JEE Main 2023 · 11 April · evening · Single correct · +4 / −1

If the system of linear equations
7 x+11 y+α z=13 ; ; 5 x+4 y+7 z=β ; ; 175 x+194 y+57 z=361
has infinitely many solutions, then α+β+2 is equal to :

  1. (a) 6
  2. (b) 4
  3. (c) 5
  4. (d) 3
Answer and solution

Answer (b) 4

Solution

Maths sir · 4 steps

Infinitely many solutions means the third equation is just a mix of the first two. Find that mix.

  1. What infinite means. The first two equations are independent, since 7/5 ≠ 11/4. For infinitely many solutions, the third equation must add no new condition. So it must equal a times the first plus b times the second, with the right-hand sides matching as well.
  2. Match x and y. Matching the x and y coefficients gives 7a + 5b = 175 and 11a + 4b = 194. Multiply the first by 4 and the second by 5, then subtract: 27a = 270. So a = 10, and then 5b = 175 - 70 = 105 gives b = 21.

    10 (7x + 11y + α z - 13) + 21 (5x + 4y + 7z - β) = 175x + 194y + 57z - 361

  3. Match z and constants. For z: 10α + 147 = 57, so α = -9. For the constants: 130 + 21β = 361, so 21β = 231 and β = 11.
  4. So. α + β + 2 = -9 + 11 + 2 = 4, which is option (b).

The idea to keep. For infinitely many solutions in 3 variables, write one equation as a times one plus b times another. Find a, b from two coefficients, then match the rest, including the constants.

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

Study Determinants

Asked about 190 times in JEE Main since 2020. This question is from 2023.