SATHEE: Matrices and Determinants 4 Question 8

Matrices and Determinants 4 Question 8

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

(2019 Main, 12 Jan I)

$\begin{aligned} (1 + α) x + β y + z = 2 \\ α x + (1 + β) y + z = 3 \\ a x + β y + 2 z = 2 \end{aligned}$

has a unique solution, is

(a) $(2, 4)$

(b) $(- 4, 2)$

(d) $(- 3, 1)$

Show Answer

Answer:

Correct Answer: 8. (a)

Solution:

Given system of linear equations,

$(1 + α) x + β y + z = 2$

$α x + (1 + β) y + z = 3$

$α x + β y + 2 z = 2$

has a unique solution, if

$| \begin{array}{ccc} 1 + α & β & 1 \\ α & (1 + β) & 1 \\ α & β & 2 \end{array} | \neq 0$

Apply $R_{1} \to R_{1} - R_{3}$ and $R_{2} \to R_{2} - R_{3}$

$| \begin{array}{ccc} 1 & 0 & - 1 \\ 0 & 1 & - 1 \\ α & β & 2 \end{array} | \neq 0$

$\begin{array}{rr} \Rightarrow & 1 (2 + β) - 0 (0 + α) - 1 (0 - α) \neq 0 \\ \Rightarrow & α + β + 2 \neq 0 \end{array}$

Note that, only $(2, 4)$ satisfy the Eq. (i).

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Matrices and Determinants 4 Question 8

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

Answer:

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