For the system of linear equations $x+y+z=6$ $\alpha x+\beta y+7z=3$ $x+2y+3z=14$, which of the following is NOT true?

(D) For every point $(\alpha, \beta) \neq(7,7)$ on the line $x-2y+7=0$, the system has infinitely many solutions.

(A) If $\alpha=\beta=7$, then the system has no solution

(B) If $\alpha=\beta$ and $\alpha \neq 7$ then the system has a unique solution.

(C) There is a unique point ($\alpha, \beta$) on the line $x+2y+18=0$ for which the system has infinitely many solutions

Matrices — JEE Main · 2023 · 31 Jan 2023 · Shift 1 Question with Solution

For the system of linear equations

x + y + z = 6

α x + β y + 7 z = 3

x + 2 y + 3 z = 14

,
which of the following is NOT true?

(A)If $α = β = 7$ , then the system has no solution
(B)If $α = β$ and $α \neq = 7$ then the system has a unique solution.
(C)There is a unique point ( $α, β$ ) on the line $x + 2 y + 18 = 0$ for which the system has infinitely many solutions
(D)For every point $(α, β) \neq = (7, 7)$ on the line $x - 2 y + 7 = 0$ , the system has infinitely many solutions.✓ Correct

Answer: (D)

Solution

Subtracting the first equation from the third gives

y + 2 z = 8

, so

y = 8 - 2 z

and

x = z - 2

. Substituting into the second equation,

α (z - 2) + β (8 - 2 z) + 7 z = 3

, which rearranges to

(α - 2 β + 7) z = 2 α - 8 β + 3

. The system has a unique solution when

α - 2 β + 7 \neq = 0

; no solution when

α - 2 β + 7 = 0

but

2 α - 8 β + 3 \neq = 0

; and infinitely many solutions only when

α - 2 β + 7 = 0

and

2 α - 8 β + 3 = 0

hold together. These two conditions meet at the single point

(α, β) = (- \frac{25}{2}, - \frac{11}{4})

, which indeed lies on

x + 2 y + 18 = 0

. On the line

x - 2 y + 7 = 0

, that is

α - 2 β + 7 = 0

, every point except this one makes

2 α - 8 β + 3 \neq = 0

and so gives no solution rather than infinitely many. The statement that every such point yields infinitely many solutions is therefore the false one.

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Single correct option

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Matrices

Solution

More JEE Main questions on Matrices