Class 12 · Chapter 4 · Algebra unit (10 of 80 marks)

Determinants Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 58 board-style questions.

Practise this chapter Join freeFree account, no card: +5 practice questions on every chapter, with instant feedback.

58 questions
17 multiple choice, 4 assertion–reason, 15 very short answer, 12 short answer, 7 long answer, 3 case study
About 8 hours to master

Unit II Algebra (Matrices + Determinants): 10 marks of the 80-mark paper — CBSE Curriculum 2025-26 Mathematics (041) (official PDF).

NCERT textbook: Chapter 4 on ncert.nic.in (PDF)

Revision notes

Determinants — revision notes

1. Evaluating determinants

$| \begin{matrix} a & b \\ c & d \end{matrix} | = a d - b c$ .
Order 3: expand along any row or column, $Δ = \sum a_{i j} A_{i j}$ , where the cofactor $A_{i j} = (- 1)^{i + j} M_{i j}$ and the minor $M_{i j}$ is the determinant left after deleting row $i$ and column $j$ . Choose the row/column with most zeros.
Elements of one row times the cofactors of a different row always sum to $0$ .

2. Area of a triangle and collinearity

Area $= \frac{1}{2} | | \begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | |$ . The determinant can be negative — take the modulus; if an area is given, use $\pm$ and keep both cases.
Three points are collinear iff this determinant is $0$ . Line through $(x_{1}, y_{1}), (x_{2}, y_{2})$ : $| \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | = 0$ .

3. Adjoint and inverse

$adj A$ = transpose of the matrix of cofactors. For order 2, $adj [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ .
$A (adj A) = (adj A) A = | A | I$ ; $A^{- 1} = \frac{1}{| A |} adj A$ exists iff $| A | \neq 0$ (non-singular).
For order $n$ : $| k A | = k^{n} | A |$ , $| A B | = | A | | B |$ , $| A^{T} | = | A |$ , $| A^{- 1} | = \frac{1}{| A |}$ , $| adj A | = | A |^{n - 1}$ , $(A B)^{- 1} = B^{- 1} A^{- 1}$ , $(A^{T})^{- 1} = (A^{- 1})^{T}$ .

4. Linear systems $A X = B$

$| A | \neq 0$ : unique solution $X = A^{- 1} B$ (consistent).
$| A | = 0$ : find $(adj A) B$ . If it is $\neq O$ , no solution (inconsistent). If it is $= O$ , the system has either infinitely many solutions or none — check the equations directly.

Worked example 1

Area of the triangle with vertices $(0, 0)$ , $(4, 1)$ , $(1, 3)$ : $\frac{1}{2} | \begin{matrix} 0 & 0 & 1 \\ 4 & 1 & 1 \\ 1 & 3 & 1 \end{matrix} | = \frac{1}{2} (12 - 1) = \frac{11}{2}$ sq units.

Worked example 2

$A = [\begin{matrix} 1 & 2 \\ 3 & 5 \end{matrix}]$ : $| A | = - 1$ , $A^{- 1} = - [\begin{matrix} 5 & - 2 \\ - 3 & 1 \end{matrix}] = [\begin{matrix} - 5 & 2 \\ 3 & - 1 \end{matrix}]$ . So $x + 2 y = 4, 3 x + 5 y = 11$ gives $X = A^{- 1} [\begin{matrix} 4 \\ 11 \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}]$ .

Worked example 3

If $| A | = 3$ for order 3: $| 2 adj A | = 2^{3} | A |^{2} = 72$ .

Common errors

Sign pattern of cofactors $\begin{matrix} + & - & + \\ - & + & - \\ + & - & + \end{matrix}$ forgotten, especially for $A_{12}, A_{21}, A_{23}, A_{32}$ .
Writing the matrix of cofactors instead of its transpose as $adj A$ .
$| k A | = k | A |$ instead of $k^{n} | A |$ .
Forgetting the modulus (or one of the two cases) in area problems.
Writing $X = B A^{- 1}$ instead of $X = A^{- 1} B$ ; for $X A = C$ the answer is $X = C A^{- 1}$ .
Concluding “no solution” from $| A | = 0$ alone.

