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The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9 x $$-$$ y + 4z = 8

JEE · Math · previous-year question

  1. A.does not have any solution
  2. B.has a solution ($$\alpha$$, $$\beta$$, $$\gamma$$) satisfying $$\alpha$$ + $$\beta$$2 + $$\gamma$$3 = 12
  3. C.has a unique solutioncorrect
  4. D.has infinitely many solutions

Answer

C. has a unique solution

Explanation

$$\Delta = \left| {\begin{matrix} 2 & 3 & 2 \\ 3 & 2 & 2 \\ 1 & { - 1} & 4 \\ \end{matrix} } \right| = - 20 \ne 0$$ $$ \therefore $$ unique solution $${\Delta _x} = \left| {\begin{matrix} 9 & 3 & 2 \\ 9 & 2 & 2 \\ 8 & { - 1} & 4 \\ \end{matrix} } \right| = 0$$ $${\Delta _y} = \left| {\begin{matrix} 2 & 9 & 2 \\ 3 & 9 & 2 \\ 1 & 8 & 4 \\ \end{matrix} } \right| = - 20$$ $${\Delta _z} = \left| {\begin{matrix} 2 & 3 & 9 \\ 3 & 2 & 9 \\ 1 & { - 1} & 8 \\ \end{matrix} } \right| = - 40$$ $$ \therefore $$ $$x = {{{\Delta _x}} \over \Delta } = 0$$ $$y = {{{\Delta _y}} \over \Delta } = 1$$ $$z = {{{\Delta _z}} \over \Delta } = 2$$ Unique solution : (0, 1, 2)

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