Mathematics
Algebra
Easy

Question

space If space the space system space of space equations space 2 x plus 3 y minus z equals 0 comma space x plus k y minus 2 z equals 0 space a n d space 2 x minus y plus z equals 0 space has space a space non minus trivial solution space left parenthesis straight x comma straight y comma straight z right parenthesis
t h e n space x over y plus y over z plus z over x plus k space space is space equal space to :  space

  1. 1 half
  2. 3 over 4
  3. negative 1 fourth
  4. negative 4

The correct answer is: 1 half

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L e t space alpha space a n d space beta space b e space t h e space r o o t s space o f space t h e space e q u a t i o n space x squared space plus space x space plus space 1 space equals space 0. space T h e n space f o r space y not equal to 0 space i n space R comma space open vertical bar table row cell y plus 1 end cell alpha beta row alpha cell y plus beta end cell 1 row beta 1 cell y plus alpha end cell end table close vertical bar space i s space e q u a l space t o space colon

L e t space alpha space a n d space beta space b e space t h e space r o o t s space o f space t h e space e q u a t i o n space x squared space plus space x space plus space 1 space equals space 0. space T h e n space f o r space y not equal to 0 space i n space R comma space open vertical bar table row cell y plus 1 end cell alpha beta row alpha cell y plus beta end cell 1 row beta 1 cell y plus alpha end cell end table close vertical bar space i s space e q u a l space t o space colon

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I f space t h e space f u n c t i o n space straight f colon straight R minus left curly bracket 1 comma negative 1 right curly bracket not stretchy rightwards arrow straight A space defined space by space straight f left parenthesis straight x right parenthesis equals fraction numerator straight x squared over denominator 1 minus straight x squared end fraction comma space is space surjective comma space then space straight A space is space equal space to space colon

I f space t h e space f u n c t i o n space straight f colon straight R minus left curly bracket 1 comma negative 1 right curly bracket not stretchy rightwards arrow straight A space defined space by space straight f left parenthesis straight x right parenthesis equals fraction numerator straight x squared over denominator 1 minus straight x squared end fraction comma space is space surjective comma space then space straight A space is space equal space to space colon

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If left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 1 row 0 1 end table right square bracket left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 2 row 0 1 end table right square bracket left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 3 row 0 1 end table right square bracket horizontal ellipsis horizontal ellipsis. left square bracket table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row 1 cell n minus 1 end cell row 0 1 end table right square bracket equals left square bracket table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row 1 78 row 0 1 end table right square bracket then the inverse of  left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 n row 0 1 end table right square bracket is

If left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 1 row 0 1 end table right square bracket left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 2 row 0 1 end table right square bracket left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 3 row 0 1 end table right square bracket horizontal ellipsis horizontal ellipsis. left square bracket table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row 1 cell n minus 1 end cell row 0 1 end table right square bracket equals left square bracket table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row 1 78 row 0 1 end table right square bracket then the inverse of  left square bracket table attributes columnalign left left columnspacing 1em rowspacing 4 pt end attributes row 1 n row 0 1 end table right square bracket is

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