UPSC NDA 2 2021 Mathematics Solved Paper

Concept:Correlation coefficient $r$ equals the geometric mean of the two regression coefficients.Explanation:Given regression lines: $3x - 4y + 8 = 0$ and $4x - 3y - 1 = 0$.First line: $3x - 4y + 8 = 0 \Rightarrow 4y = 3x + 8 \Rightarrow y = \frac{3}{4}x + 2$.Regression coefficient $b_{yx} = \frac{3}{4}$.Second line: $4x - 3y - 1 = 0 \Rightarrow 4x = 3y + 1 \Rightarrow x = \frac{3}{4}y + \frac{1}{4}$.Regression coefficient $b_{xy} = \frac{3}{4}$.Correlation coefficient: $r = \pm \sqrt{b_{yx} \times b_{xy}} = \pm \sqrt{\frac{3}{4} \times \frac{3}{4}} = \pm \frac{3}{4}$.Both slopes positive, so $r = \frac{3}{4}$. Statement 1 is correct.To find means, solve the two equations simultaneously.Substitute $y = \frac{3}{4}x + 2$ into $4x - 3y - 1 = 0$:$4x - 3(\frac{3}{4}x + 2) - 1 = 0 \Rightarrow 4x - \frac{9}{4}x - 6 - 1 = 0 \Rightarrow \frac{7}{4}x = 7 \Rightarrow x = 4$.Then $y = \frac{3}{4}(4) + 2 = 3 + 2 = 5$.So means are $\bar{x} = 4$, $\bar{y} = 5$, not $3$ and $4$. Statement 2 is incorrect.Answer:Only statement 1 is correct (Option A).