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Maths-

General

Easy

Question

The value of a for which the system of equationsx + y + z = 0ax + (a + 1)y + (a + 2)z = 0 a³x + (a + 1)³y + (a + 2)³z = 0has a non-zero solution is -

1
0
–1
None of these

The correct answer is: –1

The system of equations will have a non-zero solution if and only if
$straight capital delta$ = $open vertical bar table row 1 1 1 row a cell a plus 1 end cell cell a plus 2 end cell row cell a to the power of 3 end exponent end cell cell left parenthesis a plus 1 right parenthesis to the power of 3 end exponent end cell cell left parenthesis a plus 2 right parenthesis to the power of 3 end exponent end cell end table close vertical bar$ = 0
Using $C subscript 3 end subscript rightwards arrow C subscript 3 end subscript minus C subscript 2 end subscript comma C subscript 2 end subscript rightwards arrow C subscript 2 end subscript minus C subscript 1 end subscript$ , we get
$straight capital delta$ = $open vertical bar table row 1 0 0 row a 1 1 row cell a to the power of 3 end exponent end cell cell 3 a to the power of 2 end exponent plus 3 a plus 1 end cell cell 3 a to the power of 2 end exponent plus 9 a plus 7 end cell end table close vertical bar$ = 0
$rightwards double arrow$ 6a + 6 = 0 or a + 1 = 0
or a = –1.

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