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3 and 4 .Determinants and Matrices
medium
જો $x = cy + bz,\,\,y = az + cx,\,\,z = bx + ay$ (કે જ્યાં $ x, y, z $ બધા શૂન્ય ન હોય) તો $x = 0$, $y = 0$, $z = 0$ સિવાય નો ઉકેલ હોય તો $ a, b $ અને $c$ વચ્ચેનો સંબંધ મેળવો.
A
${a^2} + {b^2} + {c^2} + 3abc = 0$
B
${a^2} + {b^2} + {c^2} + 2abc = 0$
C
${a^2} + {b^2} + {c^2} + 2abc = 1$
D
${a^2} + {b^2} + {c^2} - bc - ca - ab = 1$
(IIT-1978)
Solution
(c) The system of homogeneous equations
$x – cy – bz = 0$
$cx – y + az = 0$
$bx + ay – z = 0$
has a non-trivial solution (since $x,\,y,\,z$ are not all zero)
If $\Delta = \left| {\,\begin{array}{*{20}{c}}1&{ – c}&{ – b}\\c&{ – 1}&a\\b&a&{ – 1}\end{array}\,} \right|\, = 0$
i.e., if $(1 – {a^2})\, + c( – c – ab) – b(ac + b) = 0$
i.e., if ${a^2} + {b^2} + {c^2} + 2abc = 1$.
Standard 12
Mathematics
