SATHEE: Matrices and Determinants 4 Question 11

Matrices and Determinants 4 Question 11

11. If the system of equations

$\begin{matrix} x + y + z = 5 \\ x + 2 y + 3 z = 9 \\ x + 3 y + α z = β \end{matrix}$

has infinitely many solutions, then $β - α$ equals

(2019 Main, 10 Jan I)

(a) 8

(b) 18

(d) 5

Show Answer

Answer:

Correct Answer: 11. (a)

Solution:

Since, the system of equations has infinitely many solution, therefore $D = D_{1} = D_{2} = D_{3} = 0$

Here,

$\begin{aligned} D & = | \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & α \end{array} | = 1 (2 α - 9) - 1 (α - 3) + 1 (3 - 2) \\ = α - 5 \\ and D_{3} & = | \begin{array}{lll} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & β \end{array} | = 1 (2 β - 27) - 1 (β - 9) + 5 (3 - 2) \end{aligned}$

$= β - 13$

$\begin{aligned} Now, D = 0 \\ \Rightarrow α - 5 = 0 \Rightarrow α = 5 \\ and D_{3} = 0 \Rightarrow β - 13 = 0 \\ \Rightarrow β = 13 \\ ∴ β - α = 13 - 5 = 8 \end{aligned}$

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Matrices and Determinants 4 Question 11

11. If the system of equations

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