CBSE Class 12 Mathematics: CBSE Class 12 Maths Question Paper 2025 with Solutions PDF & In-Depth Analysis

The CBSE Class 12 Mathematics examination is a defining milestone for students preparing for their 2025 and 2026 board examinations and competitive entrance tests. Accessing the official CBSE Class 12 Maths Question Paper 2025 with Solutions PDF provides authentic exposure to question typologies, competency-based assessments, and the official step-marking scheme implemented by the Central Board of Secondary Education (CBSE). Whether you are targeting a perfect 100 in CBSE Mathematics or looking to solidify your fundamentals across Calculus, Algebra, and Vectors, working through solved 2025 papers is the most reliable strategy for board exam 12 success.

CBSE Class 12 Maths 2025 Exam Structure and Question Blueprint

The CBSE Class 12 Mathematics theory paper carries 80 marks to be completed within a duration of 3 hours (with an additional 15 minutes of dedicated reading time). The question paper is structured into five distinct sections:

Section	Question Type	Number of Questions	Marks per Question	Total Marks
Section A	Multiple Choice Questions (MCQs) & Assertion-Reason	20 (18 MCQs + 2 A&R)	1 Mark	20 Marks
Section B	Very Short Answer (VSA)	5 Questions	2 Marks	10 Marks
Section C	Short Answer (SA)	6 Questions	3 Marks	18 Marks
Section D	Long Answer (LA)	4 Questions	5 Marks	20 Marks
Section E	Case Study / Source-Based Integrated Units	3 Questions	4 Marks	12 Marks
Total		38 Questions	-	80 Marks

Key Concepts and Chapter-Wise Weightage Analysis

The CBSE Class 12 Mathematics syllabus is divided into six core units. Understanding the weightage distribution helps prioritize revision hours effectively:

• Calculus (35 Marks): The highest-scoring unit in the syllabus, covering Continuity and Differentiability, Applications of Derivatives (tangents, increasing/decreasing functions, maxima-minima), Integrals (definite and indefinite), Applications of Integrals (area bounded by curves), and Differential Equations (order, degree, integrating factors).
• Vectors and Three-Dimensional Geometry (14 Marks): Focuses on dot products, cross products, scalar triple concepts, direction cosines/ratios, vector equations of lines, and the shortest distance between skew lines.
• Algebra (Matrices and Determinants) (10 Marks): Encompasses matrix multiplication properties, transpose, symmetric and skew-symmetric matrices, adjoint of a matrix, determinant properties, and solving systems of linear equations using matrix inversion: X = A-1B.
• Relations and Functions (8 Marks): Covers reflexive, symmetric, transitive relations, equivalence classes, one-one (injective) and onto (bijective) functions, and principal value branches of Inverse Trigonometric Functions.
• Probability (8 Marks): Includes Conditional Probability, Multiplication Theorem, Independent Events, Bayes' Theorem, and Probability Distributions.
• Linear Programming (5 Marks): Involves graphical optimization of objective functions subject to linear constraints, identifying bounded and unbounded feasible regions, and testing corner points.

Essential Mathematical Formulas & Properties

Here are fundamental equations that are frequently tested in the CBSE Class 12 board examination:

1. Matrix Inverse Formula:
   A-1 = (1 / |A|) × adj(A), provided |A| ≠ 0 (non-singular matrix).
2. Definite Integral King's Property:
   ∫ab f(x) dx = ∫ab f(a + b - x) dx
   Special case for limits 0 to a: ∫0a f(x) dx = ∫0a f(a - x) dx
3. Shortest Distance Between Skew Lines:
   Given lines r̅ = a̅1 + λb̅1 and r̅ = a̅2 + μb̅2:
   d = | ((b̅1 × b̅2) · (a̅2 - a̅1)) / |b̅1 × b̅2| |
4. Bayes' Theorem:
   For a set of mutually exclusive and exhaustive events E1, E2, ..., En and an associated event A:
   P(Ei | A) = [ P(Ei) × P(A | Ei) ] / ∑ [ P(Ek) × P(A | Ek) ]

Important CBSE Class 12 Maths 2025 Questions with Step-by-Step Solutions

The following curated questions reflect the actual difficulty, question formats, and step-marking criteria observed in the 2025 CBSE Class 12 Mathematics board examination.

Section A: 1-Mark Multiple Choice Question (Determinants & Matrices)

Question 1: If A is a square matrix of order 3 × 3 such that |A| = 4, find the value of |2 · adj(A)|.

