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3 and 4 .Determinants and Matrices
hard
यदि $R$ में किन्हीं $\alpha$ तथा $\beta$ के लिए, निम्न तीन समतलों $x+4 y-2 z=1$, $x+7 y-5 z=\beta$, $x+5 y+\alpha z=5$ का प्रतिच्छेदन, $R ^{3}$ में एक रेखा है, तो $\alpha+\beta$ का मान है
A
$10$
B
$-10$
C
$2$
D
$0$
(JEE MAIN-2020)
Solution
For planes to intersect on a line
$\Rightarrow$ there should be infinite solution of the given system of equations for infinite solutions
$\Delta=\left|\begin{array}{ccc}{1} & {4} & {-2} \\ {1} & {7} & {-5} \\ {1} & {5} & {\alpha}\end{array}\right|=0 \Rightarrow 3 \alpha+9=0 \Rightarrow \alpha=-3$
$\Delta_{z}=\left|\begin{array}{ccc}{1} & {4} & {1} \\ {1} & {7} & {\beta} \\ {1} & {5} & {5}\end{array}\right|=0 \Rightarrow 13-\beta=0 \Rightarrow \beta=13$
Also for $\alpha=-3$ and $b=13 \Delta_{x}=\Delta_{y}=0$
$\alpha+\beta=-3+13=10$
Standard 12
Mathematics
