Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • If the system of linear equations <img alt="\begin{gathered} 2 x+y-z=3 \\ x-y-z=\alpha \\ 3 x+3 y+\beta z=3 \end{gathered}" src="https://entrancecorner.codecogs.com/gif.latex?%5Cbegin%7Bgathered%7D%202%20x&plus;y-z%3D3%20%5C%5C%20x-y-z%3D%5Calp

If the system of linear equations
\begin{gathered} 2 x+y-z=3 \\ x-y-z=\alpha \\ 3 x+3 y+\beta z=3 \end{gathered}
has infinitely many solution, then \alpha+\beta-\alpha \beta is equal to_________.
 

Answers (1)

best_answer

For infinitely many solutions :
\Delta = 0= \Delta _{1}= \Delta _{2}= \Delta _{3}
\Delta = 0\Rightarrow \begin{vmatrix} 2 & 1 & -1\\ 1& -1 & -1\\ 3& 3 & \beta \end{vmatrix}= 0
\Rightarrow 2\left ( -\beta +3 \right )-1\left ( \beta +3 \right )-1\left ( 3+3 \right )= 0
\Rightarrow -3\beta -3= 0\Rightarrow \beta = -1
\Delta_{1} = 0\Rightarrow \begin{vmatrix} 3 & 1 & -1\\ \alpha & -1 & -1\\ 3& 3 & -1 \end{vmatrix}= 0
\Rightarrow 3\left ( 1+3 \right )-1\left ( -\alpha +3 \right )-1\left ( 3\alpha +3 \right )= 0
\Rightarrow 12+\alpha -3-3\alpha -3= 0
\Rightarrow 2\alpha= 6\Rightarrow \alpha = 3
\alpha = 3,\beta = -1
\alpha +\beta -\alpha \beta -3-1+3= 5

Posted by

Kuldeep Maurya

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2025 Session 2 City Slip (OUT) Live: NTA exam city intimation slip link available; admit card date
anam-khan

AESL expands Aakash Digital 2.0 to offer AI-powered coaching for NEET, JEE, Olympiads
anam-khan

Bihar Board extends BSEB Super 50 free coaching application last date to March 26

Latest Question