QJEEM20Maebda: JEE Main 2020 Mathematics - Class 12 JEEM20Maebda Solution: If for some and in R, the intersection of the following three…

For planes to intersect on a line there should be infinite solution of the given system of equations.

For infinite solutions

$\Delta$ = $$\left|\begin{array}{ccc}1& 4& -2\\ 1& 7& -5\\ 1& 5& \alpha \end{array}\right|$$ = 0

$\Rightarrow$ 1(7 $\alpha$ + 25) – 4( $\alpha$ + 5) – 2(5 – 7) = 0

$\Rightarrow$ 7 $\alpha$ + 25 – 4 $\alpha$ – 20 + 4 = 0

$\Rightarrow$ 3 $\alpha$ + 9 = 0

$\Rightarrow$ $\alpha$ = -3

Also $\Delta$ _z = 0

$\Rightarrow$ $$\left|\begin{array}{ccc}1& 4& 1\\ 1& 7& \beta \\ 1& 5& 5\end{array}\right|$$ = 0

$\Rightarrow$ 1(35 – 5 $\beta$ ) – 4(5 – $\beta$ ) + 1(5 – 7) = 0

$\Rightarrow$ 35 - 5 $\beta$ - 20 + 4 $\beta$ - 2 = 0

$\Rightarrow$ $\beta$ = 13

$\therefore$ $\alpha$ + $\beta$ = -3 + 13 = 10