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JEE Main 20252 AprilMorning Shift
Exam
JEE Main
Year
2025
Date
2 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Determinants

JEE Main 2025 · 2 April · Morning Shift · Mathematics · Determinants

If the system of linear equations aligned & 3 x+y+β z=3 & 2 x+α...

If the system of linear equations
3x+y+βz=32x+αy−z=−3x+2y+z=4\begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4\end{aligned}
has infinitely many solutions, then the value of 22β−9α22 \beta-9 \alpha is :

Options

  1. A49
  2. B31
  3. C43
  4. D37

Correct answer

B. 31

Step-by-step solution

\Delta=\left|31β2α−1121\begin{array}{ccc}3 & 1 & \beta \\ 2 & \alpha & -1 \\ 1 & 2 & 1\end{array}\right|=0
3α+4β−αβ+3=03 \alpha+4 \beta-\alpha \beta+3=0
\Delta_3=\left|3132α−3124\begin{array}{ccc}3 & 1 & 3 \\ 2 & \alpha & -3 \\ 1 & 2 & 4\end{array}\right|=0
9α+19=09 \alpha+19=0
α=−199,β=611\alpha=\frac{-19}{9}, \beta=\frac{6}{11}
⇒22β−9α=31\Rightarrow 22 \beta-9 \alpha=31
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