JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

For the system of Linear Equations:

$2x - y + 3z = 5$

$3x + 2y - z = 7$

$4x + 5y + \alpha z = \beta$

Which of the following is NOT correct?

Choose the correct answer:

Explanation

Matrix Equations:

Since, Augmented matrix of given system of equation can be written as

$[A : B] = \begin{bmatrix} 2 & -1 & 3 & : & 5 \\ 3 & 2 & -1 & : & 7 \\ 4 & 5 & \alpha & : & \beta \end{bmatrix}$

Applying $R_1 \rightarrow R_1/2$ :

$[A : B] \sim \begin{bmatrix} 1 & \frac{-1}{2} & \frac{3}{2} & : & \frac{5}{2} \\ 3 & 2 & -1 & : & 7 \\ 4 & 5 & \alpha & : & \beta \end{bmatrix}$

Applying $R_2 \rightarrow R_2 - 3R_1$ and $R_3 \rightarrow R_3 - 4R_1$ :

$\Rightarrow [A : B] \sim \begin{bmatrix} 1 & \frac{-1}{2} & \frac{3}{2} & : & \frac{5}{2} \\ 0 & \frac{7}{2} & \frac{-11}{2} & : & \frac{-1}{2} \\ 0 & 7 & \alpha - 6 & : & \beta - 10 \end{bmatrix}$

Applying $R_3 \rightarrow R_3 - 2R_2$ :

$\Rightarrow [A : B] \sim \begin{bmatrix} 1 & \frac{-1}{2} & \frac{3}{2} & : & \frac{5}{2} \\ 0 & \frac{7}{2} & \frac{-11}{2} & : & \frac{-1}{2} \\ 0 & 0 & \alpha + 5 & : & \beta - 9 \end{bmatrix}$

Analysis:

Unique solution: if Rank of $A$ = Rank of $[A : B] = 3 \Rightarrow \alpha \neq -5$ and $\beta \in \mathbb{R}$ .
Infinite solution: if Rank of $A$ = Rank of $[A : B] < 3 \Rightarrow \alpha = -5$ and $\beta = 9$ .
Inconsistent: if Rank of $A <$ Rank of $[A : B] \Rightarrow \alpha = -5$ and $\beta \neq 9$ .

Choose the correct answer:

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