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Determinants

Question
CBSEENMA12034350
Wired Faculty App

Let space space space space space straight A equals space open square brackets table row 1 cell sin space straight theta end cell 1 row cell negative sin space straight theta end cell 1 cell sin space straight theta end cell row cell negative 1 end cell cell negative sin space straight theta end cell 1 end table close square brackets space comma space w h e r e space 0 space element of space ө space 2 space pi. space T h e n
  • (A) Det(A) = 0
  • (B) Det(A) ∈ (2, ∞)
  • (C) Det(A) ∈ (2, 4)
  • (D) Det(A) ∈ [2, 4]

Solution

D.

(D) Det(A) ∈ [2, 4] space space space space space straight A equals space open square brackets table row 1 cell space space space sin space straight theta end cell 1 row cell negative sin space straight theta end cell 1 cell space space sin space straight theta end cell row cell negative 1 end cell cell space minus sin space straight theta end cell 1 end table close square brackets open vertical bar space straight A space close vertical bar space equals space open vertical bar table row 1 cell space sin space straight theta end cell cell space space 1 end cell row cell negative sin space straight theta end cell cell space 1 end cell cell space space sin space straight theta end cell row cell negative 1 end cell cell space space minus sin space straight theta end cell 1 end table close vertical bar space space space space space space space space equals space space 1 cross times space open vertical bar table row 1 cell space space sin space straight theta end cell row cell negative sin space straight theta end cell 1 end table close vertical bar minus sin space straight theta space open vertical bar table row cell negative sin space straight theta end cell cell space space sin space straight theta end cell row cell negative 1 end cell 1 end table close vertical bar plus 1 space cross times space open vertical bar table row cell negative sin space straight theta end cell 1 row cell negative 1 end cell cell sin space straight theta end cell end table close vertical bar space space space space space space space space equals space space space 1 cross times left parenthesis 1 plus sin space straight theta squared space straight theta right parenthesis minus sin space straight theta space left parenthesis negative sin space straight theta space space plus sin space straight theta right parenthesis plus 1 cross times left parenthesis sin squared space straight theta plus 1 right parenthesis space space space space space space space equals space left parenthesis 1 plus sin squared space straight theta right parenthesis minus 0 plus left parenthesis sin squared space straight theta plus 1 right parenthesis equals 2 plus 2 space sin squared space straight theta space equals 2 space left parenthesis 1 plus sin squared straight theta right parenthesis we space know space that space space space space space space space space space space minus 1 space less or equal than space sin space straight theta space less or equal than 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta therefore space space space space space space space space space space space 0 space less or equal than space sin space to the power of 2 space end exponent less or equal than 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta rightwards double arrow space space space space space space 1 plus 0 space less or equal than space 1 plus sin space squared space straight theta less or equal than space 1 plus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta space space space space space space space space space space space rightwards double arrow space space space space space space 1 plus 0 space less or equal than space 1 plus sin space squared space straight theta less or equal than 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta space space space space space space space space rightwards double arrow space space space space space space 2 less or equal than 2 space left parenthesis 1 plus sin space squared space straight theta right parenthesis less or equal than 4 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta space space space space space space space space rightwards double arrow space space space space space space 2 space less or equal than space space space open vertical bar space straight A space close vertical bar space less or equal than 4 therefore space left parenthesis straight D right parenthesis space is space correct space answer. space space space space space space space space

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