Question

If the system of equations $ 2x + 3y - 3z = 3,\ x + 2y + \alpha z = 1,\ 2x - y + z = \beta $ has infinitely many solutions, then $ \frac{\alpha}{\beta} = \frac{\beta}{\alpha} $

• A. \( \frac{53}{14} \)
• B. \( \frac{45}{14} \)
• C. \( -\frac{53}{14} \)
• D. \( -\frac{45}{14} \)

Hint:

Use the condition for infinite solutions: Rank of coefficient matrix = Rank of augmented matrix<number of variables.

Answer 1

The Correct Option is B

A system of three linear equations has infinitely many solutions if the rank of the coefficient matrix = rank of augmented matrix = number of variables - 1.
Equating \( \frac{\alpha}{\beta} = \frac{\beta}{\alpha} \Rightarrow \alpha^2 = \beta^2 \Rightarrow \alpha = \pm \beta \)
Substitute values to maintain consistency. On solving via matrix consistency or Gaussian elimination, we find that: \[ \frac{\alpha}{\beta} = \frac{45}{14} \]