Last Minute Revision for CBSE Class 12 Maths Exam 2026

Last Minute Tips for CBSE Class 12 Maths Exam 2026 focus on helping students quickly revise the most important concepts before the exam. This guide provides a fast revision of key CBSE Class 12 Maths topics such as Relations and Functions, Inverse Trigonometry, Matrices, Determinants, Calculus (Continuity, Differentiation, Integration, and Applications), 3D Geometry, Vectors, Linear Programming, and Probability. 

The emphasis is on important formulas, key properties, and problem-solving shortcuts that help students recall concepts quickly during the exam. By revising these essential areas and practicing a few important questions, students can improve accuracy, save time, and boost confidence for the CBSE Class 12 Maths Exam 2026 on 9th March 2026.

Relations and Functions Introduction

Understanding types of relations and functions is fundamental for advanced mathematical concepts.

Types of Relations

• Reflexive Relation: (a, a) is in the relation.

• Symmetric Relation: If (a, b) is in, then (b, a) is also in.

• Transitive Relation: If (a, b) and (b, c) are in, then (a, c) is also in.

• Equivalence Relation: Reflexive, Symmetric, and Transitive.
  To disprove symmetry/transitivity, try to find a counterexample. If none exists, it is symmetric/transitive.

Types of Functions

• One-One Function (Injective): f(x1) = f(x2) implies x1 = x2.

• Many-One Function: Not one-one (f(x1) = f(x2) but x1 ≠ x2). Finding two distinct x-values with the same output proves many-one.

• On-To Function (Surjective): Codomain equals range.

• In-To Function: Codomain does not equal range. One element in the codomain not in the range proves it.

Inverse Trigonometric Functions Introduction

Grasping inverse trigonometric functions' domains, ranges, and properties is crucial for calculus.

Domains and Principal Value Ranges

Properties of Inverse Trigonometric Functions

• Negative Arguments:

• sin⁻¹(-x) = -sin⁻¹x, cosec⁻¹(-x) = -cosec⁻¹x, tan⁻¹(-x) = -tan⁻¹x.

• cos⁻¹(-x) = π - cos⁻¹x, sec⁻¹(-x) = π - sec⁻¹x, cot⁻¹(-x) = π - cot⁻¹x.

• Reciprocal Properties: sin⁻¹(1/x) = cosec⁻¹x, cos⁻¹(1/x) = sec⁻¹x, tan⁻¹(1/x) = cot⁻¹x (if x > 0).

• Complementary Properties: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, cosec⁻¹x + sec⁻¹x = π/2.

• Composition: sin(sin⁻¹x) = x. sin⁻¹(sinx) = x only if x is in its principal interval.

Sum/Difference Formulas for Inverse Tangent

• tan⁻¹x + tan⁻¹y: tan⁻¹((x+y) / (1-xy)) (if xy < 1), π + tan⁻¹((x+y) / (1-xy)) (if xy > 1).

• tan⁻¹x - tan⁻¹y: tan⁻¹((x-y) / (1+xy)).

2tan⁻¹x Formulas

• sin⁻¹(2x / (1+x²))

• cos⁻¹((1-x²) / (1+x²))

• tan⁻¹(2x / (1-x²))
  These conversions are highly useful in differentiation problems.

Matrices Introduction

Matrices are key for data manipulation and solving linear systems.

Basic Matrix Operations

• Addition/Subtraction: Orders must be identical.

• Multiplication: A (m x n), B (p x q); AB possible if n = p. Result is (m x q).

Transpose of a Matrix

• Definition: Interchanging rows and columns (Aᵀ or A').

• Properties: (Aᵀ)ᵀ = A, (kA)ᵀ = kAᵀ, (A+B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ (Reversal Law).

Symmetric and Skew-Symmetric Matrices

• Symmetric: Aᵀ = A. Elements symmetric about the main diagonal are equal (aᵢⱼ = aⱼᵢ).

• Skew-Symmetric: Aᵀ = -A.

• Diagonal elements are always zero.

