Class 12 Maths Chapter 4: Determinants – Properties, NCERT Solutions & CBSE PYQs

 Here’s your comprehensive guide for Chapter 4: Determinants (Class 12 CBSE Maths), covering:

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🧠 A. Key Concepts & Properties

1. What Is a Determinant?

• Assigns a scalar value to a square matrix.

• For $1 \times 1$: $|[a]| = a$;

• For $2 \times 2$, $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$;

• For $3 \times 3$, use expansion by minors and cofactors. 

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2. Properties of Determinants

(Useful for simplifying evaluations) :

1. Transpose Invariance: $|A| = |A^T|$

2. Sign Swap: Swapping two rows flips the sign

3. Zero/Equal Row: Any row (or column) of zeros or two identical rows ⇒ $|A| = 0$

4. Scalar Factor: Multiplying a row by $k$ scales $|A|$ by $k$

5. Linearity: Split row entries into sums to split determinant

6. Triangular Matrix: Determinant equals product of diagonal entries

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3. Minors, Cofactors & Expansion

• Minor $M_{ij}$: Determinant obtained by deleting row $i$ and column $j$.

• Cofactor $A_{ij} = (-1)^{i+j} M_{ij}$.

• Determinant via cofactor expansion: $\sum_j a_{ij} A_{ij}$. 

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4. Applications

• Area of a Triangle with coordinates $(x_1,y_1), (x_2,y_2), (x_3,y_3)$:
  
  

• Adjoint and Inverse of a matrix develop using cofactors.

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📘 B. NCERT Exercise Solutions Overview

Exercise	Topics Covered
4.1 – 4.2	Determinant evaluation (1×1, 2×2, 3×3)
4.3	Minor/cofactor methods, expansion
4.4 – 4.5	Properties and solving equations
4.6	Adjoint and inverse calculations
4.7	Applications: triangle area, linear equations

(Complete worked solutions are available via Cuemath, Teachoo, Mathongo) 

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📝 C. CBSE PYQs with Model Answers

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✅ PYQ 2019 (All India)

Q: If $A = \begin{pmatrix}0 & -1\\0 & 2\end{pmatrix}$, $B = \begin{pmatrix}3 & 5\\0 & 0\end{pmatrix}$, find $|AB|$.
Solution:

$\Rightarrow |AB| = 0$. Demonstrates that a zero row/column ⇒ zero determinant (Property).

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✅ PYQ 2018 (Delhi)

Q: If $A$ is skew-symmetric and $A = \begin{pmatrix}0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0\end{pmatrix}$, find $a, b$.
Solution:
Skew-symmetric ⇒ $A^T = -A$: equate entries → $a = -2$, $b = -1$. 

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✅ PYQ 2016 (All India)

Q: Given $A = P + Q$, where $P$ is symmetric & $Q$ skew-symmetric, find $P$ for $A = \begin{pmatrix}3 & 5\\7 & 9\end{pmatrix}$.
Solution:

$$P = \frac{A + A^T}{2} = \begin{pmatrix}3 & 6\\6 & 9\end{pmatrix}$$

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✅ PYQ 2017 (All India)

Q: Prove that $\det \begin{pmatrix}1 & x & x^2\\ x^2 & 1 & x\\ x & x^2 & 1\end{pmatrix} = (1 - x^3)^2$.
Solution Outline:
Use row/col operations and factorization recognizing $1-x^3$.

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🎓 D. Exam Strategy & Tips

• Always state the property used (e.g., “by property ...”).

• For inverses, use adjoint and determinant formula.

• For row/column operations, indicate each step clearly.

• Triangle area questions gain full marks with correct formula and application.

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#Tags:

#Class12Maths, #Determinants, #NCERTSolutions, #MatrixDeterminant, #CBSEPYQs, #Adjoint, #InverseMatrix, #MathsChapter4, #TriangleArea