JEE Mains Maths · Determinants

Infinitely Many Solutions: One Equation Holds the Parameters

Systems of three linear equations where two equations are fully known and the parameters sit in the third: infinitely many solutions means the third is a combination of the first two.

All Determinants notes JEE Mains Maths strategy

Why this matters

Eighteen PYQs, sixteen of them multiple choice, and three from 2026. Fifteen put both unknown constants in one equation and ask for them, often feeding them into a follow-up; three have a coefficient determinant that is zero for every value, so only the right-hand side decides. Two ideas cover the page.

Concept 1 of 2: The third equation as a combination

Two independent equations in three unknowns describe a line of solutions. The system keeps all of them exactly when the third equation adds no new condition, that is, when it is

p

times the first plus

q

times the second. Match the

x

and

y

coefficients to find

p

and

q

; the

z

coefficient and the right-hand side then give the two constants. This is quicker than computing four determinants.

Definition

Infinitely many solutions: $E_3=pE_1+qE_2$ , including the right-hand side.
Match two coefficients to find $p,q$ , then read off the rest.
Same result as $\Delta=0$ and $\Delta_x=\Delta_y=\Delta_z=0$ .

Dependent third equation

E_3=pE_1+qE_2

Worked example

For which

\lambda,\mu

has

x+y+z=4

x+2y+3z=7

x+2y+\lambda z=\mu

infinitely many solutions?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 8 Apr 2026 Shift 2 · Q53Moderate

Example 1 · Determinants · Infinitely Many Solutions: One Equation Holds the Parameters

If the system of linear equations :

x+y+z= 6

x+ 2y+ 5z= 10

2x + 3y +\lambda z =\mu

. has infinitely many solutions, then the value of

\lambda+\mu

equals.

Match the right-hand side too

Matching the coefficients alone gives

\Delta=0

, which also allows no solution. The right-hand side of the third equation must be the same combination of the other two.

Drill 14 more on the third equation as a combination

Concept 2 of 2: When the determinant vanishes for every value

Sometimes the coefficients make

\Delta=0

whatever the parameter — two columns equal, or one row a fixed combination of the others. Then a unique solution is impossible, and the system has infinitely many solutions or none, decided by the right-hand sides alone.

Definition

$\Delta\equiv0$ : never a unique solution.
Infinitely many exactly when the right-hand sides obey the same relation as the rows.
Otherwise no solution.

Dependent rows

R_3=pR_1+qR_2\ \Rightarrow\ \text{consistent exactly when } b_3=pb_1+qb_2

Worked example

For which

k

does

x+y+z=1

x+2y+z=2

2x+3y+2z=k

have a solution?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 29 Jan 2025 · Q136Moderate

Example 2 · Determinants · Infinitely Many Solutions: One Equation Holds the Parameters

Let

\alpha,\beta(\alpha \neq \beta)

be the values of m , for which the equations

x + y + z = 1;x + 2y + 4z = m

and

x + 4y + 10z =m^{2}

have infinitely many solutions. Then the value of

\sum_{n = 1}^{10} \left( n^{\alpha}+n^{\beta} \right)

is equal to :

Look for dependence before expanding

Expanding a determinant with a parameter and finding it identically 0 wastes time. Check first whether a row is a sum or multiple of others.

Drill 2 more on when the determinant vanishes for every value

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 third equation as a combination
Dependent third equation
$E_3=pE_1+qE_2$
When the determinant vanishes for every value
Dependent rows
$R_3=pR_1+qR_2\ \Rightarrow\ \text{consistent exactly when } b_3=pb_1+qb_2$

Watch out for (2)

Match the right-hand side too
→ The third equation as a combination
Look for dependence before expanding
→ When the determinant vanishes for every value

Test yourself on Determinants

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