• Determinant of an odd-order skew-symmetric matrix is zero.

Expressing a Matrix as Sum of Symmetric and Skew-Symmetric

Any square matrix A can be uniquely expressed as A = P + Q:

• P = ½ (A + Aᵀ) (symmetric part)

• Q = ½ (A - Aᵀ) (skew-symmetric part)

Determinants Introduction

Determinants provide scalar values crucial for matrix invertibility and system solutions.

Area of a Triangle

Area = ½ | det([ x₁ y₁ 1 ], [ x₂ y₂ 1 ], [ x₃ y₃ 1 ]) |.

Collinearity: If points are collinear, Area = zero.

Adjoint and Inverse of a Matrix Introduction

Adjoint and inverse are vital for matrix equations and properties.

Adjoint of a Matrix

Adj A is the transpose of its cofactor matrix.

1. Find Minor (Mᵢⱼ).

2. Find Cofactor (Cᵢⱼ): Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ.

3. Form Cofactor Matrix, then take transpose.

Inverse of a Matrix

• A⁻¹ = (1 / det(A)) * adj(A).

• Invertible if det(A) ≠ 0.

Properties of Determinants and Inverses

• det(Aᵀ) = det(A), det(A⁻¹) = 1 / det(A).

• det(adj(A)) = (det(A))ⁿ⁻¹ (n is order, highly important).

• A * adj(A) = adj(A) * A = det(A) * I (crucial).

• (AB)⁻¹ = B⁻¹A⁻¹, det(AB) = det(A) * det(B).

• det(kA) = kⁿ det(A) (k scalar, n order, very important). When a scalar 'k' is multiplied by a matrix 'A', and then its determinant is taken, 'k' comes out with a power equal to the order of the matrix.

Continuity and Differentiability Introduction

These calculus concepts define function smoothness and behavior.

Continuity of a Function

f(x) is continuous at x = a if lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = f(a).

Differentiability of a Function

f(x) is differentiable at x = a if its Left Hand Derivative (LHD) equals Right Hand Derivative (RHD).

Relationship: Differentiability implies continuity, but the converse is not always true.

Solving System of Linear Equations

• Matrix Method: For AX = B, the solution is X = A⁻¹B.

• Smart Approach for A⁻¹: If AZ = kI, then A⁻¹ = (1/k)Z. When multiplying two matrices (e.g., 3x3), often only one or two elements of the product matrix need to be calculated to identify the pattern.

Differentiation Introduction

Differentiation measures function change, tangent slopes, and is key for optimization.

Standard Differentiation Formulas

• Trigonometric: d/dx (sinx) = cosx; d/dx (cosx) = -sinx; d/dx (tanx) = sec²x; etc.

• Exponential/Logarithmic: d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x; d/dx (aˣ) = aˣ log_e a.

• Inverse Trigonometric: d/dx (sin⁻¹x) = 1 / √(1-x²); d/dx (cos⁻¹x) = -1 / √(1-x²); d/dx (tan⁻¹x) = 1 / (1+x²); etc. The derivatives of co-functions are the negative of their counterparts.

Rules of Differentiation

• Constant Multiple: d/dx [k * f(x)] = k * d/dx [f(x)]

• Sum/Difference: d/dx [f(x) ± g(x)] = d/dx [f(x)] ± d/dx [g(x)]

• Product: d/dx [f(x) * g(x)] = f(x)g'(x) + g(x)f'(x)

• Quotient: d/dx [f(x) / g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]²

Chain Rule and Implicit/Logarithmic Differentiation

• Chain Rule: If y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx).

• Implicit Differentiation: Differentiate both sides of an implicit equation with respect to x, treating y as a function of x.

• Logarithmic Differentiation: Essential for functions of the form [f(x)]^(g(x)). Take natural logarithm, use log properties, then differentiate.

Applications of Derivatives Introduction

Derivatives analyze rates of change, optimization, and function behavior.

Rate of Change and Tangent Slope

• Rate of Change: dy/dx represents the instantaneous rate of change.

