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JEE Main 20249 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
9 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Determinants

JEE Main 2024 · 9 April · Morning Shift · Mathematics · Determinants

Let λ, μ ∈ R . If the system of equations aligned & 3 x+5 y+λ z=3 &...

Let λ,μ∈R\lambda, \mu \in \mathbf{R}. If the system of equations 3x+5y+λz=37x+11y−9z=297x+155y−189z=μ\begin{aligned} & 3 x+5 y+\lambda z=3 \\ & 7 x+11 y-9 z=2 \\ & 97 x+155 y-189 z=\mu \end{aligned} has infinitely many solutions, then μ+2λ\mu+2 \lambda is equal to :

Options

  1. A24
  2. B25
  3. C22
  4. D27

Correct answer

B. 25

Step-by-step solution

3x+5y+λz=37x+11y−9z=297x+155y−189z=μ93x+155y+31λz=9397x+155y−189z=μ−++−−4x+(31λ+189)z=93−μ1085x+1705y−1395z=3101067x+1705y−2079z=11μ−+−18x+684z=310−11μ\begin{array}{l} 3 x+5 y+\lambda z=3 \\ 7 x+11 y-9 z=2 \\ 97 x+155 y-189 z=\mu \\ 93 x+155 y+31 \lambda z=93 \\ 97 x+155 y-189 z=\mu \\ -\quad+\quad+\quad- \\ \hline-4 x+(31 \lambda+189) z=93-\mu \\ 1085 x+1705 y-1395 z=310 \\ 1067 x+1705 y-2079 z=11 \mu \\ -\quad+\quad-\quad \\ \hline 18 x+684 z=310-11 \mu \end{array}−36x+9(31λ+189)z=9(93−μ)36x+1368z=2(310−11μ)(279λ+3069)z=1457−31μ\begin{aligned} & -36 x+9(31 \lambda+189) z=9(93-\mu) \\ & 36 x+1368 z=2(310-11 \mu) \\ & (279 \lambda+3069) z=1457-31 \mu \end{aligned}forforinfinite solutions - λ=−3069279=−34131μ=145731\begin{aligned} & \lambda=\frac{-3069}{279}=\frac{-341}{31} \\ & \mu=\frac{1457}{31} \end{aligned} \mu+2 \lambda= \frac{1457-682}{31}=\frac{775}{31}=$25
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