UPSC NDA II 2015 Mathematics Paper

Concept:Concurrent lines are two or more lines that pass through a single common point.Explanation:The given lines are $4x + 4y = 1$, $8x - 3y = 2$, and $y = 0$.First, find the intersection of $4x + 4y = 1$ and $y = 0$.Substitute $y = 0$ into $4x + 4y = 1$: $4x + 4(0) = 1$ → $4x = 1$ → $x = \frac{1}{4}$.So the intersection point is $\left( \frac{1}{4}, 0 \right)$.Next, find the intersection of $8x - 3y = 2$ and $y = 0$.Substitute $y = 0$: $8x - 3(0) = 2$ → $8x = 2$ → $x = \frac{1}{4}$.So this intersection point is also $\left( \frac{1}{4}, 0 \right)$.Since all three pairs of lines meet at the same point $\left( \frac{1}{4}, 0 \right)$, the lines are concurrent.Answer:Concurrent.