Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Determinants

NDA 2016 · Mathematics · Determinants

Paragraph: For the system of linear equations 2 x+3 y+5 z=9,7 x+3...

Paragraph: For the system of linear equations 2x+3y+5z=9,7x+3y−2z=82 x+3 y+5 z=9,7 x+3 y-2 z=8 and 2x+3y+λz=μ2 x+3 y+\lambda z=\mu
Question: Under what condition does the above system of equations have infinitely many solutions?

Options

  1. Aλ=5\lambda=5 and μ≠9\mu \neq 9
  2. Bλ=5\lambda=5 and μ=9\mu=9
  3. Cλ=9\lambda=9 and μ=5\mu=5
  4. Dλ=9\lambda=9 and μ≠5\mu \neq 5

Correct answer

B. λ=5\lambda=5 and μ=9\mu=9

Step-by-step solution

A=[23573−223λ] and B=[98μ](Adj⁡A)=[3λ+615−3λ−21−(7λ+4)2λ−1039150−15]\begin{aligned} & A=\left[\begin{array}{ccc} 2 & 3 & 5 \\ 7 & 3 & -2 \\ 2 & 3 & \lambda \end{array}\right] \text{ and } B=\left[\begin{array}{l} 9 \\ 8 \\ \mu \end{array}\right] \\ & (\operatorname{Adj} A)=\left[\begin{array}{ccc} 3 \lambda+6 & 15-3 \lambda & -21 \\ -(7 \lambda+4) & 2 \lambda-10 & 39 \\ 15 & 0 & -15 \end{array}\right] \end{aligned} For infinitely many solutions : (Adj⁡A)B=0⇒(\operatorname{Adj} A) B=0 \Rightarrow\left[27λ+54+120−24λ−21μ−63λ−36+16λ−80+39μ135+0−15μ\begin{array}{c}27 \lambda+54+120-24 \lambda-21 \mu \\ -63 \lambda-36+16 \lambda-80+39 \mu \\ 135+0-15 \mu\end{array}\right]=\left[000\begin{array}{l}0 \\ 0 \\ 0\end{array}\right] Hence, μ=9\mu=9 and λ=5\lambda=5.
Practice Determinants on Quantrex Academy →

More from Determinants

Let p=(x+y+z) and q=xyz . If vmatrix x and 1 and 1 1 and y and 1 1 and 1 and z vmatrix is… NDA 2026Let f(x)= vmatrix 3x^2 and x and - x 6 and -1 and 0 q and q^2 and q^3 vmatrix where q is… NDA 2026If M= bmatrix 2 & 0 & 0 0 & 2 & 0 0 & 0 & 2 bmatrix , then what is the value of… NDA 2026If vmatrix a-b and p-q and x-y b-c and q-r and y-z c-a and r-p and z-x vmatrix =k vmatrix… NDA 2026If A is a square matrix such that |A|=-2 , then |AA^ T | , where A^ T is the transpose of… NDA 2026The system of equations 2x - 3y - 5 = 0 , 15y - 10x + 50 = 0 NDA 2025The value of the determinant vmatrix a and b and c l and m and n p and q and r vmatrix is… NDA 2025If a , b , c are the sides of a triangle ABC and p is the perimeter of the triangle, then… NDA 2025Consider the following statements: Statement-I: If X is an n × n matrix, then (mX) = m^n… NDA 2025If Δ = vmatrix a and b and c d and e and f g and h and i vmatrix and A , B , C , D , G… NDA 2025If A^2 + B^2 + C^2 = 0 , then what is the value of vmatrix 1 and C and B C and 1 and A B… NDA 2025Consider the following statements in respect of the determinant Δ = vmatrix k(k+2) and… NDA 2025If vmatrix 2 & 3+i & -1 3-i & 0 & i-1 -1 & -1-i & 1 vmatrix = A + iB where i = -1 , then… NDA 2025Consider the following in respect of a non-singular matrix M : I. |M^2| = |M|^2 II. |M| =… NDA 2025In obtaining the solution of the system of equations x + y + z = 7 , x + 2y + 3z = 16 and… NDA 2025If Z=(1)/(3)| array ccc i & 2 i & 1 2 i & 3 i & 2 3 & 1 & 3 array |=x+i y ; i= -1 then… NDA 2024 Full Determinants list Determinants JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs