QJEEM22Ma95ad: JEE Main 2022 Mathematics - Class 12 JEEM22Ma95ad Solution: The number of values of for which the system of equations : x + …

Question

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The number of values of $\alpha$ for which the system of equations :

x + y + z = $\alpha$

$\alpha$ x + 2 $\alpha$ y + 3z = $-$ 1

x + 3 $\alpha$ y + 5z = 4

is inconsistent, is

Accepted Answer

\Delta=\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ \alpha &amp; 2 \alpha &amp; 3 \\ 1 &amp; 3 \alpha &amp; 5\end{array}\right|

\begin{aligned}&amp;=1(10 \alpha-9 \alpha)-1(5 \alpha-3)+1\left(3 \alpha^{2}-2 \alpha\right) \\\\&amp;=\alpha-5 \alpha+3+3 \alpha^{2}-2 \alpha \\\\&amp;=3 \alpha^{2}-6 \alpha+3\end{aligned}

For inconsistency

\Delta=0

i.e.

\alpha=1

Now check for

\alpha=1

x+y+z=1\quad\quad...(i)

x+2 y+3 z=-1\quad\quad...(ii)

x+3 y+5 z=4\quad\quad...(iii)

By (ii)

\times 2-

(i)

\times 1

x+3 y+5 z=-3

so equations are inconsistent for

\alpha=1

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