UPSC NDA I 2023 Mathematics Solved Paper

Concept:Diagonal from origin passes through the intersection point of two given lines, so find that point and then the line joining it to the origin.Explanation:Given lines: $x=0$, $y=0$, $x+y=1$, and $6x+y=3$.The diagonal through origin uses the intersection of $x+y=1$ and $6x+y=3$.From $x+y=1$, we get $y=1-x$.Substitute into $6x+y=3$: $6x+(1-x)=3 \Rightarrow 5x=2 \Rightarrow x=\frac{2}{5}$.Then $y=1-x=1-\frac{2}{5}=\frac{3}{5}$.So the intersection point is $C\left(\frac{2}{5},\frac{3}{5}\right)$.Origin is $O(0,0)$.Using two-point form: $y-y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1)$ with $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=\left(\frac{2}{5},\frac{3}{5}\right)$ gives $y-0=\frac{\frac{3}{5}-0}{\frac{2}{5}-0}(x-0)$.Simplify: $y=\frac{\frac{3}{5}}{\frac{2}{5}}x = \frac{3}{2}x \Rightarrow 2y=3x \Rightarrow 3x-2y=0$.Answer:$3x-2y=0$ (Option C)