Question

Let the system of linear equations
x + y + az = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, -y*) are collinear points, then the sum of absolute values of all possible values of α is

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (C)

2

Answer 2

Given system of equations

$x + y + az = 2$…(i)

$3x + y + z = 4$ …(ii)

$x + 2z = 1$ …(iii)

Solving (i), (ii) and (iii), we get

$x = 1,$ $y = 1$ , $z = 0$ (and for unique solution $a ≠–3$)

Now, $(α, 1), (1, α)$ and $(1, –1)$ are collinear

$∴$ $\begin{vmatrix} \alpha&1 & 1\\\1&\alpha&1\\\1&-1 &1 \end{vmatrix}=0$

⇒ $α(α + 1) – 1(0) + 1(–1 – α) = 0$

$⇒ α^2 – 1 = 0$

$∴ α = ±1$

$∴$ Sum of absolute values of $α = 1 + 1 = 2$

Hence, the correct option is (C): $2$