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ICSE Class 12 Mathematics★ Full ISC Class 12 Mathematics syllabus · 100 marks · Board: ISC

Mathematics — Calculus, Vectors, 3D Geometry, Probability & Linear Programming

ISC Class 12 Mathematics: RELATIONS and FUNCTIONS (advanced). INVERSE TRIGONOMETRY. MATRICES and DETERMINANTS. The CALCULUS sequence — continuity, derivatives, integrals, differential equations. VECTOR ALGEBRA. 3D GEOMETRY. LINEAR PROGRAMMING. And PROBABILITY (Bayes' Theorem, random variables, binomial distribution). ISC Class 12 Mathematics.

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By the end of this chapter you'll be able to…

1Apply differentiation techniques — chain rule, implicit, parametric, logarithmic — and use derivatives for tangents, normals, maxima/minima, and rate-of-change problems
2Evaluate definite and indefinite integrals using substitution, integration by parts (ILATE), partial fractions, and standard formulas; compute areas between curves
3Solve first-order differential equations by variable-separable, homogeneous, and linear integrating factor methods
4Perform vector operations (dot product, cross product, scalar triple product) and solve 3D geometry problems involving lines and planes in vector and Cartesian forms
5Apply Bayes' theorem, calculate expected value and variance of discrete random variables, and use the binomial distribution to model random experiments

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Why this chapter matters

ISC Mathematics is one of the most marks-scoring subjects for students who master the techniques — the paper is purely computational and every step earns marks. Calculus (differentiation, integration, differential equations) accounts for ~45 marks. Vectors and 3D Geometry together ~15 marks. Probability ~10 marks. Linear programming ~5 marks. Matrices and determinants ~15 marks. Inverse trigonometry ~5 marks. Students who practise integration techniques exhaustively score 85+ consistently.

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Mathematics — Calculus, Vectors, 3D, Probability & More

1. Relations and Functions

Types of Relations

Reflexive: (a,a)∈R ∀a. Symmetric: (a,b)∈R ⇒ (b,a)∈R. Transitive: (a,b)∈R∧(b,c)∈R ⇒ (a,c)∈R. Equivalence = Reflexive + Symmetric + Transitive.

Functions

One-One (Injective) : f(x₁)=f(x₂) ⇒ x₁=x₂. Onto (Surjective) : Range = Codomain. Bijective: Both = INVERTIBLE.
Composition: (f∘g)(x) = f(g(x)). f is invertible ⇔ f is bijective.

2. Inverse Trigonometric Functions

Function	Domain	Principal Value Range
sin⁻¹x	[-1, 1]	[-π/2, π/2]
cos⁻¹x	[-1, 1]	[0, π]
tan⁻¹x	R	(−π/2, π/2)

Key Identities: sin⁻¹x + cos⁻¹x = π/2. tan⁻¹x + cot⁻¹x = π/2. tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)].

3. Matrices and Determinants

Matrix Operations

Multiplication: AB ≠ BA (generally). (AB)′ = B′A′.
Inverse: A⁻¹ = adj(A)/|A| (|A| ≠ 0).

Determinants — Properties

|A′| = |A|. Interchanging rows → sign changes. Identical rows → |A|=0.
Cramer's Rule: For AX = B, xᵢ = |Aᵢ|/|A|.

4. Continuity and Differentiability

Continuity at x = a

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

Differentiability

f is differentiable if lim(h→0) [f(x+h)−f(x)]/h EXISTS. Differentiability ⇒ Continuity. Converse NOT true (|x| at x=0).

Differentiation Techniques

Chain rule. Logarithmic differentiation. Implicit functions. Parametric forms. Second-order derivatives.

Rolle's Theorem

If f continuous on [a,b], differentiable on (a,b), and f(a)=f(b) → ∃c∈(a,b): f′(c)=0.

Mean Value Theorem (MVT)

∃c∈(a,b): f′(c) = [f(b)−f(a)]/(b−a).

5. Application of Derivatives

Rate of Change. Tangents and Normals.

Increasing/Decreasing: f′(x) > 0 ⇒ increasing. f′(x) < 0 ⇒ decreasing.

