JEE Mains Maths · Determinants

Classifying a System: Unique, Infinite or None

Deciding for every value of the parameters whether a linear system has one solution, infinitely many or none, and finding the one solution by Cramer's rule.

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Why this matters

Nineteen PYQs, eighteen of them multiple choice. Eleven give several statements about a system and ask which is not correct; eight ask for the unique solution, or for the chance that a system with random coefficients has one. Two ideas cover the page.

Concept 1 of 2: The full classification

Build the whole table once. Find the parameter values with

\Delta=0

; everywhere else the solution is unique. At those values, test consistency: infinitely many if the right-hand sides fit, none if they do not. Then check each statement against the table, remembering that a statement about 'all' values fails if one value breaks it.

Definition

$\Delta\neq0$ : unique.
$\Delta=0$ , consistent: infinitely many.
$\Delta=0$ , inconsistent: none.
A 'for all' statement needs every case; a 'there exists' statement needs one.

Three cases

\Delta\neq0\Rightarrow\text{unique};\quad\Delta=0\Rightarrow\text{infinite or none}

Worked example

Classify

x+y+z=3

x+2y+2z=5

x+2y+\lambda z=\mu

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 30 Jan 2024 · Q151Moderate

Example 1 · Determinants · Classifying a System: Unique, Infinite or None

Consider the system of linear equations

x + y + z = 5,x + 2y +\lambda^{2}z = 9

x + 3y + \lambda z = \mu

, where

\lambda,\mu \in R

. Then, which of the following statement is NOT correct?

Two options can be wrong

In 'which is NOT correct' questions, check every option against the table, not just until one fails. A booklet may print two incorrect statements; the key then picks one.

Drill 10 more on the full classification

Concept 2 of 2: The unique solution

When

\Delta\neq0

the solution is

x=\frac{\Delta_x}{\Delta}

y=\frac{\Delta_y}{\Delta}

z=\frac{\Delta_z}{\Delta}

, or it is found by elimination. When the coefficients are random — dice, or a choice from a set — the probability of a unique solution is the share of choices with

\Delta\neq0

Definition

Cramer: $x=\frac{\Delta_x}{\Delta}$ and so on.
Probability of a unique solution: $P(\Delta\neq0)$ .
Count the choices with $\Delta=0$ , then take the complement.

Cramer's rule

x=\frac{\Delta_x}{\Delta},\ y=\frac{\Delta_y}{\Delta},\ z=\frac{\Delta_z}{\Delta}\quad(\Delta\neq0)

Worked example

Solve

x+y=3

2x-y=0

by Cramer's rule.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 15 Apr 2023 · Q74Moderate

Example 2 · Determinants · Classifying a System: Unique, Infinite or None

Let the system of linear equations

- x + 2y - 9z = 7

- x + 3y + 7z = 9

- 2x + y + 5z = 8

- 3x + y + 13z = \lambda

has a unique solution

x = \alpha,y = \beta,z = \gamma

. Then the distance of the point

(\alpha,\beta,\gamma)

from the plane

2x - 2y + z = \lambda

Count ordered choices

With two dice, the pairs

(a,b)

and

(b,a)

are different outcomes. Count the pairs making

\Delta=0

as ordered pairs out of 36.

Drill 7 more on the unique solution

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 full classification
Three cases
$\Delta\neq0\Rightarrow\text{unique};\quad\Delta=0\Rightarrow\text{infinite or none}$
The unique solution
Cramer's rule
$x=\frac{\Delta_x}{\Delta},\ y=\frac{\Delta_y}{\Delta},\ z=\frac{\Delta_z}{\Delta}\quad(\Delta\neq0)$

Watch out for (2)

Two options can be wrong
→ The full classification
Count ordered choices
→ The unique solution

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.

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