UPSC NDA II 2018 Mathematics Solved Paper

Concept:Find the point where the two given lines intersect, then write the equation of a line parallel to $4x+5y-6=0$ passing through that point.Explanation:Step 1: Simplify the given lines: $\frac{x}{2} + \frac{y}{3} = 1$ becomes $3x + 2y = 6$. $\frac{x}{3} + \frac{y}{2} = 1$ becomes $2x + 3y = 6$.Step 2: Solve the system by elimination. Multiply the first equation by $2$ and the second by $3$: $6x + 4y = 12$ and $6x + 9y = 18$.Subtract: $(6x+4y) - (6x+9y) = 12 - 18$ $\Rightarrow -5y = -6$ $\Rightarrow y = \frac{6}{5}$.Substitute $y = \frac{6}{5}$ into $3x + 2y = 6$: $3x + 2\left(\frac{6}{5}\right) = 6$ $\Rightarrow 3x = 6 - \frac{12}{5} = \frac{18}{5}$ $\Rightarrow x = \frac{6}{5}$.So the intersection point is $\left(\frac{6}{5}, \frac{6}{5}\right)$.Step 3: A line parallel to $4x + 5y - 6 = 0$ has the form $4x + 5y + k = 0$. Plug the point into this equation: $4\left(\frac{6}{5}\right) + 5\left(\frac{6}{5}\right) + k = 0$ $\Rightarrow \frac{24}{5} + 6 + k = 0$ $\Rightarrow \frac{24}{5} + \frac{30}{5} + k = 0$ $\Rightarrow \frac{54}{5} + k = 0$ $\Rightarrow k = -\frac{54}{5}$.Step 4: The required line is $4x + 5y - \frac{54}{5} = 0$. Multiply by $5$ to clear the fraction: $20x + 25y - 54 = 0$.Answer:$20x + 25y - 54 = 0$ (Option A).