• Slope of Tangent: dy/dx evaluated at a specific point.

Increasing and Decreasing Functions

• Strictly Increasing: f'(x) > 0.

• Increasing: f'(x) ≥ 0.

• Strictly Decreasing: f'(x) < 0.

• Decreasing: f'(x) ≤ 0.

Maxima and Minima (First Derivative Test)

1. Find f'(x), set to 0 for critical points (α).

2. Analyze sign change of f'(x) around α:

• Positive to negative: local maximum.

• Negative to positive: local minimum.

• No sign change: point of inflection.

Maxima and Minima (Second Derivative Test)

1. Find f'(x), set to 0 for critical points (α).

2. Evaluate f''(α):

• f''(α) > 0: local minimum. (Positive = Minimum, opposite of intuition.)

• f''(α) < 0: local maximum. (Negative = Maximum, opposite of intuition.)

• f''(α) = 0: test fails; use First Derivative Test.

Absolute Maximum and Minimum in a Closed Interval

1. Find critical points where f'(x) = 0 within the interval.

2. Identify endpoints of the interval.

3. Evaluate f(x) at all critical points and endpoints.

4. Largest value is absolute maximum; smallest is absolute minimum.

Integration Introduction

Integration finds areas, volumes, and solves differential equations.

Indefinite vs. Definite Integration

Standard Integration Formulas

• ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ -1), ∫1 dx = x + C.

• ∫cosx dx = sinx + C, ∫sinx dx = -cosx + C.

• ∫sec²x dx = tanx + C, ∫cosec²x dx = -cotx + C.

• ∫eˣ dx = eˣ + C, ∫(1/x) dx = ln|x| + C, ∫aˣ dx = aˣ / ln(a) + C.

• ∫tan²x dx = tanx - x + C, ∫cot²x dx = -cotx - x + C.

Integration with Linear Functions

If a function f(ax+b) is integrated, where (ax+b) is linear:

∫f(ax+b) dx = [F(ax+b) / a] + C (where F is the antiderivative of f). In integration, we divide by the derivative of the inner linear function.

Inverse Trigonometric Form Integrals

• ∫(1 / √(a²-x²)) dx = sin⁻¹(x/a) + C

• ∫(1 / (a²+x²)) dx = (1/a) tan⁻¹(x/a) + C

• ∫(1 / (x√(x²-a²))) dx = (1/a) sec⁻¹(|x|/a) + C

Special Algebraic Form Integrals

• ∫(1 / (x²-a²)) dx = (1 / 2a) ln |(x-a) / (x+a)| + C

• ∫(1 / (a²-x²)) dx = (1 / 2a) ln |(a+x) / (a-x)| + C

Integrals of Quadratic Forms in Denominator with Square Root

• ∫(1 / √(x²+a²)) dx = ln |x + √(x²+a²)| + C

• ∫(1 / √(x²-a²)) dx = ln |x + √(x²-a²)| + C

Integration by Parts

∫u dv = uv - ∫v du. Use ILATE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) to choose 'u' (differentiable) and 'dv' (integrable).

Special Integrals (eˣ [f(x) + f'(x)])

∫eˣ [f(x) + f'(x)] dx = eˣ f(x) + C (very powerful and frequently tested).

Integration by Partial Fractions

Applicable for P(x)/Q(x) where degree P(x) < degree Q(x).

• Linear Non-Repeated Factors: A/(x-a) + B/(x-b)

• Linear Repeated Factors: A/(x-a) + B/(x-a)² + C/(x-b)

• Irreducible Quadratic Factors: (Ax+B)/(x²+ax+b) + C/(x-c)

Integrals of Root Quadratic Forms (in Numerator)

• ∫√(a²-x²) dx = (x/2)√(a²-x²) + (a²/2)sin⁻¹(x/a) + C

• ∫√(x²+a²) dx = (x/2)√(x²+a²) + (a²/2)ln|x + √(x²+a²)| + C

• ∫√(x²-a²) dx = (x/2)√(x²-a²) - (a²/2)ln|x + √(x²-a²)| + C

Definite Integration Introduction

Definite integrals calculate exact areas and accumulated quantities between limits.