Solution:

1. Recall the determinant property for scalar multiplication of an n × n matrix: |k · M| = kn |M|. Here n = 3, so |2 · adj(A)| = 23 × |adj(A)| = 8 × |adj(A)|.
2. Recall the adjoint determinant property: |adj(A)| = |A|n-1.
3. For n = 3 and |A| = 4: |adj(A)| = 43 - 1 = 42 = 16.
4. Substituting this value: |2 · adj(A)| = 8 × 16 = 128.

Answer: 128

Section B: 2-Mark Question (Vector Algebra)

Question 2: Find a unit vector perpendicular to both the vectors a̅ = 2î + ĵ - k̂ and b̅ = î - ĵ + 2k̂.

Solution:

A vector perpendicular to both a̅ and b̅ is given by their cross product a̅ × b̅.

Step 1: Calculate the cross product:

a̅ × b̅ = | î   ĵ   k̂ |
          | 2   1  -1 |
          | 1  -1   2 |
= î( (1)(2) - (-1)(-1) ) - ĵ( (2)(2) - (-1)(1) ) + k̂( (2)(-1) - (1)(1) )
= î(2 - 1) - ĵ(4 + 1) + k̂(-2 - 1)
= î - 5ĵ - 3k̂

Step 2: Find the magnitude:

|a̅ × b̅| = √( 12 + (-5)2 + (-3)2 ) = √(1 + 25 + 9) = √35

Step 3: Compute the unit vector:

n̂ = ± (a̅ × b̅) / |a̅ × b̅| = ± (î - 5ĵ - 3k̂) / √35 = ± [ (1/√35)î - (5/√35)ĵ - (3/√35)k̂ ]

Section C: 3-Mark Question (Definite Integrals)

Question 3: Evaluate the definite integral: I = ∫0π/2 [ (sin4 x) / (sin4 x + cos4 x) ] dx.

Solution:

Let the given integral be:

(1)   I = ∫0π/2 [ sin4 x / (sin4 x + cos4 x) ] dx

Applying the definite integral property ∫0a f(x) dx = ∫0a f(a - x) dx:

I = ∫0π/2 [ sin4(π/2 - x) / (sin4(π/2 - x) + cos4(π/2 - x)) ] dx

Since sin(π/2 - x) = cos x and cos(π/2 - x) = sin x:

(2)   I = ∫0π/2 [ cos4 x / (cos4 x + sin4 x) ] dx

Adding equations (1) and (2):

2I = ∫0π/2 [ (sin4 x + cos4 x) / (sin4 x + cos4 x) ] dx

2I = ∫0π/2 1 dx = [ x ]0π/2 = π/2 - 0 = π/2

I = π / 4

Section D: 5-Mark Question (Matrix Method for System of Linear Equations)

Question 4: Solve the following system of linear equations using the matrix method:

2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3

Solution:

The system can be represented in matrix form as AX = B, where:

A = [ [2, -3, 5], [3, 2, -4], [1, 1, -2] ],   X = [ [x], [y], [z] ],   B = [ [11], [-5], [-3] ]

Step 1: Compute the determinant |A|:

|A| = 2[(2)(-2) - (-4)(1)] - (-3)[(3)(-2) - (-4)(1)] + 5[(3)(1) - (2)(1)]
|A| = 2[-4 + 4] + 3[-6 + 4] + 5[3 - 2]
|A| = 2(0) + 3(-2) + 5(1) = 0 - 6 + 5 = -1

Since |A| = -1 ≠ 0, the matrix A is non-singular and has a unique solution given by X = A-1B.

Step 2: Calculate the cofactors of elements of matrix A:

• C11 = + [ (2)(-2) - (-4)(1) ] = -4 + 4 = 0
• C12 = - [ (3)(-2) - (-4)(1) ] = - (-6 + 4) = 2
• C13 = + [ (3)(1) - (2)(1) ] = 3 - 2 = 1
• C21 = - [ (-3)(-2) - (5)(1) ] = - (6 - 5) = -1
• C22 = + [ (2)(-2) - (5)(1) ] = -4 - 5 = -9
• C23 = - [ (2)(1) - (-3)(1) ] = - (2 + 3) = -5
• C31 = + [ (-3)(-4) - (5)(2) ] = 12 - 10 = 2
• C32 = - [ (2)(-4) - (5)(3) ] = - (-8 - 15) = 23
• C33 = + [ (2)(2) - (-3)(3) ] = 4 + 9 = 13

Step 3: Determine adj(A) and A-1:

adj(A) = [ [0, -1, 2], [2, -9, 23], [1, -5, 13] ]
A-1 = (1 / |A|) × adj(A) = (1 / -1) × [ [0, -1, 2], [2, -9, 23], [1, -5, 13] ] = [ [0, 1, -2], [-2, 9, -23], [-1, 5, -13] ]

