NIMCET 2020 — Mathematics PYQ

Concept:
Cramer's rule for Linear Equations of Three Variables:
$a_{1}x+b_{1}y+c_{1}z=d_1,a_{2}x+b_{2}y+c_{2}z=d_2,a_{3}x+b_{3}y+c_{3}z=d_3$
$D=\left|\begin{array}{ccc}{a}_1& amp;a_2& amp;a_3\\ b_1& amp;b_2& amp;b_3\\ c_1& amp;c_2& amp;c_3\end{array}\right|$
$D_x=\left|\begin{array}{ccc}{d}_1& amp;a_2& amp;a_3\\ b_1& amp;b_2& amp;b_3\\ c_1& amp;c_2& amp;c_3\end{array}\right|$ $D_y=\left|\begin{array}{ccc}{a}_1& amp;d_2& amp;a_3\\ b_1& amp;b_2& amp;b_3\\ c_1& amp;c_2& amp;c_3\end{array}\right|$ $D_z=\left|\begin{array}{ccc}{a}_1& amp;a_2& amp;a_3\\ d_1& amp;b_2& amp;b_3\\ d_1& amp;c_2& amp;c_3\end{array}\right|$
- If $D\ne 0$: a unique solution (consistent).
The solution is: $x=\frac{D_x}{D}$, $y=\frac{D_y}{D}$, $z=\frac{D_z}{D}$.
- If $D=0$: either infinitely many solutions (consistent and dependent) or no solution (inconsistent). To find out if the system is dependent or inconsistent, another method, such as elimination or Rouché–Capelli theorem, will have to be used.