Maxima and Minima

First Derivative Test: Critical points (f′=0). Check sign change.
Second Derivative Test: f″>0 ⇒ local MINIMUM. f″<0 ⇒ local MAXIMUM.

6. Integrals

Indefinite Integration — Antiderivative + C.

Methods

Substitution. Integration by Parts: ∫u dv = uv − ∫v du. ILATE rule. Partial Fractions.

Definite Integral — ∫ₐᵇ f(x)dx = F(b)−F(a). (FTC)

Properties: ∫ₐᵇ = −∫ᵦᵃ. ∫₋ₐᵃ f(x)dx = 0 (odd) = 2∫₀ᵃ (even).

Area Under Curves

Area between y=f(x) (upper) and y=g(x) (lower) from a to b: ∫ₐᵇ [f(x)−g(x)] dx.

7. Differential Equations

Order and Degree.

Solving First-Order, First-Degree:

Variable Separable: dy/dx = f(x)g(y) → ∫(1/g)dy = ∫f dx.
Homogeneous: y=vx substitution → separable.
Linear: dy/dx + Py = Q. IF = e^∫Pdx. Solution: y·IF = ∫Q·IF dx + C.

8. Vector Algebra

Dot Product: a·b = |a||b|cos θ = a₁b₁+a₂b₂+a₃b₃. a·b=0 ⇔ a⟂b.

Cross Product: a×b = |a||b| sin θ n̂. |a×b| = area of parallelogram. a×b=0 ⇔ a∥b.

Scalar Triple Product: [a b c] = a·(b×c) = volume of parallelepiped. =0 ⇒ coplanar.

9. Three-Dimensional Geometry

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

Line: r = a + λb. (x−x₁)/a = (y−y₁)/b = (z−z₁)/c.

Plane: r·n = d. ax+by+cz+d=0. Intercept form: x/a + y/b + z/c = 1.

Distances and Angles

Distance of point from plane: |ax₁+by₁+cz₁+d|/√(a²+b²+c²).
Angle between two planes: cos θ = |n₁·n₂|/(|n₁||n₂|).

10. Linear Programming

Graphical Method

Graph constraints as lines. Shade FEASIBLE REGION. 2. Find corner points. 3. Evaluate objective function Z at each corner. 4. Optimal value at a corner (if bounded).

11. Probability

Conditional Probability: P(A|B) = P(A∩B)/P(B).

Bayes' Theorem

P(Bᵢ|A) = [P(Bᵢ)·P(A|Bᵢ)] / ΣⱼP(Bⱼ)·P(A|Bⱼ). 'Reverses the conditional — given the EVIDENCE, what's the probability of the CAUSE?'

Random Variables

Discrete: Probability mass function. P(X=xᵢ) = pᵢ. Σpᵢ = 1.
Mean μ = Σxᵢpᵢ = E(X). Variance σ² = E(X²) − [E(X)]².

Binomial Distribution

n independent trials. Two outcomes (Success p, Failure q=1−p). P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ (r=0,1,...,n). Mean = np. Variance = npq.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Integration Techniques — Comprehensive

SUBSTITUTION: ∫f(g(x))g′(x)dx — let u = g(x). BY PARTS (ILATE): ∫u dv = uv − ∫v du. ILATE priority: Inverse trig > Logarithmic > Algebraic > Trigonometric > Exponential. PARTIAL FRACTIONS: For rational functions — decompose into simpler fractions before integrating. STANDARD FORMS: ∫1/(a²−x²) dx = (1/2a)ln|(a+x)/(a−x)|+C. ∫1/√(a²−x²) dx = sin⁻¹(x/a)+C. ∫1/(a²+x²) dx = (1/a)tan⁻¹(x/a)+C. DEFINITE INTEGRAL PROPERTY: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx (King's property). AREA = ∫ₐᵇ |f(x)−g(x)| dx.

ISC integration is 25-30 marks across the paper. The integration by parts ILATE rule must be second nature. For ∫eˣ[f(x) + f′(x)]dx = eˣf(x) + C — memorise this directly.