Board-exam tips

In the 5-mark “solve using matrix method” question, marks are given for $| A |$ , adj A (or cofactors), $A^{- 1}$ and the solution separately — show each clearly, and substitute your answer back into one equation.
Properties of determinants (row/column operations) are not in the current syllabus: expand directly.

Topics in this chapter: Results on adjoint, inverse and |kA| · Minors and cofactors · Area of a triangle using determinants · Consistency of a system of linear equations · Adjoint and inverse of a matrix · Evaluating determinants · Solving linear systems by the matrix method.

Formulas for this chapter: Chapter 4 on the Class 12 formula sheet (free PDF and A3 poster).

Determinants Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 195 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceEvaluating determinants

If $| \begin{matrix} x & 3 \\ 4 & x \end{matrix} | = | \begin{matrix} 5 & 2 \\ 1 & 1 \end{matrix} |$ , then $x$ equals

Show worked answer

$\begin{aligned} x^{2} - 12 = 5 - 2 = 3 \\ \Rightarrow & x^{2} = 15 \\ \Rightarrow & x = \pm \sqrt{15} \end{aligned}$ A1
Correct option: (a) $\pm \sqrt{15}$

Q2·1 mark·Multiple choiceAdjoint and inverse

$A$ is a $3 \times 3$ matrix with $| A | = 4$ . Then $| adj A |$ equals

Q3·1 mark·Assertion–reasonAdjoint and inverse

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The matrix $[\begin{matrix} 2 & 4 \\ 3 & 6 \end{matrix}]$ has no inverse.

Reason (R): A square matrix is invertible if and only if its determinant is non-zero.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerArea of a triangle

Using determinants, find the area of the triangle with vertices $(1, 2)$ , $(4, 6)$ and $(7, 2)$ .

Show worked answer

Answer:

12

square units

$\begin{aligned} Δ & = \frac{1}{2} | \begin{array}{c} 1 & 2 & 1 \\ 4 & 6 & 1 \\ 7 & 2 & 1 \end{array} | \\ = \frac{1}{2} [1 (6 - 2) - 2 (4 - 7) + 1 (8 - 42)] \end{aligned}$ M1
$= \frac{1}{2} (4 + 6 - 34) = - 12$ ; area $= 12$ square units A1

Q5·2 marks·Short answerArea of a triangle

The area of the triangle with vertices $(k, 0)$ , $(6, 0)$ and $(0, 3)$ is $6$ square units. Find the possible values of $k$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerAdjoint and inverse

Find the adjoint and the inverse of $A = [\begin{matrix} 3 & - 2 \\ 4 & 1 \end{matrix}]$ .

Show worked answer

Answer:

A^{- 1} = \frac{1}{11} [\begin{matrix} 1 & 2 \\ - 4 & 3 \end{matrix}]

$| A | = 3 + 8 = 11 \neq 0$ A1
$adj A = [\begin{matrix} 1 & 2 \\ - 4 & 3 \end{matrix}]$ A1
$A^{- 1} = \frac{1}{11} [\begin{matrix} 1 & 2 \\ - 4 & 3 \end{matrix}]$ A1

Q7·3 marks·Short answerEvaluating determinants

Expand $| A |$ along its first row for $A = [\begin{matrix} 2 & - 1 & 3 \\ 0 & 4 & 1 \\ 5 & 2 & - 2 \end{matrix}]$ , showing the three cofactors you use.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerSolving linear systems

Use the matrix method to solve: $x + y + z = 6$ , $2 x - y + z = 3$ , $x + 2 y - z = 2$ .

