PYQ Vault

JEE Mains Maths · Determinants

Homogeneous Systems and Non-trivial Solutions

Systems with every right-hand side zero: they always have the zero solution, and have others exactly when the coefficient determinant is zero.

All Determinants notesJEE Mains Maths strategy

Why this matters

Ten PYQs, nine of them multiple choice, and two from 2026. Five have an angle in the coefficients and ask for the values of the angle giving a non-trivial solution; five have an algebraic constant instead. Both reduce to Δ = 0. Two ideas cover the page.

Concept 1 of 2: An angle in the coefficients

A homogeneous system always has x=y=z=0x=y=z=0. It has another solution exactly when Δ=0\Delta=0. With sin⁡θ\sin\theta or cos⁡θ\cos\theta in the coefficients, Δ=0\Delta=0 becomes a trigonometric equation; solve it and list the solutions in the interval asked for.

Definition

  • Homogeneous: all right-hand sides 0; the trivial solution always exists.
  • Non-trivial solution exactly when Δ=0\Delta=0 (then infinitely many).
  • Solve Δ(θ)=0\Delta(\theta)=0 in the given interval.

Non-trivial solutions

Ax=0 has x≠0 exactly when ∣A∣=0A\mathbf x=\mathbf0\ \text{has }\mathbf x\neq\mathbf0\ \text{exactly when}\ |A|=0

Worked example

For which θ∈[0,2π)\theta\in[0,2\pi) has x+(sin⁡θ)y=0x+(\sin\theta)y=0, (sin⁡θ)x+y=0(\sin\theta)x+y=0 a non-trivial solution?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 2 · Q54Moderate

Example 1 · Determinants · Homogeneous Systems and Non-trivial Solutions

The sum of all possible values of θ∈[0,2π]\theta\in \lbrack 0,2\pi\rbrack, for which the system of equations :
xcos3θ−8y−12z=0xcos3\theta- 8y- 12z= 0
xcos2θ+3y+3z=0xcos2\theta+ 3y+ 3z= 0
x+y+3z=0x+y+ 3z= 0
has a non-trivial solution, is equal to:

Open or closed interval

Trig solutions often sit at the ends, 00 or π\pi. Check whether the interval includes them before counting.
Drill 4 more on an angle in the coefficients

Concept 2 of 2: An algebraic constant

With a constant kk in the coefficients, expand Δ\Delta — row operations first when the rows are similar — and solve Δ=0\Delta=0. Follow-up questions then use the non-trivial solution: express x:y:zx:y:z from two equations.

Definition

  • Non-trivial solution: Δ(k)=0\Delta(k)=0.
  • The ratio x:y:zx:y:z comes from any two independent equations.
  • Symmetric systems: add all rows first.

Symmetric determinant

∣k111k111k∣=(k+2)(k−1)2\begin{vmatrix}k&1&1\\1&k&1\\1&1&k\end{vmatrix}=(k+2)(k-1)^2

Worked example

For which kk has x+2y+3z=0x+2y+3z=0, 2x+y+kz=02x+y+kz=0, x−y+z=0x-y+z=0 a non-trivial solution?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 6 April 2024 · Q81Moderate

Example 2 · Determinants · Homogeneous Systems and Non-trivial Solutions

Let αβγ=45;α,β,γ∈R\alpha\beta\gamma = 45;\alpha,\beta,\gamma \in R. If x(α,1,2)+y(1,β,2)x(\alpha,1,2) + y(1,\beta,2) +z(2,3,γ)=(0,0,0)+ z(2,3,\gamma) = (0,0,0) for some x,y,z∈R,xyz≠x,y,z \in R,xyz \neq 0 , then 6α+4β+γ6\alpha + 4\beta + \gamma is equal to

Every root counts

(k+2)(k−1)2=0(k+2)(k-1)^2=0 has two distinct values. A question asking for the sum or number of values needs both.
Drill 4 more on an algebraic constant

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)

  • An angle in the coefficients

    Non-trivial solutions

    Ax=0 has x≠0 exactly when ∣A∣=0A\mathbf x=\mathbf0\ \text{has }\mathbf x\neq\mathbf0\ \text{exactly when}\ |A|=0
  • An algebraic constant

    Symmetric determinant

    ∣k111k111k∣=(k+2)(k−1)2\begin{vmatrix}k&1&1\\1&k&1\\1&1&k\end{vmatrix}=(k+2)(k-1)^2

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 10 questions