Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20257 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
7 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Determinants

JEE Main 2025 · 7 April · Evening Shift · Mathematics · Determinants

Let the system of equations aligned & x+5 y-z=1 & 4 x+3 y-3 z=7 &...

Let the system of equations
<br/>x+5y−z=14x+3y−3z=724x+y+λz=μ<br/>\begin{aligned}<br/>& x+5 y-z=1 \\ & 4 x+3 y-3 z=7 \\ & 24 x+y+\lambda z=\mu<br/>\end{aligned}
λ,μ∈R\lambda, \mu \in \mathrm{R}, have infinitely many solutions. Then the number of the solutions of this system, If x,y,zx, y, z are integers and satisfy 7≤x+y+z≤777 \leq x+y+z \leq 77, is

Options

  1. A3
  2. B6
  3. C5
  4. D4

Correct answer

A. 3

Step-by-step solution

For infinitely many solutionΔ=0∣15−143−3241λ∣=0⇒1(3λ+3)−5(4λ+72)−1(4−72)=0⇒−17λ+3−4×72−4=0⇒17λ=−289\begin{aligned}& \Delta=0 \\ & \left|\begin{array}{ccc}1 & 5 & -1 \\ 4 & 3 & -3 \\ 24 & 1 & \lambda\end{array}\right|=0 \\ & \Rightarrow 1(3 \lambda+3)-5(4 \lambda+72)-1(4-72)=0 \\ & \Rightarrow-17 \lambda+3-4 \times 72-4=0 \\ & \Rightarrow 17 \lambda=-289\end{aligned}Δ1=0⇒∣15−173−3μ1−17∣=0⇒1(−51+3)−5(−119+3μ)−1(7−3μ)=0⇒−48+595−15μ−7+3μ=0⇒12μ=540\begin{aligned} & \Delta 1=0 \\ & \Rightarrow\left|\begin{array}{ccc}1 & 5 & -1 \\ 7 & 3 & -3 \\ \mu & 1 & -17\end{array}\right|=0 \\ & \Rightarrow 1(-51+3)-5(-119+3 \mu)-1(7-3 \mu)=0 \\ & \Rightarrow-48+595-15 \mu-7+3 \mu=0 \\ & \Rightarrow 12 \mu=540\end{aligned}x+5y−z=14x+3y−3z=724x+y−17z=45 Let z=1x+5y=1+λ]×44x+3y=7+3λ4x+20y=4+4λ−17y=3−λ−17\begin{aligned} & x+5 y-z=1 \\ & 4 x+3 y-3 z=7 \\ & 24 x+y-17 z=45 \\ & \text{ Let } z=1 \\ & x+5 y=1+\lambda] \times 4 \\ & 4 x+3 y=7+3 \lambda \\ & 4 x+20 y=4+4 \lambda \\ & \frac{-17 y=3-\lambda}{-17}\end{aligned} \begin{aligned} & \begin{aligned} \mathrm{y} & =\frac{\lambda-3}{17}, \mathrm{x}=1+\lambda-\frac{5 \lambda-15}{17} \\ & =\frac{32-12 \lambda}{17} \\ 7 & \leq \frac{\lambda-3}{17}+\frac{32+12 \lambda}{17}+\lambda \leq 77 \\ 7 & \leq \frac{30 \lambda+29}{17} \leq 77 \\ 3 & \leq \lambda \leq 42 \\ \lambda & =3,20,37\end{aligned}\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 2026The sum of all possible values of θ ∈ [0, 2π] , for which the system of equations : x 3θ… JEE Main 2026If f: N arrow Z is defined by f(n) = vmatrix n & -1 & -5 -2n^2 & 3(2k+1) & 2k+1 -3n^3 &… JEE Main 2026Consider the system of linear equations in x, y, z : x + 2y + tz = 0 , 6x + y + 5tz = 0… JEE Main 2026If the system of equations: x+y+z=5 x+2y+3z=9 x+3y+λ z=μ has infinitely many solutions… JEE Main 2026If the system of equations x + 5y + 6z = 4 , 2x + 3y + 4z = 7 , x + 6y + az = b has… JEE Main 2026Let α, β ∈ ℝ be such that the system of linear equations x + 2y + z = 5 2x + y + α z = 5… JEE Main 2026The system of linear equations x+y+z=6 2 x+5 y+a z=36 x+2 y+3 z=b has JEE Main 2026Among the statements : I: If | array ccc 1 & α & β α & 1 & γ β & γ & 1 array |=| array… JEE Main 2026Let n be the number obtained on rolling a fair die. If the probability that the system… JEE Main 2026If 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 2026Let 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 2025If y(x)=| array ccc x & x & x+ x+1 27 & 28 & 27 1 & 1 & 1 array |, x ∈ ℝ , then (d^2… JEE Main 2025If 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 2025If 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 2025Let α, β(α ≠ β) be the values of m , for which the equations x+y+z=1 ; x+2 y+4 z=m and… JEE Main 2025 Full Determinants list Determinants JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs