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Diketahui  dan  adalah akar-akar persamaan kuadrat...

Diketahui begin mathsize 14px style m end style dan begin mathsize 14px style n end style adalah akar-akar persamaan kuadrat begin mathsize 14px style x squared – space 2 x plus k equals 0 end style. Jika matriks begin mathsize 14px style A equals open parentheses table row cell m squared minus 1 end cell cell 2 minus n squared end cell row 4 4 end table close parentheses end style adalah matriks singular, maka begin mathsize 14px style k equals space horizontal ellipsis end style

  1. begin mathsize 14px style 5 end style 

  2. begin mathsize 14px style 5 over 3 end style 

  3. begin mathsize 14px style 3 end style 

  4. begin mathsize 14px style 1 half end style 

  5. begin mathsize 14px style 4 over 3 end style 

Jawaban:

Diketahui begin mathsize 14px style m end style dan begin mathsize 14px style n end style adalah akar-akar persamaan kuadrat begin mathsize 14px style x squared – space 2 x plus k equals 0 end style. Jika matriks begin mathsize 14px style A equals open parentheses table row cell m squared minus 1 end cell cell 2 minus n squared end cell row 4 4 end table close parentheses end style adalah matriks singular, maka begin mathsize 14px style k equals space horizontal ellipsis end style

DIiketahui begin mathsize 14px style A equals open parentheses table row cell m squared minus 1 end cell cell 2 minus n squared end cell row 4 4 end table close parentheses end style dan A adalah matriks singular, artinya nilai determinannya = 0, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell d e t space A end cell equals 0 row cell 4 open parentheses m squared minus 1 close parentheses minus 4 open parentheses 2 minus n squared close parentheses end cell equals 0 row cell 4 m squared minus 4 minus 8 plus 4 n squared end cell equals 0 row cell 4 m squared plus 4 n squared minus 12 end cell equals 0 row cell 4 m squared plus 4 n squared end cell equals 12 row cell m squared plus n squared end cell equals 3 end table end style 

Karena begin mathsize 14px style m end style dan begin mathsize 14px style n end style adalah akar-akar persamaan kuadrat begin mathsize 14px style x squared – space 2 x plus k equals 0 end style, maka berlaku:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m plus n end cell equals cell negative b over a end cell row blank equals cell negative fraction numerator negative 2 over denominator 1 end fraction end cell row blank equals 2 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m cross times n end cell equals cell c over a end cell row blank equals cell k over 1 end cell row blank equals k end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses m plus n close parentheses squared end cell equals cell m squared plus n squared plus 2 m n end cell row cell 2 squared end cell equals cell 3 plus 2 k end cell row 4 equals cell 3 plus 2 k end cell row cell negative 2 k end cell equals cell 3 minus 4 end cell row cell negative 2 k end cell equals cell negative 1 end cell row k equals cell 1 half end cell end table end style

Jadi, jawaban yang tepat adalah D

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