Properties of Definite Integrals

1. ∫(a to b) f(x) dx = - ∫(b to a) f(x) dx. (Swapping limits changes sign).

2. ∫(a to b) f(x) dx = ∫(a to c) f(x) dx + ∫(c to b) f(x) dx. (Splitting over intermediate point c).

3. King Property: ∫(a to b) f(x) dx = ∫(a to b) f(a+b-x) dx. (Replacing x with a+b-x). A common application is ∫(0 to a) f(x) dx = ∫(0 to a) f(a-x) dx.

4. Queen Property (0 to 2a):

• = 2 ∫(0 to a) f(x) dx if f(2a-x) = f(x).

• = 0 if f(2a-x) = -f(x).

5. Symmetric Limits (-a to a):

• = 2 ∫(0 to a) f(x) dx if f(x) is an even function (f(-x) = f(x)).

• = 0 if f(x) is an odd function (f(-x) = -f(x)).

Applications of Integrals: Area Introduction

Integrals calculate areas bounded by curves and axes.

Area between a Curve and the X-axis

Area = ∫(a to b) |f(x)| dx. Area is always positive; modulus must be applied.

Area between a Curve and the Y-axis

Area = ∫(a to b) |g(y)| dy (for x = g(y)).

Area between Two Curves

• With respect to x: ∫(a to b) [Upper Curve - Lower Curve] dx.

• With respect to y: ∫(a to b) [Right Curve - Left Curve] dy.

Differential Equations Introduction

Differential equations model dynamic processes.

Order and Degree of a Differential Equation

• Order: Order of the highest order derivative present.

• Degree: Power of the highest order derivative, if all differential coefficients are in polynomial form. Otherwise, the degree is not defined.

Methods of Solving Differential Equations

1. Variable Separable: Separate x-terms with dx and y-terms with dy, then integrate both sides.

2. Homogeneous Differential Equation: All terms in the equation have the same degree. Use substitution y = vx (or x = ty).

3. Linear Differential Equation:

• Standard Form 1: dy/dx + Py = Q. Calculate Integrating Factor (IF) = e^(∫P dx). Solution: y * IF = ∫(Q * IF) dx + C.

• Standard Form 2: dx/dy + Px = Q. Calculate IF = e^(∫P dy). Solution: x * IF = ∫(Q * IF) dy + C.

Vectors Introduction

Vectors have magnitude and direction, essential for physics and geometry.

Basic Vector Operations

• Magnitude of a Vector: For a = ai + bj + ck, |a| = √(a² + b² + c²).

• Unit Vector: â = a / |a|. The unit vector gives the direction of the vector.

Direction Cosines and Direction Ratios

• Direction Ratios (DRs): The coefficients of i, j, k (a, b, c).

• Direction Cosines (DCs): l = a/|a|, m = b/|a|, n = c/|a|.

• Property: The sum of squares of direction cosines is 1: l² + m² + n² = 1.

Scalar (Dot) Product

• a ⋅ b = |a| |b| cos θ.

• Perpendicular Vectors: If a ⋅ b = 0, a and b are perpendicular.

• Parallel/Collinear Vectors: Their respective coefficients (DRs) are proportional.

Vector (Cross) Product

• a × b = |a| |b| sin θ n̂.

• Vector Perpendicular to Two Given Vectors: a × b gives a vector perpendicular to both a and b. If asked for unit vectors, the answer is always two (± (a × b) / |a × b|).

Projection of a Vector

Projection of vector a on vector b: (a ⋅ b) / |b|.

Area using Cross Product

• Area of Triangle: If adjacent sides are a and b, Area = (1/2) |a × b|.

• Area of Parallelogram (sides): If adjacent sides are a and b, Area = |a × b|.

• Area of Parallelogram (diagonals): If diagonals are d₁ and d₂, Area = (1/2) |d₁ × d₂|.