Differential Equations — Solution Methods

VARIABLE SEPARABLE: dy/dx = f(x)/g(y) → ∫g(y)dy = ∫f(x)dx + C. HOMOGENEOUS: dy/dx = F(y/x) → substitute y = vx, dy/dx = v + x(dv/dx) → separable. LINEAR FIRST ORDER: dy/dx + P(x)y = Q(x) → Integrating Factor = e^∫P dx. Solution: y × IF = ∫(Q × IF)dx + C. BERNOULLI: dy/dx + Py = Qyⁿ → divide by yⁿ, substitute z = y^(1−n).

Always find the GENERAL SOLUTION first (with arbitrary constant C), then apply initial conditions to find PARTICULAR SOLUTION. State the order (highest derivative) and degree (power of highest derivative) before solving.

Vectors — Key Products and Applications

DOT PRODUCT: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃. Projection of a on b = (a·b)/|b|. a⟂b ⟺ a·b = 0. CROSS PRODUCT: |a×b| = |a||b|sinθ. Direction: right-hand rule. a×b = det|i j k; a₁ a₂ a₃; b₁ b₂ b₃|. Area of parallelogram = |a×b|. SCALAR TRIPLE PRODUCT: [a b c] = a·(b×c) = det(3×3 matrix). Volume of parallelepiped = |[a b c]|. Coplanar vectors ⟺ [a b c] = 0.

For ISC 3D questions: equation of LINE through point A with direction b: r = a + λb (vector), (x−x₁)/l = (y−y₁)/m = (z−z₁)/n (Cartesian). Angle between lines: cosθ = |l₁l₂+m₁m₂+n₁n₂|/1. Shortest distance between skew lines: d = |[(a₂−a₁)·(b₁×b₂)]|/|b₁×b₂|.

Probability — Bayes' Theorem and Distributions

CONDITIONAL PROBABILITY: P(A|B) = P(A∩B)/P(B). TOTAL PROBABILITY: P(A) = ΣP(Bᵢ)·P(A|Bᵢ). BAYES' THEOREM: P(Bᵢ|A) = P(Bᵢ)·P(A|Bᵢ) / ΣP(Bⱼ)·P(A|Bⱼ). BINOMIAL DISTRIBUTION: P(X=r) = ⁿCᵣ·pʳ·qⁿ⁻ʳ. Mean μ = np. Variance σ² = npq. EXPECTED VALUE: E(X) = Σxᵢpᵢ. Var(X) = E(X²) − [E(X)]² = Σxᵢ²pᵢ − (Σxᵢpᵢ)².

Bayes' theorem questions in ISC: typically a disease diagnosis scenario. 'A test is 95% accurate. 1% of population has the disease. A person tests positive — what's the probability they have the disease?' This type of problem is asked every few years and solved by careful Bayes application.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting the constant of integration C in indefinite integrals

✓ Every indefinite integral MUST have +C. ISC deducts marks for missing C. The constant is not cosmetic — it represents the FAMILY of functions (all antiderivatives) that differ by a constant. In particular solutions, C is found from the initial condition.

WATCH OUT

✗ Using the wrong sign in the 3D distance of a point from a plane formula

✓ Distance of point (x₁,y₁,z₁) from plane ax+by+cz+d=0 is |ax₁+by₁+cz₁+d|/√(a²+b²+c²). The absolute value ensures POSITIVE distance. Do NOT include a minus sign accidentally. Also check: the plane equation must be in the form ax+by+cz+d=0 (with d on the left side, not as a separate constant).

WATCH OUT

✗ Applying ILATE rule incorrectly in integration by parts

✓ ILATE = Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. The FIRST function in the list (leftmost) becomes 'u' (differentiated). The SECOND becomes 'dv' (integrated). So ∫x·sin(x)dx: x is Algebraic (A) and sin x is Trigonometric (T). A comes before T in ILATE, so u = x, dv = sin(x)dx. For ∫x·eˣdx: u = x (algebraic), dv = eˣdx.

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· inverse-trig

Prove that sin⁻¹(3/5) + sin⁻¹(4/5) = π/2.

