JEE MAIN - Mathematics (2020 - 4th September Morning Slot - No. 4)
If the system of equations
x - 2y + 3z = 9
2x + y + z = b
x - 7y + az = 24,
has infinitely many solutions, then a - b is equal to.........
x - 2y + 3z = 9
2x + y + z = b
x - 7y + az = 24,
has infinitely many solutions, then a - b is equal to.........
Answer
5
Explanation
D = 0
$$\left| {\matrix{ 1 & { - 2} & 3 \cr 2 & 1 & 1 \cr 1 & { - 7} & a \cr } } \right| = 0$$
$$1(a + 7) + 2(2a - 1) + 3( - 14 - 1) = 0$$
$$a + 7 + 4a - 2 - 45 = 0$$
$$5a = 40$$
$$a = 8$$
$${D_1} = \left| {\matrix{ 9 & { - 2} & 3 \cr b & 1 & 1 \cr {24} & { - 7} & 8 \cr } } \right| = 0$$
$$ \Rightarrow 9(8 + 7) + 2(8b - 24) + 3( - 7b - 24) = 0$$
$$ \Rightarrow 135 + 16b - 48 - 21b - 72 = 0$$
$$ \Rightarrow $$ $$15 = 5b$$
$$ \Rightarrow b = 3$$
$$a - b = 5$$
$$\left| {\matrix{ 1 & { - 2} & 3 \cr 2 & 1 & 1 \cr 1 & { - 7} & a \cr } } \right| = 0$$
$$1(a + 7) + 2(2a - 1) + 3( - 14 - 1) = 0$$
$$a + 7 + 4a - 2 - 45 = 0$$
$$5a = 40$$
$$a = 8$$
$${D_1} = \left| {\matrix{ 9 & { - 2} & 3 \cr b & 1 & 1 \cr {24} & { - 7} & 8 \cr } } \right| = 0$$
$$ \Rightarrow 9(8 + 7) + 2(8b - 24) + 3( - 7b - 24) = 0$$
$$ \Rightarrow 135 + 16b - 48 - 21b - 72 = 0$$
$$ \Rightarrow $$ $$15 = 5b$$
$$ \Rightarrow b = 3$$
$$a - b = 5$$
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