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JEE Mains Maths · Determinants

Infinitely Many Solutions: Parameters in Two Equations

Systems with infinitely many solutions where the unknown constants sit in different equations: set the coefficient determinant to zero, then use a Cramer determinant, or eliminate step by step.

All Determinants notesJEE Mains Maths strategy

Why this matters

Eighteen PYQs, fifteen of them multiple choice. Fourteen solve Δ = 0 together with the Cramer determinant that holds only one of the constants; four are quicker by eliminating to a single equation in z. Two ideas cover the page.

Concept 1 of 2: The coefficient determinant and one Cramer determinant

Infinitely many solutions needs Δ=0\Delta=0 and Δx=Δy=Δz=0\Delta_x=\Delta_y=\Delta_z=0. When one constant is a coefficient and the other a right-hand side, Δ=0\Delta=0 gives the coefficient; then pick the Cramer determinant (the one with a column replaced by the right-hand sides) that contains the other constant, and set it to 0.

Definition

  • Unique: Δ≠0\Delta\neq0, with x=ΔxΔx=\frac{\Delta_x}{\Delta} and so on.
  • Infinitely many: Δ=Δx=Δy=Δz=0\Delta=\Delta_x=\Delta_y=\Delta_z=0 (three equations).
  • Solve Δ=0\Delta=0 first; it usually holds only the coefficient.

Cramer's rule

x=ΔxΔ,y=ΔyΔ,z=ΔzΔx=\frac{\Delta_x}{\Delta},\quad y=\frac{\Delta_y}{\Delta},\quad z=\frac{\Delta_z}{\Delta}

Worked example

For which α,β\alpha,\beta has x+y+z=3x+y+z=3, x+2y+αz=4x+2y+\alpha z=4, x+3y+5z=βx+3y+5z=\beta infinitely many solutions?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 2 Apr 2025 · Q133Moderate

Example 1 · Determinants · Infinitely Many Solutions: Parameters in Two Equations

If the system of equation
2x+λy+3z=52x+\lambda y+ 3z= 5
3x+2y−z=73x+ 2y-z= 7
4x+5y+μz=94x+ 5y+\mu z= 9
has infinitely many solutions, then (λ2+μ2)\left( \lambda^{2}+\mu^{2} \right) is equal to :

Confirm consistency

Δ=0\Delta=0 with one vanishing Cramer determinant is the usual shortcut, but the definition asks for all of them to vanish. When two options differ only in the right-hand constant, confirm by eliminating once.
Drill 13 more on the coefficient determinant and one cramer determinant

Concept 2 of 2: Elimination instead of determinants

Subtract equations to remove xx, then yy, until one equation in zz remains, of the form kz=ckz=c. There is a unique solution when k≠0k\neq0, infinitely many when k=c=0k=c=0, and none when k=0≠ck=0\neq c. This reads off every case at once, which suits questions that ask about several cases together.

Definition

  • Reduce to kz=ckz=c.
  • k≠0k\neq0: unique; k=0, c=0k=0,\ c=0: infinitely many; k=0, c≠0k=0,\ c\neq0: none.

The last equation decides

kz=c:k≠0 unique; k=c=0 infinite; k=0≠c nonekz=c:\quad k\neq0\ \text{unique};\ k=c=0\ \text{infinite};\ k=0\neq c\ \text{none}

Worked example

Classify x+y+z=1x+y+z=1, x+2y+3z=2x+2y+3z=2, x+3y+az=bx+3y+az=b.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 23 Jan 2025 · Q69Moderate

Example 2 · Determinants · Infinitely Many Solutions: Parameters in Two Equations

If the system of equations
(λ−1)x+(λ−4)y+λz=5λx+(λ−1)y+(λ−4)z=7 (λ+1)x+(λ+2)y−(λ+2)z=9\begin{matrix} & (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \\ & \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \\ & \ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \end{matrix}
has infinitely many solutions, then λ2+λ\lambda^{2}+ \lambda is equal to

Keep the right-hand sides

Elimination must carry the constants along. Dropping them loses the difference between infinitely many solutions and none.
Drill 3 more on elimination instead of determinants

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • The coefficient determinant and one Cramer determinant

    Cramer's rule

    x=ΔxΔ,y=ΔyΔ,z=ΔzΔx=\frac{\Delta_x}{\Delta},\quad y=\frac{\Delta_y}{\Delta},\quad z=\frac{\Delta_z}{\Delta}
  • Elimination instead of determinants

    The last equation decides

    kz=c:k≠0 unique; k=c=0 infinite; k=0≠c nonekz=c:\quad k\neq0\ \text{unique};\ k=c=0\ \text{infinite};\ k=0\neq c\ \text{none}

Watch out for (2)

Test yourself on Determinants

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Determinants chapter testDrill all 18 questions