Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Matrices and Determinants question

2024 · 8 Apr · Shift 2 · Q39
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Matrices and Determinants
  5. /2024 · 8 Apr · Shift 2 · Q39

Matrices and Determinants question

2024 · 8 Apr · Shift 2 · Q39

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1
If the system of equations x+4y−z=λ,7x+9y+μz=−3,5x+y+2z=−1x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1x+4y−z=λ,7x+9y+μz=−3,5x+y+2z=−1 has infinitely many solutions, then (2μ+3λ)(2 \mu+3 \lambda)(2μ+3λ) is equal to :
  1. A
    −2-2−2
  2. B
    2
  3. C
    3
  4. D
    −3-3−3
View written solutionFree

Correct answer: D

  1. Write the system in matrix form:
{x+4y−z=λ7x+9y+μz=−35x+y+2z=−1\begin{cases} x+4y-z=\lambda \\ 7x+9y+\mu z=-3 \\ 5x+y+2z=-1 \end{cases}⎩⎨⎧​x+4y−z=λ7x+9y+μz=−35x+y+2z=−1​

The coefficient matrix is

A=(14−179μ512)A=\begin{pmatrix} 1&4&-1\\ 7&9&\mu\\ 5&1&2 \end{pmatrix}A=​175​491​−1μ2​​

For the system to have infinitely many solutions, we need:

  • det⁡(A)=0\det(A)=0det(A)=0,
  • and the system must be consistent, i.e. the dependent equations must also satisfy the same dependence on the constants.
  1. First compute det⁡(A)\det(A)det(A):
det⁡(A)=∣14−179μ512∣\det(A)=\begin{vmatrix} 1&4&-1\\ 7&9&\mu\\ 5&1&2 \end{vmatrix}det(A)=​175​491​−1μ2​​

Expanding along the first row,

det⁡(A)=1∣9μ12∣−4∣7μ52∣+(−1)∣7951∣\det(A)=1\begin{vmatrix}9&\mu\\1&2\end{vmatrix}-4\begin{vmatrix}7&\mu\\5&2\end{vmatrix}+(-1)\begin{vmatrix}7&9\\5&1\end{vmatrix}det(A)=1​91​μ2​​−4​75​μ2​​+(−1)​75​91​​ =1(18−μ)−4(14−5μ)−1(7−45)=1(18-\mu)-4(14-5\mu)-1(7-45)=1(18−μ)−4(14−5μ)−1(7−45) =18−μ−56+20μ+38=18-\mu-56+20\mu+38=18−μ−56+20μ+38 =19μ=19\mu=19μ

For infinitely many solutions,

19μ=0  ⟹  μ=019\mu=0 \implies \mu=019μ=0⟹μ=0
  1. Now substitute μ=0\mu=0μ=0 into the system:
{x+4y−z=λ7x+9y=−35x+y+2z=−1\begin{cases} x+4y-z=\lambda \\ 7x+9y=-3 \\ 5x+y+2z=-1 \end{cases}⎩⎨⎧​x+4y−z=λ7x+9y=−35x+y+2z=−1​

Since infinitely many solutions require one equation to be dependent on the other two, let us check whether the second row is a linear combination of the first and third rows.

Let

R2=aR1+bR3R_2=aR_1+bR_3R2​=aR1​+bR3​

Then comparing coefficients of x,y,zx,y,zx,y,z:

a+5b=7a+5b=7a+5b=7 4a+b=94a+b=94a+b=9 −a+2b=0-a+2b=0−a+2b=0

From

−a+2b=0  ⟹  a=2b-a+2b=0 \implies a=2b−a+2b=0⟹a=2b

Substitute into a+5b=7a+5b=7a+5b=7:

2b+5b=7  ⟹  7b=7  ⟹  b=12b+5b=7 \implies 7b=7 \implies b=12b+5b=7⟹7b=7⟹b=1

Hence

a=2a=2a=2

Check in the yyy-equation:

4a+b=4(2)+1=94a+b=4(2)+1=94a+b=4(2)+1=9

Correct.

So indeed,

R2=2R1+R3R_2=2R_1+R_3R2​=2R1​+R3​

For consistency, the constants must satisfy the same relation:

−3=2λ+(−1)-3=2\lambda+(-1)−3=2λ+(−1) −3=2λ−1-3=2\lambda-1−3=2λ−1 2λ=−22\lambda=-22λ=−2 λ=−1\lambda=-1λ=−1
  1. Now compute:
2μ+3λ=2(0)+3(−1)=−32\mu+3\lambda=2(0)+3(-1)=-32μ+3λ=2(0)+3(−1)=−3
  1. Evaluate options:
  • A: −2-2−2 ❌
  • B: 222 ❌
  • C: 333 ❌
  • D: −3-3−3 ✅

Therefore, the correct answer is

−3\boxed{-3}−3​
PreviousNext

More from Matrices and Determinants

  • Let λ,μ∈R. If the system of equations ​3x+5y+λz=37x+11y−9z=297x+155y−189z=μ​ has infinitely many solutions, then μ+2λ is equal to…2024 · MCQ
  • Let A be a non-singular matrix of order 3. If det(3adj(2adj((detA)A)))=3−13⋅2−10 and det(3adj(2A))=2m⋅3n…2024 · Numerical
  • Let B=[11​35​] and A be a 2×2 matrix such that AB−1=A−1. If BCB−1=A and C4+αC2+βI=O, then 2β−α is equal to2024 · MCQ
  • Consider the matrices : A=[23​−5m​],B=[20m​] and X=[xy​]. Let the set of all m, for which…2024 · Numerical
  • Consider the matrix f(x)=​cosxsinx0​−sinxcosx0​001​​. Given below are two statements : Statement I : f(−x) is the inverse of the matrix f(x)…2024 · MCQ
  • Let A=​211​010​101​​,B=[B1​,B2​,B3​], where B1​,B2​,B3​ are column matrics, and AB1​=​100​​,AB2​=​230​​,AB3​=​321​​…2024 · Numerical
  • The values of α, for which ​112α+3​23​31​3α+1​α+23​α+31​0​​=0, lie in the interval2024 · MCQ
  • Let A be a 2×2 real matrix and I be the identity matrix of order 2. If the roots of the equation ∣A−xI∣=0 be −1 and 3, then the sum of the diagonal elements of the matrix A2 is2024 · Numerical