NCERT Solutions for Class 12 Mathematics Chapter 4: Determinants – Free PDF Download

Complete, verified solutions to the NCERT Class 12 Maths Chapter 4 (Determinants) exercise questions. These Class 12 Mathematics Chapter 4 solutions are also useful as quick revision notes before exams.

NCERT Exercise Solutions

4.1 Evaluate the determinant |2 4; -5 -1|.
Ans: (2)(-1) – (4)(-5) = -2 + 20 = 18.

4.2 Evaluate |cosθ -sinθ; sinθ cosθ|.
Ans: cos²θ – (-sin²θ) = cos²θ + sin²θ = 1.

4.3 If A=[[1,2],[4,2]], find |A| and |2A|. Verify |2A|=4|A| for a 2×2 matrix.
Ans: |A|=(1)(2)-(2)(4)=2-8=-6. 2A=[[2,4],[8,4]], |2A|=(2)(4)-(4)(8)=8-32=-24. Check: 4|A|=4(-6)=-24=|2A|. ✓ (For an n×n matrix, |kA|=kⁿ|A|; here n=2 so factor is 2²=4.)

4.4 Find the values of x for which |3 x;x 1| = |3 2;4 1|.
Ans: LHS = 3-x². RHS = 3-8 = -5. So 3-x²=-5 ⇒ x²=8 ⇒ x=±2√2.

4.5 By using properties of determinants, prove |x x² yz; y y² zx; z z² xy| = (x-y)(y-z)(z-x)(xy+yz+zx).
Ans: Standard property-based proof: apply R₁→R₁-R₂, R₂→R₂-R₃ to factor out (x-y) and (y-z), then expand the remaining determinant to reveal the (z-x)(xy+yz+zx) factor — a classic symmetric-determinant identity proved via row operations rather than direct expansion.

Frequently Asked Questions

What is the area of a triangle using determinants?
Area = (1/2)|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|, expressed as a 3×3 determinant.

How does |kA| relate to |A| for an n×n matrix?
|kA| = kⁿ|A|.

Written by Satish

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