the line $\sqrt{3}x - y - 2 = 0$, rearrange into slope-intercept form: $y = \sqrt{3}x - 2$. Thus, its slope $m_2 = \sqrt{3}$. The angle $\theta_2$ satisfies $\tan\theta_2 = \sqrt{3}$, so $\theta_2 = 60^{\circ}$. The acute angle between the two lines is the absolute difference of their angles: $\theta = |\theta_1 - \theta_2| = |90^{\circ} - 60^{\circ}| = 30^{\circ}$. Alternatively, using the formula with $\infty$ slope leads to the same result.Answer:$30^{\circ}$ (Option B).