Class 12 Maths · Chapter 4
Unit: Algebra
Tentative Study Share: 5.0 Marks
Chapter 4: Determinants — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Determinants.
NCERT Exercises in Chapter 4: Determinants
📘 NCERT Concept Notes — Determinants
01 · A determinant is a scalar
Only square matrices have determinants. For expansion, pair entries with their own cofactors and keep the alternating signs. Do not confuse the determinant with the matrix itself.
02 · Coordinates can become area
A coordinate determinant gives signed double area. Absolute value removes orientation; collinear points give zero.
03 · Adjoint turns cofactors into an inverse
Transpose the cofactor matrix to get the adjoint. The determinant decides whether division is possible. Singular matrices need a separate consistency analysis.
04 · Matrix equations and consistency
Write the unknowns as one column and keep equation order fixed. An invertible coefficient matrix gives one solution. A zero determinant alone cannot tell you whether a solution exists.
⚡ Key Formulas & Results — Determinants
Order-two determinant
What is the determinant of a 2×2 matrix?
\(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\)
Minors and cofactors
How does a cofactor differ from a minor?
\(C_{ij}=(-1)^{i+j}M_{ij}\)
Cofactor expansion
What is a row expansion of the determinant?
\(\det A=\sum_j a_{ij}C_{ij}\)
Triangle area
How is triangle area expressed as a determinant?
\(\Delta=\frac12\left|\det\begin{pmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{pmatrix}\right|\)
Adjoint and inverse
Which matrix is transposed to form adj A?
\(\operatorname{adj}A=[C_{ij}]^T,\quad A^{-1}=\frac{\operatorname{adj}A}{\det A}\)
Derive the inverse formula
Why does dividing adj A by the determinant produce an inverse?
\(A\operatorname{adj}A=\operatorname{adj}A\,A=(\det A)I\)
Solve AX=B
How do you isolate X without commuting matrices?
\(X=A^{-1}B\)
Derive the matrix solution
Why does left multiplication give the unique solution?
\(AX=B\ \Rightarrow\ X=A^{-1}B\)
When det A is zero
Does a singular coefficient matrix always mean no solution?
\(\det A=0\quad\not\Rightarrow\quad\text{inconsistent}\)
🎯 CBSE 2026 Board PYQs from Determinants
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q4
If Δ₁ = | 1 0 0; 0 2 0; 0 0 3 | and Δ₂ = | 0 2 0; 1 0 0; 0 0 6 |, then
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q5
One of the values of x for which | cos x sin x; −cos x sin x | = 1 is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q15
If Δ₁ = | 1 0 0; 0 2 0; 0 0 3 | and Δ₂ = | 0 2 0; 1 0 0; 0 0 6 |, then
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q16
One of the values of x for which | cos x sin x; −cos x sin x | = 1 is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q8
If Δ₁ = | 1 0 0; 0 2 0; 0 0 3 | and Δ₂ = | 0 2 0; 1 0 0; 0 0 6 |, then
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q9
One of the values of x for which | cos x sin x; −cos x sin x | = 1 is
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q6
The determinant of [[−1,−2,5],[−2,a,−1],[0,4,2a]] is −86. Find the sum of the possible a values.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q17
The determinant of [[−1,−2,5],[−2,a,−1],[0,4,2a]] is −86. Find the sum of the possible a values.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q10
The determinant of [[−1,−2,5],[−2,a,−1],[0,4,2a]] is −86. Find the sum of the possible a values.
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q6
If B adj(B)=(1/3)I₃, find det(B⁻¹).
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q17
If A adj(A)=2026 I₃, find det(adj(A)).
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q10
If A adj(A)=3I₃, find det(2A).
CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Q4
If (2,3), (0,4) and (p,2) are collinear, find p.
CBSE Class 12 Mathematics 2026 Main Paper 65/4/2 · Q3
If (2,3), (0,4) and (p,2) are collinear, find p.
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Q8
If (2,3), (0,4) and (p,2) are collinear, find p.
CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Q5
Which statement is false when A is nonsingular?
CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Q7
A=(3,1), B=(−2,1), C=(0,k) form a triangle of area 5. Find the possible k values.
CBSE Class 12 Mathematics 2026 Main Paper 65/5/2 · Q6
Which statement is false when A is nonsingular?
