IPMAT Indore 2022 · Quantitative Ability · Question 18

Concurrent means all three lines pass through one common point. So find where the first two meet, then force the third through that point.

Step 1 — intersect the first two lines.

x-y=1 \text{and}2x+3y=12

From the first, x=y+1. Substituting:

2\left( y+1 \right)+3y=12 \Rightarrow 5y=10 \Rightarrow y=2, x=3

The point of intersection is \left( 3, 2 \right).

Step 2 — make the third line pass through it.

2\left( 3 \right)-3\left( 2 \right)+k=0 \Rightarrow 6-6+k=0 \Rightarrow k=0

Check: with k=0 the third line is 2x=3y, and \left( 3,2 \right) gives 6=6 ✔

Answer: Option 3 — 0

Shortcut. Solve two, substitute into the third — never set up the 3\times 3 determinant. The determinant condition

\left∣ \begin{matrix}1 & -1 & -1 \\ 2 & 3 & -12 \\ 2 & -3 & k\end{matrix}\right∣=0

is correct but slower and far more error-prone than finding \left( 3,2 \right) and plugging in.

Here the arithmetic is especially kind: at \left( 3,2 \right) the first two terms of the third line are 6 and -6, which cancel, so k=0 on sight.