If the system of linear equations \(7 x+11 y+\alpha z=13\) \(5 x+4 y+7 z=\beta\) \(175 x+194 y+57 z=361\) has infinitely many solutions, then \(\alpha+\beta+2\) is equal to

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Matrices

If the system of linear equations

7 x + 11 y + α z = 13

5 x + 4 y + 7 z = β

175 x + 194 y + 57 z = 361

has infinitely many solutions, then

α + β + 2

is equal to

(A)4✓ Correct
(B)3
(C)5
(D)6

Answer: (A)

Solution

For infinitely many solutions, the third equation must be a linear combination of the first two. Multiply equation (i) by 10 and equation (ii) by 21, then subtract equation (iii):

10 (7 x + 11 y + α z) + 21 (5 x + 4 y + 7 z) - (175 x + 194 y + 57 z) = 10 (13) + 21 β - 361

70 x + 110 y + 10 α z + 105 x + 84 y + 147 z - 175 x - 194 y - 57 z = 130 + 21 β - 361

0 + 0 + (10 α + 147 - 57) z = 130 + 21 β - 361

(10 α + 90) z = 21 β - 231

For infinitely many solutions, both sides must be zero:

10 α + 90 = 0 \Rightarrow α = - 9

21 β - 231 = 0 \Rightarrow β = 11

Thus

α + β + 2 = - 9 + 11 + 2 = 4

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Single correct option

Practise more: all JEE Main PYQs on Matrices →

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Matrices

Solution

More JEE Main questions on Matrices