📝 Chapterwise Practice Questions — Determinants (110 Questions)
Class 12 Determinants · Mcq Single Correct
If \(\begin{vmatrix}2 & 4\\5 & 1\end{vmatrix}\)=\(\begin{vmatrix}2x & 4\\6 & x\end{vmatrix}\), find all…
Class 12 Determinants · Mcq Single Correct
The area of the triangle with vertices (−3,0),(3,0),(0,k) is 9 square units. Find k.
Class 12 Determinants · Mcq Single Correct
Let f(x)=\(\begin{vmatrix}0 & x−a & x−b\\x+a & 0 & x−c\\x+b & x+c & 0\end{vmatrix}\). Which equality…
Class 12 Determinants · Mcq Single Correct
If \(\begin{vmatrix}2 & 3 & 2\\x & x & x\\4 & 9 & 1\end{vmatrix}\)+3=0, find x.
Class 12 Determinants · Mcq Single Correct
If x=−4 is a root of \(\begin{vmatrix}x & 2 & 3\\1 & x & 1\\3 & 2 & x\end{vmatrix}\)=0, find the sum of…
Class 12 Determinants · Mcq Single Correct
For a 3×3 real matrix A, det(Aᵀ)=−3. Find det(AAᵀ).
Class 12 Determinants · Mcq Single Correct
For A=\(\begin{bmatrix}α & 2\\2 & α\end{bmatrix}\), det(A³)=27. Find all real α.
Class 12 Determinants · Mcq Single Correct
For which real k is A=\(\begin{bmatrix}k & 8\\4 & 2k\end{bmatrix}\) singular?
Class 12 Determinants · Mcq Single Correct
Evaluate \(\begin{vmatrix}x+y & y+z & z+x\\z & x & y\\1 & 1 & 1\end{vmatrix}\).
Class 12 Determinants · Mcq Single Correct
For a nonsingular 2×2 real matrix A, det(A)=det(kA). Find the sum of all possible real k.
Class 12 Determinants · Mcq Single Correct
Find the full set of real a for which \(\begin{bmatrix}1 & 2 & 1\\2 & 3 & 1\\3 & a & 1\end{bmatrix}\)…
Class 12 Determinants · Mcq Single Correct
For P=\(\begin{bmatrix}1 & -1 & 2\\0 & 2 & -3\\3 & 2 & 4\end{bmatrix}\), let Cᵢⱼ denote a cofactor…
Class 12 Determinants · Mcq Single Correct
For A=\(\begin{bmatrix}2 & 5\\-11 & 7\end{bmatrix}\), find (adj A)ᵀ.
Class 12 Determinants · Mcq Single Correct
A 3×3 matrix A satisfies A adj(A)=3I₃. Find det(A)+det(adj A).
Class 12 Determinants · Mcq Single Correct
For A=diag(−2,−2,−2), find det(adj A).
Class 12 Determinants · Mcq Single Correct
A is a real skew-symmetric matrix of order 3 and det(A)=x. Find 2023ˣ.
Class 12 Determinants · Mcq Single Correct
Find the inverse of \(\begin{bmatrix}-4 & 3\\7 & -5\end{bmatrix}\).
Class 12 Determinants · Mcq Single Correct
For a 2×2 matrix A with det(A)=2, find det(4A⁻¹).
Class 12 Determinants · Mcq Single Correct
Vertices (a,b),(c,d),(e,f) form a triangle of area Δ. Find the square of \(\begin{vmatrix}a & c & e\\b…
Class 12 Determinants · Mcq Single Correct
For cosθ≠0, let \(\begin{bmatrix}1 & -\tan\theta\\\tan\theta & 1\end{bmatrix}\)\(\begin{bmatrix}1 &…
Class 12 Determinants · Assertion Reason
Assertion: A determinant is a scalar associated with a square matrix. Reason: A determinant is itself a…
Class 12 Determinants · Assertion Reason
Assertion: For A=\(\begin{bmatrix}5−x & x+1\\2 & 4\end{bmatrix}\), A is singular at x=3. Reason: A…
Class 12 Determinants · Assertion Reason
Assertion: If A is 3×3, det(A)≠0 and det(5A)=Kdet(A), then K=125. Reason: For any n×n matrix A and…
Class 12 Determinants · Assertion Reason
Assertion: If \(\begin{vmatrix}x & 2\\18 & x\end{vmatrix}\)=\(\begin{vmatrix}6 & 2\\18 &…