JEE MAIN - Mathematics (2021 - 27th August Morning Shift - No. 16)
If the system of linear equations
2x + y $$-$$ z = 3
x $$-$$ y $$-$$ z = $$\alpha$$
3x + 3y + $$\beta$$z = 3
has infinitely many solution, then $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ is equal to _____________.
2x + y $$-$$ z = 3
x $$-$$ y $$-$$ z = $$\alpha$$
3x + 3y + $$\beta$$z = 3
has infinitely many solution, then $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ is equal to _____________.
Answer
5
Explanation
2 $$\times$$ (i) $$-$$ (ii) $$-$$ (iii) gives :
$$-$$ (1 + $$\beta$$)z = 3 $$-$$ $$\alpha$$
For infinitely many solution
$$\beta$$ + 1 = 0 = 3 $$-$$ $$\alpha$$ $$\Rightarrow$$ ($$\alpha$$, $$\beta$$) = (3, $$-$$1)
Hence, $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ = 5
$$-$$ (1 + $$\beta$$)z = 3 $$-$$ $$\alpha$$
For infinitely many solution
$$\beta$$ + 1 = 0 = 3 $$-$$ $$\alpha$$ $$\Rightarrow$$ ($$\alpha$$, $$\beta$$) = (3, $$-$$1)
Hence, $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ = 5
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