QJEEM23Ma7997: JEE Main 2023 Mathematics - Class 12 JEEM23Ma7997 Solution: If the system of equations has infinitely many solutions, then…

Question

Verified Solution • 2025–26 Session moderate

If the system of equations

$x+2y+3z=3$

$4x+3y-4z=4$

$8x+4y-\lambda z=9+\mu$

has infinitely many solutions, then the ordered pair ( $\lambda,\mu$ ) is equal to :

Accepted Answer

For infinite many solution,

\Delta=0

and

\Delta_x=0

\begin{aligned}&amp; \Delta=\left|\begin{array}{ccc}1 &amp; 2 &amp; 3 \\4 &amp; 3 &amp; -4 \\8 &amp; 4 &amp; -\lambda\end{array}\right|=0 \\\\&amp; \Rightarrow 1(-3 \lambda+16)-2(-4 \lambda+32)+3(16-24)=0 \\\\&amp; \Rightarrow 16-3 \lambda+8 \lambda-64-24=0 \Rightarrow 5 \lambda=72 \\\\&amp; \therefore \lambda=\frac{72}{5} \\\\&amp; \Delta_x=\left|\begin{array}{ccc}3 &amp; 2 &amp; 3 \\4 &amp; 3 &amp; -4 \\9+\mu &amp; 4 &amp; -\lambda\end{array}\right|=0\end{aligned}

\begin{aligned}&amp; \Rightarrow \quad 3(-3 \lambda+16)-2(-4 \lambda+36+4 \mu)+3(16-27-3 \mu)=0 \\\\&amp; \Rightarrow-9 \lambda+48+8 \lambda-72-8 \mu-33-9 \mu=0 \\\\&amp; \Rightarrow-\lambda-17 \mu=57 \\\\&amp; \Rightarrow-17 \mu=57+\lambda \\\\&amp; \therefore -\mu=\frac{57+\frac{72}{5}}{17}\\\\&amp; \Rightarrow \mu=\frac{-357}{85}=\frac{-21}{5}\end{aligned}

\text { Thus, }(\lambda, \mu) \equiv\left(\frac{72}{5}, \frac{-21}{5}\right)

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