Question

An ordered pair (α. β) for which the system of linear equations

(1 + α)x + βy + z = 2

αx + (1 + β)y + z = 3

αx + β + 2z = 2 has a unique solution, is
1. (2, 4)
2. (-4, 2)
3. (1, -3)
4. (-3, 1)

Answer 1

Correct Answer - Option 1 : (2, 4)

From the system of linear equations,

(1 + α)x + βy + z = 2

αx + (1 + β)y + z = 3

αx + β + 2z = 2

Has a unique solution, if

\(\Rightarrow \left| {\begin{array}{*{20}{c}} {1 + \alpha }&\beta &1\\ \alpha &{\left( {1 + \beta } \right)}&1\\ \alpha &\beta &2 \end{array}} \right| \ne 0\)

On applying, R₁ → R₁ – R₃ and R₂ → R₂ – R₃

\(\Rightarrow \left| {\begin{array}{*{20}{c}} 1&0&{ - 1}\\ 0&1&{ - 1}\\ \alpha &\beta &2 \end{array}} \right| \ne 0\)

⇒ 1(2 + β) – 0(0 + α) – 1(0 – α) ≠ 0

⇒ α + β + 2 ≠ 0

The coordinates in option (a) is the only point that satisfies the above equation.