Maths-
General
Easy

Question

The system of equations alpha x plus y plus z equals alpha minus 1 comma x plus alpha y plus z equals alpha minus 1 comma x plus y plus alpha z equals alpha minus 1 has no solution, if alpha is

  1. 1
  2. not negative 2
  3. either negative 2 or 1
  4. negative 2

hintHint:

In this question, we have to find the alpha, if the system of equations alpha x plus y plus z equals alpha minus 1 comma x plus alpha y plus z equals alpha minus 1 comma x plus y plus alpha z equals alpha minus 1 has no solution. As we know that if the solution is zero, so, D=0 and D subscript 1 not equal to 0 comma space D subscript 2 not equal to 0 space a n d space D subscript 3 not equal to 0. Using D and D subscript 1 we can find the value of alpha.

The correct answer is: negative 2


    alpha x plus y plus z equals alpha minus 1
x plus alpha y plus z equals alpha minus 1
x plus y plus alpha z equals alpha minus 1
A s space i t space h a s space n o space s o l u t i o n comma space s o comma space D equals 0 comma space D subscript 1 not equal to 0 comma space D subscript 2 not equal to 0 space a n d space D subscript 3 not equal to 0
D equals open vertical bar table row alpha 1 1 row 1 alpha 1 row 1 1 alpha end table close vertical bar
rightwards double arrow D equals 0
rightwards double arrow alpha left parenthesis alpha squared minus 1 right parenthesis minus 1 left parenthesis alpha plus 1 right parenthesis plus 1 left parenthesis 1 minus alpha right parenthesis equals 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left square bracket alpha left parenthesis alpha plus 1 right parenthesis minus 1 minus 1 right square bracket equals 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left parenthesis alpha squared plus alpha minus 2 right parenthesis equals 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left parenthesis alpha squared plus 2 alpha minus alpha minus 2 right parenthesis equals 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left parenthesis alpha left parenthesis alpha plus 2 right parenthesis minus 1 left parenthesis alpha plus 2 right parenthesis right parenthesis equals 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left parenthesis alpha minus 1 right parenthesis left parenthesis alpha plus 2 right parenthesis equals 0
rightwards double arrow alpha equals 1 space o r space alpha space equals negative 2
D subscript 1 equals open vertical bar table row cell alpha minus 1 end cell 1 1 row cell alpha minus 1 end cell alpha 1 row cell alpha minus 1 end cell 1 alpha end table close vertical bar
rightwards double arrow D subscript 1 equals left parenthesis alpha minus 1 right parenthesis open vertical bar table row 1 1 1 row 1 alpha 1 row 1 1 alpha end table close vertical bar
rightwards double arrow D subscript 1 not equal to 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left square bracket left parenthesis alpha squared minus 1 right parenthesis minus 1 left parenthesis 1 minus alpha right parenthesis plus 1 left parenthesis 1 minus alpha right parenthesis right square bracket not equal to 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left parenthesis alpha minus 1 right parenthesis left square bracket alpha plus 1 plus 1 minus 1 right square bracket not equal to 0
rightwards double arrow left parenthesis alpha minus 1 right parenthesis left parenthesis alpha minus 1 right parenthesis left parenthesis alpha plus 1 right parenthesis not equal to 0
rightwards double arrow alpha space not equal to plus-or-minus 1
S o comma space alpha space equals space minus 2

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