a b c d |A| = ad − bc
|A|
Chapter 4  ·  Class XII Mathematics  ·  MCQ Practice

MCQ Practice Arena

Determinants

Expand, Compute, Apply — Determinants for Area, Inverse & Systems

📋 50 MCQs ⭐ 28 PYQs ⏱ 95 sec/Q
🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs
18Easy
20Medium
12Hard
28PYQs
95 secAvg Time/Q
5Topics
Easy 36% Medium 40% Hard 24%

Why Practise These MCQs?

JEE MainJEE AdvancedCBSE

Determinants sit right after Matrices for a reason — nearly every inverse and system-of-equations MCQ leans on them. JEE Advanced likes property-based simplification before expansion, while CBSE boards test adjoint/inverse computation and the area-of-triangle formula directly. This set covers both calculation speed and property recognition.

Topic-wise MCQ Breakdown

Determinant Expansion & Properties12 Q
Area of a Triangle using Determinants6 Q
Minors & Cofactors8 Q
Adjoint & Inverse of a Matrix12 Q
Applications (Consistency of Equations)12 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$|A| = a(ei-fh) - b(di-fg) + c(dh-eg)$
$A^{-1} = \dfrac{\text{adj}(A)}{|A|}$
$\text{Area} = \tfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$
$A \cdot \text{adj}(A) = |A| \cdot I$

MCQ Solving Strategy

Use row/column operations to create zeros before expanding a determinant — it saves real time on 3×3 questions. Expand along the row or column with the most zeros. Remember that |A|=0 signals a singular matrix, which usually means "no unique solution" in a system-of-equations MCQ.

⚠ Common Traps & Errors

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Expand 2×2 and simple 3×3 determinants, evaluate minors and cofactors

② Medium

Apply row/column properties to simplify before expanding, find adjoint

③ Hard

Determine consistency of equation systems, prove determinant identities

★ PYQ

JEE Advanced — property-based simplification; CBSE — inverse-via-adjoint problems

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 🎯 PYQ Bank Previous year questions ✔️❌ True / False Concept-based True & False questions 📘✔️ Exercises Step by step Solutions to textbook exercises ➡ Next Chapter Continuity and Differentiability MCQs
🎯 Knowledge Check