Identity: Modulus Square of a Vector

The square of the modulus of any vector a is equal to its dot product with itself: |a|² = a ⋅ a. This identity is very important.

Application: |a + b + c|² = |a|² + |b|² + |c|² + 2(a ⋅ b + b ⋅ c + c ⋅ a).

3D Geometry Introduction

3D Geometry extends coordinate geometry to spatial lines and planes.

Direction of a Line

• Direction Ratios (DRs): Given by any vector parallel to the line.

• Direction Cosines (DCs) (l, m, n): Related by l² + m² + n² = 1.

Angle Between Two Lines

For direction vectors b₁ and b₂, cos θ = |(*b₁ ⋅ b₂) / (|b₁| |b₂|)|*.

• Perpendicular: If b₁ ⋅ b₂ = 0.

• Parallel: If b₁ and b₂ are proportional.

Equation of a Line

• Vector Form (Through point a, parallel to b): r = a + λb.

• Vector Form (Through two points a and b): r = a + λ(b - a).

• Cartesian Form (Through point (x₁, y₁, z₁), DRs (a, b, c)): (x - x₁) / a = (y - y₁) / b = (z - z₁) / c.

• Cartesian Form (Through two points (x₁, y₁, z₁), (x₂, y₂, z₂)): (x - x₁) / (x₂ - x₁) = (y - y₁) / (y₂ - y₁) = (z - z₁) / (z₂ - z₁) .

General Point of a Line

From the Cartesian form (x - x₁) / a = (y - y₁) / b = (z - z₁) / c = λ, any point on the line can be represented as (x₁ + aλ, y₁ + bλ, z₁ + cλ). This "general point of a line" is very useful for solving problems.

Distance Between Two Lines

• Skew Lines: **r₁** = **a₁** + λ**b₁**, **r₂** = **a₂** + μ**b₂**.

• Distance d = |((**a₂** - **a₁**) ⋅ (**b₁** × **b₂**)) / |**b₁** × **b₂**||. Convert Cartesian equations to vector form.

• Parallel Lines: **r₁** = **a₁** + λ**b**, **r₂** = **a₂** + μ**b**.

• Distance d = |((**a₂** - **a₁**) × **b**) / |**b**||.

Image of a Point and Foot of Perpendicular from a Point to a Line

This topic is very important.

1. Let the given point be P, and the line L have a general point F (in terms of λ).

2. Form the vector PF.

3. Since PF is perpendicular to the direction vector b of line L, PF ⋅ b = 0. Solve this for λ.

4. Substitute λ back into F to get the Foot of Perpendicular (F).

5. To find the Image I: The foot F is the midpoint of the segment PI. Use the midpoint formula to find I.

Linear Programming Problems (LPP) Introduction

LPPs optimize a linear objective function under linear constraints.

Components of an LPP

• Objective Function (Z): A linear function to be maximized or minimized.

• Constraints: Linear inequalities (e.g., x + y ≤ 10, x ≥ 0, y ≥ 0).

• Feasible Region: The region satisfying all constraints.

• Corner Points: Vertices of the feasible region.

Solving an LPP

1. Plot the constraints to identify the feasible region.

2. Identify the corner points of the feasible region.

3. Evaluate the objective function (Z) at each corner point.

4. Bounded Region: The maximum or minimum value of Z will occur at one of the corner points.

5. Unbounded Region: Check half-plane (Ax + By > Z_max or < Z_min) for common points with the feasible region to determine if a max/min exists.

6. Special Case: If the maximum/minimum value occurs at two corner points, it occurs at every point on the line segment joining these two points.

Probability Introduction

Probability studies chance and uncertainty, analyzing random events.

Conditional Probability

• Definition: The probability of event A occurring given that event B has already occurred.

• Formula: P(A|B) = P(A ∩ B) / P(B) (where P(B) ≠ 0). Conditional probability means a restriction of the sample space.

Multiplication Theorem of Probability

• Formula: P(A ∩ B) = P(A) * P(B|A) or P(A ∩ B) = P(B) * P(A|B).

