Q120 / 170 · Manipal (MET) 2017 · Mathematics

The system $x+4 y-2 z=3,3 x+y+5 z=7$, $2 x+3 y+z=5$ has

  • A
    infinite number of solutions
  • B
    unique solution
  • C
    trivial solution
  • Correct answer
    D
    no solution

Solution

Given system of equations are and $\begin{aligned} & x+4 y-2 z=3 \\ & 3 x+y+5 z=7 \\ & 2 x+3 y+z=5 \end{aligned}$ $\begin{aligned} \therefore \Delta & =\left|\begin{array}{ccc} 1 & 4 & -2 \\ 3 & 1 & 5 \\ 2 & 3 & 1 \end{array}\right| \\ & =1(1-15)-4(3-10)-2(9-2) \\ & =-14+28-14=0 \\ \text { and } \Delta_2 & =\left|\begin{array}{lll} 1 & 3 & -2 \\ 3 & 7 & 5 \\ 2 & 5 & 1 \end{array}\right| \\ & =1 \neq 0 \end{aligned}$ $\therefore$ No solution will exist.

Hence, the correct answer is option: 'D'.

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