What is a determinant in Class 12 Mathematics?

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.

What is the determinant of a 2×2 matrix?

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

What is the determinant of a 3×3 matrix?

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

What are minors and cofactors in determinants?

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.

When is a matrix called singular or non-singular?

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

What is the formula for the inverse of a matrix?

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

How are determinants used to find the area of a triangle?

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.

How can a system of linear equations be solved using an inverse matrix?

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

What is the importance of determinants for CBSE and JEE exams?

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.

What should students remember most from the Determinants chapter?

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.

What is covered in these Class 12 Determinants MCQs?

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.

How many Determinants MCQs are included?

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

Are these Determinants MCQs based on NCERT Class 12 Mathematics?

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

Do these MCQs include answers?

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

Do these Determinants MCQs include explanations?

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

Determinants | Mathematics Class 12

10 CBSE Marks

★★★★★ Difficulty

7 Topics

Very High JEE Weight

Topics Covered

7 key topics in this chapter

Determinant of a Square Matrix

Properties of Determinants

Area of a Triangle Using Determinants

Minors and Cofactors

Adjoint of a Matrix

Inverse of a Matrix via Adjoint

Applications: Solving Systems of Linear Equations

Study Resources

📖 NCERT Notes ✏️ MCQ Practice 🏆 JEE / CBSE PYQs 📝 Exercise Solutions ✅ True / False Quiz

𝑓 Key Formulae

Essential mathematical expressions for this chapter — understand derivations, not just results.

2×2 Determinant

\[\begin{vmatrix}a&b\\c&d\end{vmatrix} = ad-bc\]

📌 Cross-multiply and subtract

Area of Triangle

\[\Delta = \tfrac{1}{2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}\]

📌 Take absolute value of the result

Adjoint

\[\text{adj}(A) = [\text{Cof}(A)]'\]

📌 Transpose of the cofactor matrix

Inverse via Adjoint

\[A^{-1} = \tfrac{1}{|A|}\,\text{adj}(A),\quad |A| \ne 0\]

📌 Only exists for non-singular matrices

Consistency Test

\[|A| \ne 0 \implies \text{unique solution } X = A^{-1}B\]

📌 Matrix method for a system of linear equations

🎯 Exam-Ready Insights

Important points to remember — curated from CBSE Board question patterns.

CBSE always includes a determinant-based area-of-triangle or collinearity check — collinear points make the determinant exactly zero.

Minors and cofactors must be computed with correct sign placement (−1)^(i+j) — a sign error invalidates the entire adjoint.

The matrix method for solving equations requires checking |A| ≠ 0 first; if |A| = 0, the system needs a separate consistency check.

Properties of determinants (row/column operations, factor extraction) can simplify a 3×3 determinant to near-triangular form before expansion.

🏆 Competitive Exam Strategy

Targeted tips for JEE Main, JEE Advanced, NEET, BITSAT, and CBSE Boards.

JEE Main

Use determinant properties (Rᵢ→Rᵢ+kRⱼ) to create zeros before expanding — this is far faster than direct cofactor expansion under time pressure.

JEE Advanced

Problems combine determinants with matrices and system consistency in a single multi-part question — track singular vs non-singular carefully.

BITSAT

Rapid evaluation of 3×3 determinants using Sarrus-style expansion saves crucial seconds.

⚠️ Common Mistakes to Avoid

Forgetting the alternating sign pattern when computing cofactors.

Expanding a 3×3 determinant along a row/column without applying available row operations first, leading to arithmetic errors.

Confusing minor (a determinant) with cofactor (a signed minor).

💡 Key Takeaways

◆

A determinant is a single scalar value associated with a square matrix, encoding whether it is invertible.

◆

Three collinear points always give an area determinant equal to zero.

◆

The inverse of a matrix exists only when its determinant is non-zero.

Determinants