▶ Show solution

Let A = sin⁻¹(3/5), so sin A = 3/5 → cos A = 4/5. Let B = sin⁻¹(4/5), so sin B = 4/5 → cos B = 3/5. Now: cos(A+B) = cos A cos B − sin A sin B = (4/5)(3/5) − (3/5)(4/5) = 12/25 − 12/25 = 0. Therefore A + B = cos⁻¹(0) = π/2. Hence sin⁻¹(3/5) + sin⁻¹(4/5) = π/2. ∎

Q2MEDIUM· integration-by-parts

Evaluate ∫ x² · eˣ dx.

▶ Show solution

Use integration by parts twice. ILATE: u = x², dv = eˣdx. Then v = eˣ, du = 2x dx. ∫x²eˣ dx = x²eˣ − ∫eˣ · 2x dx = x²eˣ − 2∫xeˣ dx. Now ∫xeˣ dx by parts: u = x, dv = eˣdx → v = eˣ, du = dx. = xeˣ − ∫eˣ dx = xeˣ − eˣ. Therefore: ∫x²eˣ dx = x²eˣ − 2(xeˣ − eˣ) + C = eˣ(x² − 2x + 2) + C.

Q3HARD· differential-equations

Solve the differential equation: dy/dx + y/x = x², given that y = 1 when x = 1.

▶ Show solution

This is a LINEAR first-order ODE. dy/dx + Py = Q where P = 1/x, Q = x². IF = e^∫(1/x)dx = e^(ln x) = x. Multiplying: d/dx(xy) = x · x² = x³. Integrating: xy = ∫x³ dx = x⁴/4 + C. General solution: y = x³/4 + C/x. Applying IC: y(1) = 1 → 1 = 1/4 + C → C = 3/4. PARTICULAR SOLUTION: y = x³/4 + 3/(4x) = (x⁴ + 3)/(4x).

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Bijective function ↔ invertible. f∘g exists if range of g ⊆ domain of f.
•sin⁻¹x + cos⁻¹x = π/2. tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] if xy < 1.
•Differentiability ⇒ continuity. Converse FALSE: |x| at x=0 is continuous but not differentiable.
•Rolle's theorem: f(a)=f(b), continuous, differentiable → ∃c: f′(c)=0.
•Integration by parts ILATE: ∫eˣ[f(x)+f′(x)]dx = eˣf(x) + C (direct formula).
•Linear ODE: dy/dx + Py = Q. IF = e^∫P dx. Solution: y·IF = ∫Q·IF dx + C.
•a·b = 0 ⟺ perpendicular. a×b = 0 ⟺ parallel. [a b c] = 0 ⟺ coplanar.
•Distance of point from plane |ax₁+by₁+cz₁+d|/√(a²+b²+c²).
•P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ) / ΣP(Bⱼ)P(A|Bⱼ). Bayes' theorem.
•Binomial distribution: E(X) = np. Var(X) = npq. P(X=r) = ⁿCᵣpʳqⁿ⁻ʳ.

CBSE marks blueprint

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Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research the Fundamental Theorem of Calculus in depth — it connects differentiation (instantaneous rate) and integration (accumulation) as INVERSE operations. Investigate how Newton and Leibniz independently discovered this in the 1660s-70s and how it transformed mathematics. The bridge: ∫ₐᵇ f'(x) dx = f(b) − f(a).
Investigate Vector Calculus — extending calculus to 3D vector fields. Gradient (∇f), divergence (∇·F), and curl (∇×F) describe how fields change in space. Maxwell's equations of electromagnetism are entirely vector calculus. This is the natural extension of ISC Class 12 vectors and calculus and the language of modern physics.
Explore the Central Limit Theorem — one of the most important results in probability. Regardless of the underlying distribution, the sum (or mean) of independent random variables tends to a normal distribution as the number of variables grows. This explains why the bell curve appears everywhere in nature and underlies all statistical inference.
Research Differential Equations beyond Class 12 — partial differential equations (PDEs) describe phenomena like heat flow (heat equation), wave propagation (wave equation), and fluid dynamics (Navier-Stokes equations). The Navier-Stokes problem is one of the seven Millennium Prize Problems — proving solutions exist and are smooth is worth $1 million.