Show worked answer

Answer:

x = 1

y = 2

z = 3

$A X = B$ with $A = [\begin{matrix} 1 & 1 & 1 \\ 2 & - 1 & 1 \\ 1 & 2 & - 1 \end{matrix}]$ ; $\begin{aligned} | A | & = 1 (1 - 2) - 1 (- 2 - 1) + 1 (4 + 1) \\ = 7 \neq 0 \end{aligned}$ A1
Cofactors give $adj A = [\begin{matrix} - 1 & 3 & 2 \\ 3 & - 2 & 1 \\ 5 & - 1 & - 3 \end{matrix}]$ M1
$A^{- 1} = \frac{1}{7} adj A$ A1
$\begin{aligned} X & = A^{- 1} B \\ = \frac{1}{7} [\begin{array}{c} - 6 + 9 + 4 \\ 18 - 6 + 2 \\ 30 - 3 - 6 \end{array}] \end{aligned}$ M1
$= [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ : $x = 1$ , $y = 2$ , $z = 3$ A1

Q9·5 marks·Long answerAdjoint and inverse

For $A = [\begin{matrix} 1 & 2 & 0 \\ 0 & 1 & 3 \\ 2 & 0 & 1 \end{matrix}]$ , find (a) $| A |$ , (b) $adj A$ , (c) $A^{- 1}$ , and (d) verify that $(adj A) A = | A | I$ .

Section E style · Case study (4 marks)

Q10·4 marks·Case studyArea of a triangle

Village wells. Three wells in a village are at $P (1, 1)$ , $Q (4, 5)$ and $R (7, 3)$ on a map grid (units in hundreds of metres).

Find the area of triangle $P Q R$ . [1 mark]
Show that $P$ , $Q$ and $S (7, 9)$ are collinear. [1 mark]
A pump is placed at $T (k, 3)$ so that triangle $P Q T$ has area $6$ square units. Find $k$ . [2 marks]

Show worked answer

$\begin{aligned} \frac{1}{2} | 1 (5 - 3) - 1 (4 - 7) + 1 (12 - 35) | & = \frac{1}{2} | - 18 | \\ = 9 \end{aligned}$ square units A1
$\begin{aligned} | \begin{array}{c} 1 & 1 & 1 \\ 4 & 5 & 1 \\ 7 & 9 & 1 \end{array} | & = - 4 + 3 + 1 \\ = 0 \end{aligned}$ , so the area is $0$ : collinear A1
$\frac{1}{2} | 10 - 4 k | = 6$ M1
$\begin{aligned} 10 - 4 k = \pm 12 \\ \Rightarrow & k = - \frac{1}{2} \end{aligned}$ or $k = \frac{11}{2}$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

Expand 2x2 and 3x3 determinants, find minors and cofactors, and find inverses of 2x2 matrices.

Read first: 1. Evaluating determinants; 3. Adjoint and inverse 6 practice questions · checkpoint: 3 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

Use determinants for area, collinearity and lines; use |kA|, |adj A| and inverse results; decide consistency of small systems.

Read first: 2. Area of a triangle and collinearity; 3. Adjoint and inverse; 4. Linear systems 22 practice questions · checkpoint: 4 questions, 9 marks, pass 75%

Practise step 2

Step 3

Full marks on long answers

Score full marks on the 5-mark matrix-method question and on determinant case studies, with every step mark visible.

Read first: 4. Linear systems; Worked example 2; Board-exam tips 6 practice questions · checkpoint: 3 questions, 14 marks, pass 70%

Practise step 3

Step 4

95+ stretch (HOTS)

Handle parameter systems (unique / none / infinitely many), adj-inverse identities and unfamiliar applications.

Read first: 3. Adjoint and inverse (results); 4. Linear systems 6 practice questions · checkpoint: 3 questions, 12 marks, pass 60%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 6 of the 58 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceResults on adjoint, inverse and |kA|

If $A$ is a square matrix of order $3$ with $| A | = 5$ , then $| - 2 A^{T} |$ equals