Mathematics — DETERMINANTS

50 Questions Class 12 MCQs
1
The determinant of a matrix of order \(1\times1\), \(A=[5]\), is
2
The value of \(\begin{vmatrix}2&3\\1&4\end{vmatrix}\) is
3
If \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\), then \(|A|\) is
4
The determinant \(\begin{vmatrix}3&0\\0&7\end{vmatrix}\) is
5
The determinant \(\begin{vmatrix}4&5\\4&5\end{vmatrix}\) is
6
If two rows of a determinant are interchanged, the value of the determinant
7
The determinant of a triangular matrix is equal to
8
The value of \(\begin{vmatrix}1&2\\3&4\end{vmatrix}\) is
9
If every element of a row of a determinant is multiplied by \(k\), then the determinant becomes
10
If \(A\) is a \(3\times3\) matrix and \(|A|=5\), then \(|2A|\) is
11
The minor \(M_{11}\) of the determinant \(\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}\) is
12
The cofactor \(C_{12}\) of a determinant is
13
The cofactor \(C_{23}\) is related to its minor \(M_{23}\) by
14
The value of \(\begin{vmatrix}1&1&1\\1&2&3\\1&3&6\end{vmatrix}\) is
15
If two rows of a determinant are proportional, then its value is
16
The determinant \(\begin{vmatrix}2&4\\3&6\end{vmatrix}\) is
17
If \(A\) is a \(3\times3\) matrix and \(|A|=4\), then \(|A^T|\) equals
18
If \(|A|=3\) for a nonsingular matrix \(A\), then \(|A^{-1}|\) is
19
If \(A\) is a \(3\times3\) matrix with \(|A|=-2\), then \(|3A|\) is
20
If \(|A|=2\) and \(|B|=5\), then \(|AB|\) is
21
The value of \(\begin{vmatrix}x&2\\3&4\end{vmatrix}\) is
22
If \(\begin{vmatrix}x&2\\3&4\end{vmatrix}=0\), then \(x\) is
23
The determinant \(\begin{vmatrix}1&a\\a&1\end{vmatrix}\) is
24
The determinant \(\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}\) is
25
If \(A\) is a \(3\times3\) matrix and \(|A|=7\), then \(|A^T A|\) is
26
The value of \(\begin{vmatrix}1&2&3\\0&4&5\\0&0&6\end{vmatrix}\) is
27
If \(A\) is a \(3\times3\) matrix with \(|A|=2\), then \(|-A|\) is
28
If \(A\) is a \(2\times2\) matrix and \(|A|=-3\), then \(|-A|\) is
29
If the elements of one row of a determinant are all multiplied by \(3\) and the elements of another row are all multiplied by \(2\), the determinant becomes
30
If \(D=\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}\), then interchanging the first and third columns gives
31
The value of \(\begin{vmatrix}1&1&1\\a&b&c\\bc&ca&ab\end{vmatrix}\) is
32
If \(D=\begin{vmatrix}a&b&c\\a&b&c\\x&y&z\end{vmatrix}\), then \(D\) is
33
The value of \(\begin{vmatrix}1&2&3\\2&4&6\\3&6&9\end{vmatrix}\) is
34
If \(D=\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}\), then replacing \(R_2\) by \(R_2+kR_1\)
35
If \(D=\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}\), then \(C_{11}\) is
36
If \(D=\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}\), then the cofactor \(C_{13}\) is
37
The area of the triangle with vertices \((0,0)\), \((4,0)\), and \((0,3)\) is
38
The area of the triangle whose vertices are \((1,2)\), \((3,4)\), and \((5,6)\) is
39
If the area of the triangle formed by \((1,2)\), \((3,4)\), and \((x,6)\) is \(4\) square units, then \(x\) can be
40
The points \((1,2)\), \((3,4)\), and \((5,k)\) are collinear when
41
If \(A\) is a \(3\times3\) matrix and \(|A|=2\), then \(|\operatorname{adj}A|\) is
42
If \(A\) is a nonsingular \(3\times3\) matrix with \(|A|=5\), then \(|\operatorname{adj}A|\) is
43
If \(A\) is a nonsingular matrix, then \(A^{-1}\) is equal to
44
If \(A\) is a \(2\times2\) matrix and \(|A|=4\), then \(|\operatorname{adj}A|\) equals
45
The system \(2x+3y=5,\ 4x+6y=10\) has
46
Using Cramer's rule, for \(2x+y=5\) and \(x-y=1\), the value of \(x\) is
47
For the system \(ax+by=c,\ dx+ey=f\), the determinant used in Cramer's rule for a unique solution is
48
The system \(x+y+z=6,\ x+2y+3z=14,\ 2x+3y+z=13\) has solution
49
If \(A\) is a \(3\times3\) matrix satisfying \(A^2=A\) and \(|A|\neq0\), then \(|A|\) must be
50
If \(A\) and \(B\) are \(3\times3\) matrices such that \(|A|=2\) and \(|B|=-3\), then \(|2AB^{-1}|\) is
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NCERT Class 12 Determinants MCQs: 50 Questions
NCERT Class 12 Determinants MCQs: 50 Questions — Complete Notes & Solutions · academia-aeternum.com
Master NCERT Class 12 Mathematics Chapter 4 – Determinants with this carefully structured collection of 50 multiple-choice questions (MCQs) designed for progressive learning and exam preparation. The questions begin with fundamental concepts such as the order and value of determinants, minors, cofactors, determinant properties, and evaluation of (2\times2) and (3\times3) determinants, and gradually move towards more challenging applications. The set also covers important NCERT concepts…
🎓 Class 12 📐 Mathematics 📖 NCERT ✅ Free Access 🏆 CBSE · JEE
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Frequently Asked Questions

A determinant is a unique number associated with a square matrix. It is denoted by |A| or det(A) and is used to study matrix properties, solve equations and find the inverse of a matrix.

For A = [[a,b],[c,d]], the determinant is |A| = ad - bc.

A 3×3 determinant can be evaluated by expanding along any row or column using the corresponding elements, minors and cofactors.

The minor Mij of an element aij is obtained by deleting its ith row and jth column. Its cofactor is Aij = (-1)^(i+j)Mij.

A square matrix A is singular if |A| = 0 and non-singular if |A| ? 0.

For a non-singular square matrix A, the inverse is A?¹ = (1/|A|) adj(A), where |A| ? 0.

For vertices (x1,y1), (x2,y2), and (x3,y3), the area is 1/2 times the absolute value of the determinant formed using the three coordinates and a final column of 1s.

If a system is written as AX = B and |A| ? 0, then A is invertible and the unique solution is X = A?¹B.

Determinants are important for CBSE and JEE because questions commonly test determinant evaluation, properties, minors, cofactors, inverse matrices, area problems and systems of linear equations.

Key results include |A| ? 0 for an invertible matrix, A?¹ = (1/|A|)adj(A), A(adj A) = |A|I, and X = A?¹B for AX = B when A is non-singular.

These MCQs cover important NCERT Class 12 Determinants topics including evaluation of determinants, properties, minors, cofactors, adjoint, inverse, area of triangles, collinearity and Cramer’s Rule.

This practice set contains 50 multiple-choice questions arranged in increasing order of difficulty.

Yes, the questions are designed around the concepts and applications covered in NCERT Class 12 Mathematics Chapter 4, Determinants.

Yes, every MCQ includes the correct answer so students can immediately check their understanding.

Yes, each question is accompanied by a concise explanation showing the relevant concept or calculation.

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