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SUBSTITUTION: Use when you can spot a function and its derivative in the integrand — the derivative cancels out after substitution. E.g., ∫2x(x²+1)⁵ dx — set u = x²+1. BY PARTS: Use when the integrand is a PRODUCT of two different types of functions (e.g., polynomial × exponential, polynomial × trigonometric, logarithm × anything). Apply ILATE to decide which is u. If substitution doesn't simplify, try by parts.

EXTREMUM (max/min): f′(x) = 0 AND f′ changes sign at that point. If f′ changes from + to −: local MAX. From − to +: local MIN. INFLECTION POINT: f″(x) = 0 AND f″ changes sign (concavity changes — from concave up to concave down or vice versa). At an inflection point, f′ may or may not be zero. If f′(x) = 0 but f′ does NOT change sign, it is NOT a maximum or minimum — it could be an inflection point.

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Related chapters

Students who read Mathematics — Calculus, Vectors, 3D Geometry, Probability & Linear Programming usually read these next.

'Relations and Functions'

'Types of relations — reflexive, symmetric, transitive, equivalence. Types of functions — one-one, onto, bijective. Composition of functions, invertible functions, and binary operations.'

'Inverse Trigonometric Functions'

'Principal values, domain and range, properties and graphs of inverse trigonometric functions. Key identities and their applications in ISC problems.'

'Matrices'

'Types of matrices, algebra including addition and multiplication, transpose, symmetric and skew-symmetric matrices, elementary row/column operations, and finding inverse using elementary operations.'

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Mathematics — Calculus, Vectors, 3D Geometry, Probability & Linear Programming

By the end of this chapter you'll be able to…

Before you start — revise these

Mathematics — Calculus, Vectors, 3D, Probability & More

1. Relations and Functions

Types of Relations

Functions

2. Inverse Trigonometric Functions

3. Matrices and Determinants

Matrix Operations

Determinants — Properties

4. Continuity and Differentiability

Continuity at x = a

Differentiability

Differentiation Techniques

Rolle's Theorem

Mean Value Theorem (MVT)

5. Application of Derivatives

Rate of Change. Tangents and Normals.

Increasing/Decreasing: f′(x) > 0 ⇒ increasing. f′(x) < 0 ⇒ decreasing.

Maxima and Minima

6. Integrals

Indefinite Integration — Antiderivative + C.

Methods

Definite Integral — ∫ₐᵇ f(x)dx = F(b)−F(a). (FTC)

Area Under Curves

7. Differential Equations

Order and Degree.

Solving First-Order, First-Degree:

8. Vector Algebra

Dot Product: a·b = |a||b|cos θ = a₁b₁+a₂b₂+a₃b₃. a·b=0 ⇔ a⟂b.

Cross Product: a×b = |a||b| sin θ n̂. |a×b| = area of parallelogram. a×b=0 ⇔ a∥b.

Scalar Triple Product: [a b c] = a·(b×c) = volume of parallelepiped. =0 ⇒ coplanar.

9. Three-Dimensional Geometry

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

Line: r = a + λb. (x−x₁)/a = (y−y₁)/b = (z−z₁)/c.

Plane: r·n = d. ax+by+cz+d=0. Intercept form: x/a + y/b + z/c = 1.

Distances and Angles

10. Linear Programming

Graphical Method

11. Probability

Conditional Probability: P(A|B) = P(A∩B)/P(B).

Bayes' Theorem

Random Variables

Binomial Distribution

Key formulas & results

Common mistakes & fixes

Practice problems

5-minute revision

CBSE marks blueprint

Where this shows up in the real world

Going beyond the textbook

Where else this chapter is tested

AI helpers for this chapter

Generate 5 more questions

Explain at a different level

📝 My notes

Questions students ask

How do I decide between integration by substitution and integration by parts?

What is the difference between a point of inflection and a point of extremum?

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Related chapters

'Relations and Functions'

'Inverse Trigonometric Functions'

'Matrices'

'Determinants'

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