JEE MAIN - Mathematics (2021 - 27th July Morning Shift - No. 17)
For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :
x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y + z = $$\beta$$. If the system has infinite solutions, then $$\alpha$$ + $$\beta$$ is equal to ______________.
x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y + z = $$\beta$$. If the system has infinite solutions, then $$\alpha$$ + $$\beta$$ is equal to ______________.
Answer
5
Explanation
For infinite solutions
$$\Delta$$ = $$\Delta$$1 = $$\Delta$$2 = $$\Delta$$3 = 0
$$\Delta$$ = $$\left| {\matrix{ 1 & 1 & { - 1} \cr 1 & 2 & \alpha \cr 2 & { - 1} & 1 \cr } } \right| = 0$$
$$\Delta = \left| {\matrix{ 3 & 0 & 0 \cr 1 & 2 & \alpha \cr 2 & { - 1} & 1 \cr } } \right| = 0$$
$$\Delta$$ = 3(2 + $$\alpha$$) = 0
$$\Rightarrow$$ $$\alpha$$ = $$-$$2
$${\Delta _2} = \left| {\matrix{ 1 & 2 & { - 1} \cr 1 & 1 & { - 2} \cr 2 & \beta & 1 \cr } } \right| = 0$$
1(1 + 2$$\beta$$) $$-$$2(1 + 4) $$-$$ ($$\beta$$ $$-$$ 2) = 0
$$\beta$$ $$-$$ 7 = 0
$$\beta$$ = 7
$$\therefore$$ $$\alpha$$ + $$\beta$$ = 5
$$\Delta$$ = $$\Delta$$1 = $$\Delta$$2 = $$\Delta$$3 = 0
$$\Delta$$ = $$\left| {\matrix{ 1 & 1 & { - 1} \cr 1 & 2 & \alpha \cr 2 & { - 1} & 1 \cr } } \right| = 0$$
$$\Delta = \left| {\matrix{ 3 & 0 & 0 \cr 1 & 2 & \alpha \cr 2 & { - 1} & 1 \cr } } \right| = 0$$
$$\Delta$$ = 3(2 + $$\alpha$$) = 0
$$\Rightarrow$$ $$\alpha$$ = $$-$$2
$${\Delta _2} = \left| {\matrix{ 1 & 2 & { - 1} \cr 1 & 1 & { - 2} \cr 2 & \beta & 1 \cr } } \right| = 0$$
1(1 + 2$$\beta$$) $$-$$2(1 + 4) $$-$$ ($$\beta$$ $$-$$ 2) = 0
$$\beta$$ $$-$$ 7 = 0
$$\beta$$ = 7
$$\therefore$$ $$\alpha$$ + $$\beta$$ = 5
Comments (0)
