Std. 12
Mathematics

A function assigns exactly one output to every input in its domain. Domain, codomain and range must be distinguished carefully.

(f∘g)(x)=f(g(x)) means apply g first and then f. Composition is generally not commutative.

An inverse reverses a one-one mapping. For a bijection, the inverse exists as a function.

A binary operation combines two members of a set and gives a result within that set; closure is essential.

A matrix is a rectangular arrangement of numbers in rows and columns. Its order is rows × columns.

AB exists when columns of A equal rows of B. Matrix multiplication is generally not commutative.

For a 2×2 matrix, |A|=ad−bc. A non-zero determinant indicates an invertible square matrix.

A⁻¹=adj(A)/|A| when |A|≠0, and AA⁻¹=I.

Matrices and determinants provide systematic methods for solving simultaneous linear equations.

At x=a, continuity requires the left-hand limit, right-hand limit and function value to agree.

f′(a)=lim(h→0)[f(a+h)−f(a)]/h gives instantaneous rate of change.

If y=f(g(x)), then dy/dx=f′(g(x))·g′(x).

Differentiate both sides with respect to x while treating y as a function of x.

If x=x(t), y=y(t), then dy/dx=(dy/dt)/(dx/dt) when dx/dt≠0.

d²y/dx² describes the rate of change of the first derivative.

The derivative represents instantaneous change. Define variables and their relationship before differentiating.

f′(x)>0 indicates increasing behaviour; f′(x)<0 indicates decreasing behaviour on the relevant interval.

Candidates for maxima/minima occur where f′=0 or f′ is undefined, provided the point belongs to the domain.

At f′(a)=0, f″(a)<0 indicates a local maximum and f″(a)>0 a local minimum when the test applies.

Under its conditions, the theorem connects average rate of change with an instantaneous derivative inside the interval.

The tangent slope is dy/dx. A normal is perpendicular to the tangent.

If F′(x)=f(x), then ∫f(x)dx=F(x)+C.

Replace a complicated inner expression with a new variable when its derivative is also present.

∫u dv=uv−∫v du. Choose u and dv so the remaining integral becomes simpler.

Decompose a rational expression into simpler fractions before integrating.

A definite integral gives signed accumulated quantity and represents area directly when the function is non-negative.

Differentiation and definite integration are inverse operations under the required conditions.

Integrating a non-negative function over an interval gives the area between the curve and the x-axis.

Find intersections first, then integrate upper minus lower over the correct interval.

The ISC syllabus includes areas involving lines, circles/parabolas/ellipses, polynomial, modulus, trigonometric, exponential and logarithmic curves.

If a curve crosses an axis, split the interval when actual geometric area is required.

A differential equation contains derivatives of a dependent variable. Order is the highest derivative present.

A general solution contains arbitrary constants; initial conditions produce a particular solution.

Rewrite as g(y)dy=f(x)dx and integrate both sides.

A first-order homogeneous equation can often be reduced using a substitution such as y=vx.

For dy/dx+Py=Q, the integrating factor is IF=e^(∫Pdx).

Differentiate a family containing arbitrary constants and eliminate those constants.

A vector has magnitude and direction; a scalar has magnitude only.

a=a₁i+a₂j+a₃k represents a vector through rectangular components.

|a|=√(a₁²+a₂²+a₃²).

a·b=|a||b|cosθ; it is zero for perpendicular vectors.

|a×b|=|a||b|sinθ; its direction is perpendicular to both.

Internal division in ratio m:n gives (ma+nb)/(m+n).

If direction cosines are l,m,n, then l²+m²+n²=1.

A line can be written as r=a+λb, where a fixes a point and b gives direction.

For direction vectors a,b, cosθ=(a·b)/(|a||b|).

A plane is determined by a point and a normal vector.

Use vectors for parallelism, perpendicularity, angles, distances and intersections in space.

P(A|B)=P(A∩B)/P(B) when P(B)>0.

P(A∩B)=P(A)P(B|A).

For independent events, P(A∩B)=P(A)P(B).

For a suitable partition, P(A)=ΣP(Bᵢ)P(A|Bᵢ).

P(Bᵢ|A)=P(Bᵢ)P(A|Bᵢ)/ΣP(Bⱼ)P(A|Bⱼ).

For discrete X, E(X)=Σxᵢpᵢ and Var(X)=E(X²)−[E(X)]².

Section B is the 20-mark alternative to Section C. It is not an additional compulsory section.

Use definite integration to calculate geometric areas after identifying intersections, limits and upper/lower curves.

These topics extend coordinate methods into space for angles, lines, planes and distances.

Section C is the 20-mark alternative to Section B. Students should follow the option selected by their school/course.

Use differentiation and integration to model rates, optimisation and accumulated quantities.

Regression describes a best-fitting linear relationship and is used for estimation and prediction.

Optimise a linear objective function subject to linear constraints. The feasible region contains all permitted solutions.