(a) $- 40$
(b) $40$
(c) $- 10$
(d) $- 80$

Q2·1 mark·Multiple choiceAdjoint and inverse of a matrix

The inverse of $A = [\begin{matrix} 2 & 3 \\ 1 & 4 \end{matrix}]$ is

(a) $\frac{1}{11} [\begin{matrix} 4 & - 3 \\ - 1 & 2 \end{matrix}]$
(b) $\frac{1}{5} [\begin{matrix} 4 & 3 \\ 1 & 2 \end{matrix}]$
(c) $\frac{1}{5} [\begin{matrix} 2 & - 3 \\ - 1 & 4 \end{matrix}]$
(d) $\frac{1}{5} [\begin{matrix} 4 & - 3 \\ - 1 & 2 \end{matrix}]$

Q3·1 mark·Multiple choiceArea of a triangle using determinants

Using determinants, the equation of the line joining $(2, 3)$ and $(- 1, 5)$ is

(a) $3 x + 2 y = 12$
(b) $2 x + 3 y = 13$
(c) $2 x - 3 y = - 5$
(d) $3 x - 2 y = 0$

Worked solution·2 marks·Very short answerEvaluating determinants

Find the values of $x$ for which $| \begin{matrix} x & - 1 & 2 \\ 3 & x & 0 \\ 1 & 2 & 1 \end{matrix} | = 18$ .

Full worked solution Free, no sign-in

Expanding along $R_{1}$ : $x (x - 0) + 1 (3 - 0) + 2 (6 - x) = x^{2} - 2 x + 15$ M1
$x^{2} - 2 x + 15 = 18 \Rightarrow x^{2} - 2 x - 3 = 0 \Rightarrow x = 3$ or $x = - 1$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S25 · Problem S89 · Problem S24 · Problem S88. All 22 →

Knowledge organiser: Determinants on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Determinants

Wrong cofactor signs (A12, A21, A23, A32). Fix: write the +/- chessboard next to the matrix before computing cofactors.
Giving the cofactor matrix, not its transpose, as adj A. Fix: write adj A = Cᵀ and check that A(adj A) = |A|I for one row.
Using |kA| = k|A|. Fix: |kA| = kⁿ|A| for order n (for order 3, cube the scalar).
Missing a case in "area = given value" questions. Fix: set the determinant equal to ±(2 × area) and keep both values.
Writing X = BA⁻¹ for AX = B. Fix: pre-multiply by A⁻¹, so X = A⁻¹B; for XA = C, X = CA⁻¹.
Saying "no solution" as soon as |A| = 0. Fix: compute (adj A)B; if it is O, check the equations for infinitely many solutions or none.
Arithmetic slips in the 5-mark system with no check. Fix: substitute the answer into all three equations (takes 30 seconds).

Exam strategy: how answers are step-marked · competency-based questions

Determinants in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 3, 4, 7, 32
Sample paper 2 (Board standard): questions 4, 6, 7, 35
Sample paper 3 (Above board): questions 4, 7, 32
Sample paper 4 (Above board): questions 4, 5, 7, 35
Sample paper 5 (95+ challenge): questions 4, 6, 7, 32

Want it against the clock? Take a timed 30-mark chapter test on Determinants, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on matrices and determinants.

Determinants: common questions

What is in CBSE Class 12 Maths Chapter 4, Determinants?

The topics are: Results on adjoint, inverse and |kA|, Minors and cofactors, Area of a triangle using determinants, Consistency of a system of linear equations, Adjoint and inverse of a matrix, Evaluating determinants, Solving linear systems by the matrix method. This page has revision notes for each, and 58 board-style practice questions with step-marked solutions.

How many marks is Determinants worth?

It is in the Algebra unit, which carries 10 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Determinants?

Wrong cofactor signs (A12, A21, A23, A32). Fix: write the +/- chessboard next to the matrix before computing cofactors. Giving the cofactor matrix, not its transpose, as adj A. Fix: write adj A = Cᵀ and check that A(adj A) = |A|I for one row.

Are the Determinants important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Determinants — revision notes

1. Evaluating determinants

2. Area of a triangle and collinearity

3. Adjoint and inverse

4. Linear systems AX=B

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Determinants Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Determinants

Determinants in our sample papers

Determinants: common questions

Work through the Route to 95 with your progress saved

4. Linear systems $A X = B$