Maths-
General
Easy

Question

The system of equations a x minus y minus z equals alpha minus 1 x minus alpha y minus z equals alpha minus 1 x minus y minus alpha z equals alpha minus 1 Has no solution if alpha is

  1. Either negative 2 or 1    
  2. negative 2    
  3. 1    
  4. Not negative 2    

The correct answer is: negative 2


    For no solution or infinitely many solutions
    open vertical bar table row alpha cell negative 1 end cell cell negative 1 end cell row 1 cell negative alpha end cell cell negative 1 end cell row 1 1 cell negative alpha end cell end table close vertical bar equals 0
    rightwards double arrow blank alpha open parentheses alpha to the power of 2 end exponent minus 1 close parentheses minus 1 open parentheses alpha minus 1 close parentheses plus 1 open parentheses 1 minus alpha close parentheses equals 0
    rightwards double arrow blank alpha open parentheses alpha to the power of 2 end exponent minus 1 close parentheses minus 2 alpha plus 2 equals 0
    rightwards double arrow blank alpha open parentheses alpha minus 1 close parentheses open parentheses alpha plus 1 close parentheses minus 2 open parentheses alpha minus 1 close parentheses equals 0
    rightwards double arrow open parentheses alpha minus 1 close parentheses open parentheses alpha to the power of 2 end exponent plus alpha minus 2 close parentheses equals 0
    rightwards double arrow open parentheses alpha minus 1 close parentheses open parentheses alpha plus 2 close parentheses open parentheses alpha minus 1 close parentheses equals 0
    rightwards double arrow open parentheses alpha minus 1 close parentheses to the power of 2 end exponent open parentheses alpha plus 2 close parentheses equals 0
    rightwards double arrow blank alpha equals 1 comma blank 1 comma blank minus 2
    But for alpha equals 1, there are infinite solutions. When alpha equals negative 2, wehave
    negative 2 x minus y minus z equals negative 3
    x plus 2 y minus z equals negative 3
    x minus y plus 2 z equals negative 3
    Adding, we get 0 equals negative 9, which is not true. Hence there is no solution

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