UPSC NDA 2 2022 Mathematics Solved Paper

Concept:The slope of a line making angle $\theta$ with the positive x-axis is $\tan \theta$. The line through origin with slope $m$ has equation $y = mx$.Explanation:Given: line passes through origin $(0,0)$ and makes $75^{\circ}$ with x-axis. Slope: $\tan 75^{\circ} = \tan(45^{\circ}+30^{\circ}) = \frac{\sqrt{3}+1}{\sqrt{3}-1} = 2+\sqrt{3}$. Statement 1: Check if point $\left(1, \frac{1}{2-\sqrt{3}}\right)$ lies on $y = (2+\sqrt{3})x$. Simplify the y-coordinate: $\frac{1}{2-\sqrt{3}} = 2+\sqrt{3}$ (rationalizing). At $x=1$, $y = 2+\sqrt{3}$. So point satisfies $y = (2+\sqrt{3})x$. Hence statement 1 is correct. Statement 2: The line through origin with slope positive ($\tan 75^{\circ} > 0$) goes only into quadrants where both $x$ and $y$ have same sign (first and third). Thus statement 2 is also correct. Answer:Both statements 1 and 2 are correct. So option C.