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Maths-

General

Easy

Question

Find the value of ‘a’ if the three equations, (a + 1)³x + (a + 2)³y = (a + 3)³, (a + 1) x + (a + 2)y = (a + 3) & x + y = 1 are consistent.

–2
2
–3
–6

hint Hint:

For the three equation to be consistent, the determinant must be 0.

The correct answer is: –2

Step by step solution:
Given three equations:
$left parenthesis a space plus space 1 right parenthesis cubed x space plus space left parenthesis a space plus space 2 right parenthesis cubed$ $space y space equals space left parenthesis a space plus space 3 right parenthesis cubed$
$left parenthesis a space plus space 1 right parenthesis space x space plus space left parenthesis a space plus space 2 right parenthesis$ $y space equals space left parenthesis a space plus space 3 right parenthesis$
$x space plus space y space equals space 1$
For the three equation to be consistent, the determinant must be 0.
Therefore, $open vertical bar table row cell open parentheses a plus 1 close parentheses cubed end cell cell open parentheses a plus 2 close parentheses cubed end cell cell open parentheses a plus 3 close parentheses cubed end cell row cell open parentheses a plus 1 close parentheses end cell cell open parentheses a plus 2 close parentheses end cell cell open parentheses a plus 3 close parentheses end cell row 1 1 1 end table close vertical bar equals 0$
$rightwards double arrow open parentheses a plus 1 close parentheses cubed left square bracket open parentheses a plus 2 close parentheses$ $negative open parentheses a plus 3 close parentheses right square bracket minus open parentheses a plus 2 close parentheses cubed$ $left square bracket open parentheses a plus 1 close parentheses minus open parentheses a plus 3 close parentheses right square bracket$ $plus open parentheses a plus 3 close parentheses cubed$ $left square bracket open parentheses a plus 1 close parentheses minus open parentheses a plus 2 close parentheses right square bracket$ $equals 0$
$rightwards double arrow open parentheses a plus 1 close parentheses cubed open parentheses negative 1 close parentheses minus$ $open parentheses a plus 2 close parentheses cubed open parentheses negative 2 close parentheses$ $plus open parentheses a plus 3 close parentheses cubed open parentheses negative 1 close parentheses equals 0$
$rightwards double arrow negative open parentheses a cubed plus 3 a squared plus 3 a plus 1 close parentheses$ $plus 2 open parentheses a cubed plus 6 a squared plus 12 a plus 8 close parentheses$ $negative 1 open parentheses a cubed plus 9 a squared plus 27 a plus 27 close parentheses$ $equals 0$
$rightwards double arrow negative a cubed minus 3 a squared minus 3 a minus 1$ $plus 2 a cubed plus 12 a squared plus 24 a plus 16$ $negative a cubed minus 9 a squared minus 27 a minus 27$ $equals 0$
$rightwards double arrow negative 6 a minus 12 equals 0$
$rightwards double arrow a equals negative 2$
Hence, option (a) is the correct option.

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