Determinants

Maths · Class 12

Simulation · Maths · Class 12

Two lines, one determinant

"Two lines, one determinant" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Consistent and inconsistent systems" in Determinants. The blue line is x + 2y = 4. The orange line is ax + 4y = c. Together they are AX = B with A = [[1, 2], [a, 4]] and B = (4, c). Watch |A| = 4 − 2a as the lines cross, then stop crossing. |A| ≠ 0: the lines cross once and X = A⁻¹B. |A| = 0: parallel lines, and (adj A)B ≠ O proves there is no solution; (adj A)B = O leaves infinitely many or none. JEE Main asked 59 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson Consistent and inconsistent systems in Determinants. Change the values and watch what happens.

Two lines, one determinantMaths · Class 12

In JEE Main

42 shifts analysed

59 questions from Determinants across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §4.6

  • A system of equations is consistent if it has at least one solution and inconsistent if it has none.
  • The system a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃ is the single matrix equation AX = B, with A the coefficient matrix, X the column (x, y, z) and B the column (d₁, d₂, d₃).
  • If |A| ≠ 0, the system has exactly one solution.
  • If |A| = 0, compute (adj A)B. If (adj A)B ≠ O, there is no solution and the system is inconsistent.
  • If |A| = 0 and (adj A)B = O, the test is not decisive: the system may have infinitely many solutions or none.
  • In two variables the picture is two lines: |A| ≠ 0 means the lines cross at one point; |A| = 0 means they are parallel (no solution) or the same line (infinitely many).
  • Example: x + 2y = 4 and 2x + 4y = 10 have |A| = 4 − 4 = 0 and adj A = [[4, −2], [−2, 1]] times (4, 10) gives (adj A)B = (−4, 2) ≠ O, so the system is inconsistent: the lines are parallel.

Questions

3 answers

What does the "Two lines, one determinant" simulation show?
The blue line is x + 2y = 4. The orange line is ax + 4y = c. Together they are AX = B with A = [[1, 2], [a, 4]] and B = (4, c). Watch |A| = 4 − 2a as the lines cross, then stop crossing.
Does JEE Main ask questions from Determinants?
Yes. JEE Main asked 59 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Consistent and inconsistent systems" in Determinants (Class 12 Maths).

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