UPSC NDA I 2018 Mathematics Solved Paper

Concept:Two lines are parallel if their slopes are equal.The intersection of two lines is found by solving their equations simultaneously.The equation of a line can be written as $y = mx + c$, where $m$ is the slope and $c$ is the y‑intercept.Explanation:Find the intersection point of lines $x + 2y - 3 = 0$ and $2x - y + 5 = 0$.Multiply the second equation by 2: $4x - 2y + 10 = 0$.Add this to the first equation: $(x + 2y - 3) + (4x - 2y + 10) = 0 \Rightarrow 5x + 7 = 0 \Rightarrow x = -\frac{7}{5}$.Substitute $x = -\frac{7}{5}$ into $2x - y + 5 = 0$: $2\left(-\frac{7}{5}\right) - y + 5 = 0 \Rightarrow -\frac{14}{5} - y + 5 = 0 \Rightarrow -y + \frac{11}{5} = 0 \Rightarrow y = \frac{11}{5}$.Thus the intersection point is $\left(-\frac{7}{5}, \frac{11}{5}\right)$.The required line is parallel to $y - x + 10 = 0$, which can be rewritten as $y = x - 10$. Its slope is $1$.Hence the required line has slope $m = 1$. Using point‑slope form: $y - \frac{11}{5} = 1\left(x + \frac{7}{5}\right)$.Simplify: $y - \frac{11}{5} = x + \frac{7}{5} \Rightarrow y = x + \frac{18}{5}$.Multiply by $5$: $5y = 5x + 18 \Rightarrow 5x - 5y + 18 = 0$.Answer:$5x - 5y + 18 = 0$ (Option C)