Independent Events

• Definition: Events A and B are independent if the occurrence of one does not affect the probability of the other.

• Formula: P(A ∩ B) = P(A) * P(B).

Mutually Exclusive Events

• Definition: Events A and B are mutually exclusive if they cannot occur simultaneously.

• Formula: P(A ∩ B) = 0.

Exhaustive Events

• Definition: A set of events is exhaustive if at least one of them must occur, covering the entire sample space.

• If events are both mutually exclusive and exhaustive, their probabilities sum to 1: P(E₁) + P(E₂) + … + P(Eₙ) = 1.

Important Probability Formulas

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

• P(A' ∩ B') = 1 - P(A ∪ B) (De Morgan's Law).

• P(A' ∪ B') = 1 - P(A ∩ B) (De Morgan's Law).

Total Probability Theorem

If E₁, E₂, …, Eₙ are mutually exclusive and exhaustive events:

P(A) = P(A|E₁)P(E₁) + P(A|E₂)P(E₂) + … + P(A|Eₙ)P(Eₙ). This represents the probability of reaching a "destination" (A) through multiple "paths" (Eᵢ).

Bayes' Theorem

If E₁, E₂, …, Eₙ are mutually exclusive and exhaustive events:

P(Eᵢ|A) = [P(A|Eᵢ)P(Eᵢ)] / P(A) (where P(A) is from the Total Probability Theorem). This calculates the probability that a specific "path" (Eᵢ) was taken to reach the "destination" (A). Tree diagrams are very helpful for visualizing these problems.

Random Variable and Probability Distribution

• Random Variable (X): A variable whose value is a numerical outcome of a random phenomenon.

• Probability Distribution Table: A table listing possible values of X and their probabilities P(x). The sum of all probabilities Σ P(x) must be equal to 1.

• Mean (Expectation) of a Random Variable E(X): E(X) = Σ [x * P(x)].

Important Trigonometric Identities for Integration and Differentiation Introduction

Mastering key trigonometric identities is vital for simplifying complex expressions in calculus.

Half-Angle Identities for Cosine

• 1 + cos(2θ) = 2 cos²θ

• 1 - cos(2θ) = 2 sin²θ

Half-Angle Identities for Sine

• 1 + sin(θ) = [sin(θ/2) + cos(θ/2)]²

• 1 - sin(θ) = [sin(θ/2) - cos(θ/2)]²

Double Angle Identity for Cosine

cos(θ) = cos²(θ/2) - sin²(θ/2)

Double Angle Identity for Sine

sin(θ) = 2 sin(θ/2) cos(θ/2)

Tan Conversion Identity

• (1 + tan x) / (1 - tan x) = tan(π/4 + x)

• (1 - tan x) / (1 + tan x) = tan(π/4 - x)
  These conversions are crucial for simplifying expressions in integration and differentiation.

Examination Preparation: Tips and Mindset for Success Introduction

Strategic planning and a focused mindset are crucial for exam performance.

General Exam Day Instructions

• Handwriting: Keep your handwriting neat and legible.

• Attempt Questions: If you tend to solve slowly, consider attempting the paper in reverse order of sections. Attempt Multiple Choice Questions (MCQs) towards the end.

• No Panic: Do not panic if others request supplementary sheets. Stay focused on your own paper. Definitely carry a water bottle.

Specific Guidelines for Rough Work

• Rough work can be done on the right side of your answer sheet or the end pages of your answer booklet. Request a supplementary sheet for extensive rough work.

• DO NOT scribble on the question paper. All writing, including rough work, must be done exclusively in the answer booklet.

Mindset & Final Motivational Boost

• Post-Completion Check: After completing, ensure you have attempted every part of each question, especially "follow-up questions" within a problem.

• Positive Mindset: Approach the exam with a positive and confident attitude. Trust that you know everything.

• Effort, Not Result: Concentrate on giving your best effort; the result is not entirely in your hands. Do not take stress about the outcome.