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

2019 · 10 Jan · Shift 1 · Q24
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. /2019 · 10 Jan · Shift 1 · Q24

Matrices and Determinants question

2019 · 10 Jan · Shift 1 · Q24

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1
If the system of equations x + y + z = 5 x + 2y + 3z = 9 x + 3y + az = β\betaβ has infinitely many solutions, then β−α\beta -\alphaβ−α equals -
  1. A
    8
  2. B
    21
  3. C
    18
  4. D
    5
View written solutionFree

Correct answer: A

  1. Write the system in augmented matrix form:
{x+y+z=5x+2y+3z=9x+3y+αz=β\begin{cases} x+y+z=5 \\ x+2y+3z=9 \\ x+3y+\alpha z=\beta \end{cases}⎩⎨⎧​x+y+z=5x+2y+3z=9x+3y+αz=β​

For infinitely many solutions, the third equation must be a linear combination of the first two, and the system must be consistent with rank <3<3<3.

  1. First eliminate xxx by subtracting the first equation from the second and third:
  • Equation (2)−(1)(2)-(1)(2)−(1):
y+2z=4y+2z=4y+2z=4
  • Equation (3)−(1)(3)-(1)(3)−(1):
2y+(α−1)z=β−52y+(\alpha-1)z=\beta-52y+(α−1)z=β−5

So we now have:

y+2z=4y+2z=4y+2z=4 2y+(α−1)z=β−52y+(\alpha-1)z=\beta-52y+(α−1)z=β−5
  1. For infinitely many solutions, these two equations in y,zy,zy,z must be dependent. So the second must be twice the first:

Doubling y+2z=4y+2z=4y+2z=4 gives

2y+4z=82y+4z=82y+4z=8

Comparing with

2y+(α−1)z=β−52y+(\alpha-1)z=\beta-52y+(α−1)z=β−5

we get:

α−1=4⇒α=5\alpha-1=4 \Rightarrow \alpha=5α−1=4⇒α=5

and

β−5=8⇒β=13\beta-5=8 \Rightarrow \beta=13β−5=8⇒β=13
  1. Therefore,
β−α=13−5=8\beta-\alpha = 13-5 = 8β−α=13−5=8
  1. Comparing with the options, the correct choice is:

8\boxed{8}8​

So option A is correct.

PreviousNext

More from Matrices and Determinants

  • Let d ∈ R, and A=​−215​4+d(sinθ)+2(2sinθ)−d​(sinθ)−2d(−sinθ)+2+2d​​,θ∈[0,2π]…2019 · MCQ
  • The number of values of θ∈(0, π) for which the system of linear equations x + 3y + 7z = 0 − x + 4y + 7z = 0 (sin3 θ)x + (cos2 θ)y + 2z = 0. has a non-trival solution, is -2019 · MCQ
  • Let A = ​2b1​bb2+1b​1b2​​ where b > 0. Then the minimum value of bdet(A)​ is -2019 · MCQ
  • Let A = ​0pp​2qq−q​r−rr​​. If AAT = I3, then ∣p∣ is :2019 · MCQ
  • If the system of linear equations 2x + 2y + 3z = a 3x – y + 5z = b x – 3y + 2z = c where a, b, c are non zero real numbers, has more one solution, then :2019 · MCQ
  • If ​a−b−c2b2c​2ab−c−a2c​2a2bc−a−b​​= (a + b + c) (x + a + b + c)2, x e 0, then x is equal to :2019 · MCQ
  • If B=​50α​2α23​11−1​​ is the inverse of a 3 × 3 matrix A, then the sum of all values of α for which det(A) + 1 = 0,…2019 · MCQ
  • If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = [25​3−1​], then AB is equal to :2019 · MCQ