Step 4: Compute X = A-1B:

[ [x], [y], [z] ] = [ [0, 1, -2], [-2, 9, -23], [-1, 5, -13] ] × [ [11], [-5], [-3] ]
x = (0)(11) + (1)(-5) + (-2)(-3) = 0 - 5 + 6 = 1
y = (-2)(11) + (9)(-5) + (-23)(-3) = -22 - 45 + 69 = 2
z = (-1)(11) + (5)(-5) + (-13)(-3) = -11 - 25 + 39 = 3

Answer: x = 1, y = 2, z = 3

Section E: 4-Mark Case-Study Based Question (Bayes' Theorem & Probability)

Question 5: An electronics manufacturing plant utilizes two automated assembly lines: Line A and Line B. Line A produces 60% of the total microchips, while Line B produces 40%. Quality control inspections reveal that 2% of the chips manufactured on Line A are defective, and 1% of the chips produced on Line B are defective. At the end of a production cycle, a microchip is selected at random and found to be defective.

1. What is the total probability that a randomly chosen microchip from the plant is defective? (2 Marks)
2. What is the probability that the selected defective chip was manufactured on Line A? (2 Marks)

Solution:

Let E1 be the event that the microchip is manufactured on Line A, and E2 be the event that it is manufactured on Line B.

Let D be the event that the selected microchip is defective.

• P(E1) = 60/100 = 0.6
• P(E2) = 40/100 = 0.4
• P(D | E1) = 2/100 = 0.02
• P(D | E2) = 1/100 = 0.01

Part (i): Total Probability of Defective Chip:

P(D) = P(E1) · P(D | E1) + P(E2) · P(D | E2)
P(D) = (0.6 × 0.02) + (0.4 × 0.01)
P(D) = 0.012 + 0.004 = 0.016 = 16 / 1000 = 2 / 125

Part (ii): Applying Bayes' Theorem for P(E1 | D):

P(E1 | D) = [ P(E1) · P(D | E1) ] / P(D)
P(E1 | D) = 0.012 / 0.016 = 12 / 16 = 3 / 4 = 0.75

Answer: (i) 0.016 (or 2/125); (ii) 0.75 (or 3/4)

CBSE Marking Scheme & Step-Marking Insights for Board Exam 12

The CBSE evaluation process follows strict step-marking criteria. Understanding how marks are allocated enables students to maximize their scores even on complex multi-step problems:

• Formula and Property Statements (0.5 to 1 Mark): Always write down the core formula, theorem, or integration property (e.g., King's Property or Bayes' theorem formula) before inserting numeric values.
• Intermediate Computations (1 to 2 Marks): In matrices and 3D geometry questions, marks are awarded for computing individual cofactors, finding the determinant |A|, and calculating cross products b̅1 × b̅2 independently.
• Final Answer with Correct Units/Signs (0.5 to 1 Mark): For definite integrals and area questions, always include square units (sq. units). In indefinite integrals, remember the arbitrary constant + C.
• Linear Programming & Area Graphs: Clearly shade the feasible region, label all axis intercepts, and tabulate the value of the objective function Z = ax + by at every corner point.

Proven Preparation Strategy for CBSE Class 12 Mathematics (2025-2026)

1. Complete NCERT & NCERT Exemplar: Every standard question in Section B, C, and D has roots in NCERT examples and exercises. Ensure all miscellaneous exercises are solved thoroughly.
2. Practice Competency-Based Case Studies: Section E requires applied conceptual understanding. Practice application-oriented questions from Probability, Maxima-Minima, and Integrals.
3. Simulate 3-Hour Timed Tests: Solve past 5 years' question papers and official sample papers under strict exam conditions to refine time allocation: spend ~25 minutes on Section A, ~20 minutes on Section B, ~35 minutes on Section C, ~50 minutes on Section D, and ~30 minutes on Section E, leaving 20 minutes for revision.
4. Maintain an Error Log: Track calculation oversights, algebraic sign errors, and missed trigonometric identities to avoid repeating them in the final examination.

Where to Practice More and Download Practice Papers

Consistent, targeted problem-solving is the single most effective way to excel in your Class 12 Mathematics board examination. To access comprehensive chapter-wise question banks, previous year question papers with detailed step-by-step solutions, and AI-curated mock tests tailored specifically to the latest CBSE pattern, visit Theorify QPTool (qptool.theorify.in). Start practicing today to build the speed, precision, and confidence needed to score a 100/100 in CBSE Class 12 Mathematics!