SATHEE: Matrices and Determinants 4 Question 2

Matrices and Determinants 4 Question 2

2. Let $λ$ be a real number for which the system of linear equations

$x + y + z = 6, 4 x + λ y - λ z = λ - 2$ and

$3 x + 2 y - 4 z = - 5$

has infinitely many solutions. Then $λ$ is a root of the quadratic equation

(2019 Main, 10 April II)

(a) $λ^{2} - 3 λ - 4 = 0$

(b) $λ^{2} + 3 λ - 4 = 0$

(d) $λ^{2} + λ - 6 = 0$

Show Answer

Answer:

Correct Answer: 2. (c)

Solution:

Given, system of linear equations

$\begin{aligned} x + y + z & = 6 \\ 4 x + λ y - λ z & = λ - 2 \end{aligned}$

and $3 x + 2 y - 4 z = - 5$

has infinitely many solutions, then $Δ = 0$

$\Rightarrow | \begin{array}{ccc} 1 & 1 & 1 \\ 4 & λ & - λ \\ 3 & 2 & - 4 \end{array} | = 0$

$\Rightarrow 1 (- 4 λ + 2 λ) - 1 (- 16 + 3 λ) + 1 (8 - 3 λ) = 0$

$\Rightarrow - 8 λ + 24 = 0 \Rightarrow λ = 3$

From, the option $λ = 3$ , satisfy the quadratic equation $λ^{2} - λ - 6 = 0$

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Matrices and Determinants 4 Question 2

2. Let λ be a real number for which the system of linear equations

Answer:

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2. Let $λ$ be a real number for which the system of linear equations