The number of values of [] for which the system of equations : x + y + z = [] []x + 2[]y +…

Question

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

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

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

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 $\alpha=1$

Solutions · 1