Let α β γ=45 ; α, β, γ ∈ ℝ . If x(α, 1,2)+y(1, β, 2)+z(2,3, γ)=(0,0,0) for some x, y, z ∈ ℝ, x y z ≠ 0 , then 6 α+4 β+γ is equal to _______

Correct answer: 0 aligned & α β γ=45, α β γ ∈ R & x(α, 1,2)+y(1, β, 2)+z(2,3, γ)=(0,0,0) & x, y, z ∈ R, xyz ≠ 0 & α x+y+2 z=0 & x+β y+3 z=0 & 2 x+2 y+γ z=0 & xyz ≠ 0 non-trivial & | array lll α & 1 & 2 1 & β & 3 2 & 2 & γ array |=0 aligned aligned & α(β γ-6)-1(γ-6)+2(2-2 β)=0 & α β γ-6 α-γ+6+4-4 β=0 & 6 α+4 β+γ=55 aligned

JEE Main 20246 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 6 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Determinants

JEE Main 2024 · 6 April · Morning Shift · Mathematics · Determinants

Let α β γ=45 ; α, β, γ ∈ ℝ . If x(α, 1,2)+y(1, β, 2)+z(2,3,...

Let

\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}

. If

x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)

for some

x, y, z \in \mathbb{R}, x y z \neq 0

, then

6 \alpha+4 \beta+\gamma

is equal to _______

Correct answer

Step-by-step solution

\begin{aligned} & \alpha \beta \gamma=45, \alpha \beta \gamma \in \mathrm{R} \\ & \mathrm{x}(\alpha, 1,2)+\mathrm{y}(1, \beta, 2)+\mathrm{z}(2,3, \gamma)=(0,0,0) \\ & \mathrm{x}, \mathrm{y}, \mathrm{z} \in \mathrm{R}, \mathrm{xyz} \neq 0 \\ & \alpha \mathrm{x}+\mathrm{y}+2 \mathrm{z}=0 \\ & \mathrm{x}+\beta \mathrm{y}+3 \mathrm{z}=0 \\ & 2 \mathrm{x}+2 \mathrm{y}+\gamma \mathrm{z}=0 \\ & \mathrm{xyz} \neq 0 \Rightarrow \text{ non-trivial } \\ & \left|\begin{array}{lll}\alpha & 1 & 2 \\ 1 & \beta & 3 \\ 2 & 2 & \gamma\end{array}\right|=0\end{aligned}

\begin{aligned} & \Rightarrow \alpha(\beta \gamma-6)-1(\gamma-6)+2(2-2 \beta)=0 \\ & \Rightarrow \alpha \beta \gamma-6 \alpha-\gamma+6+4-4 \beta=0 \\ & \Rightarrow 6 \alpha+4 \beta+\gamma=55\end{aligned}

Practice Determinants on Quantrex Academy →

More from Determinants

If the system of linear equations: x+y+z=6 , x+2y+5z=10 , 2x+3y+λ z=μ has infinitely many… JEE Main 2026 The sum of all possible values of θ ∈ [0, 2π] , for which the system of equations : x 3θ… JEE Main 2026 If f: N arrow Z is defined by f(n) = vmatrix n & -1 & -5 -2n^2 & 3(2k+1) & 2k+1 -3n^3 &… JEE Main 2026 Consider the system of linear equations in x, y, z : x + 2y + tz = 0 , 6x + y + 5tz = 0… JEE Main 2026 If the system of equations: x+y+z=5 x+2y+3z=9 x+3y+λ z=μ has infinitely many solutions… JEE Main 2026 If the system of equations x + 5y + 6z = 4 , 2x + 3y + 4z = 7 , x + 6y + az = b has… JEE Main 2026 Let α, β ∈ ℝ be such that the system of linear equations x + 2y + z = 5 2x + y + α z = 5… JEE Main 2026 The system of linear equations x+y+z=6 2 x+5 y+a z=36 x+2 y+3 z=b has JEE Main 2026 Among the statements : I: If | array ccc 1 & α & β α & 1 & γ β & γ & 1 array |=| array… JEE Main 2026 Let n be the number obtained on rolling a fair die. If the probability that the system… JEE Main 2026 If the system of equations 3 x+y+4 z=3 2 x+α y-z=-3 x+2 y+z=4 has no solution, then the… JEE Main 2026 Let the system of equations aligned & x+5 y-z=1 & 4 x+3 y-3 z=7 & 24 x+y+λ z=μ aligned λ… JEE Main 2025 Let the system of equations : aligned & 2 x+3 y+5 z=9 & 7 x+3 y-2 z=8 & 12 x+3 y-(4+λ)… JEE Main 2025 If y(x)=| array ccc x & x & x+ x+1 27 & 28 & 27 1 & 1 & 1 array |, x ∈ ℝ , then (d^2… JEE Main 2025 If the system of equation aligned & 2 x+λ y+3 z=5 & 3 x+2 y-z=7 & 4 x+5 y+μ z=9 aligned… JEE Main 2025 If the system of linear equations aligned & 3 x+y+β z=3 & 2 x+α y-z=-3 & x+2 y+z=4… JEE Main 2025 Full Determinants list Determinants JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs