JEE Main 2022 · 29 July · evening · Single correct · +4 / −1

If the system of equations
x+y+z=6 ; 2 x+5 y+α z=β ; x+2 y+3 z=14
has infinitely many solutions, then α+β is equal to

  1. (a) 8
  2. (b) 36
  3. (c) 44
  4. (d) 48
Answer and solution

Answer (c) 44

Solution

Maths sir · 4 steps

Infinitely many solutions means the middle equation must be a mix of the other two, so find that mix.

  1. What infinite means. Three planes meet along a whole line only when one equation adds no new information. Here the first and third equations are fixed, so the second must be a combination of them.
  2. Match x and y. Write equation 2 as p times equation 1 plus q times equation 3. Matching x gives p + q = 2 and matching y gives p + 2q = 5, so q = 3 and p = -1.

    -1(x+y+z=6) + 3(x+2y+3z=14) ⇒ 2x+5y+8z=36

  3. Read off the values. Comparing with 2x+5y+α z=β gives α = -1 + 9 = 8 and β = -6 + 42 = 36. As a check, Δ = 8 - α = 0 and Δ_z = 36 - β = 0 give the same values.
  4. So. α + β = 8 + 36 = 44, which is option (c).

The idea to keep. For infinitely many solutions, Δ = 0 fixes the coefficient and Δₓ = Δ_y = Δ_z = 0 fixes the constant. Writing one equation as a combination of the others does both at once.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 190 times in JEE Main since 2020. This question is from 2022.