Std. 11
Mathematics

A set is a well-defined collection of distinct objects. The objects are called elements or members of the set.

A set may be described by listing its elements or by a rule/property that every member satisfies.

Important types include empty set, finite set, infinite set, singleton set, equal sets, equivalent sets and subsets.

Union combines elements from both sets: A∪B. Intersection contains common elements: A∩B. Difference contains elements in one set but not the other.

With universal set U, the complement of A contains elements of U not in A: A′=U−A.

(A∪B)′=A′∩B′ and (A∩B)′=A′∪B′. Venn diagrams help visualise these identities.

For finite sets, n(A∪B)=n(A)+n(B)−n(A∩B). This prevents common elements from being counted twice.

A relation connects elements of one set with elements of another according to a specified rule.

A function assigns exactly one output to each input in its domain. It is commonly written {f('y=f(x)')}.

The domain is the set of permitted inputs. The range is the set of actual outputs produced by those inputs.

A one-one function maps distinct inputs to distinct outputs. An onto function covers the entire codomain. A many-one function may give the same output for different inputs.

Function composition means applying one function after another: (f∘g)(x)=f(g(x)). The order matters.

An inverse reverses a function's mapping. A function has an inverse function when it is one-one on the relevant domain and its range is used as the inverse's domain.

A graph gives a visual representation of a function. The vertical-line test helps determine whether a graph represents y as a function of x.

Angles may be measured in degrees or radians. One complete revolution is 360° or 2π radians.

sinθ=opposite/hypotenuse cosθ=adjacent/hypotenuse tanθ=opposite/adjacent for a right triangle.

sin²θ+cos²θ=1 1+tan²θ=sec²θ 1+cot²θ=cosec²θ. These connect the six trigonometric functions.

Study the periodic nature, zeros, maxima/minima and symmetry of sine, cosine and tangent functions. The period of sin x and cos x is 2π; tan x has period π.

Key formulae include sin(A±B)=sinA cosB±cosA sinB and cos(A±B)=cosA cosB∓sinA sinB.

sin2A=2sinA cosA cos2A=cos²A−sin²A=1−2sin²A=2cos²A−1 tan2A=2tanA/(1−tan²A).

General solutions of simple trigonometric equations are written using integer multiples of the relevant period. Always check the original equation before finalising.

Some equations such as x²+1=0 have no real solution. Complex numbers extend the real number system so such equations can be solved.

The imaginary unit is defined by i²=−1. A complex number has the form z=a+ib where a and b are real.

In z=a+ib, a is the real part and b is the imaginary part. The conjugate is {f('z̄=a−ib')}.

The modulus gives the distance from the origin in the complex plane: |z|=√(a²+b²).

A complex number is represented as a point/vector with real part along the horizontal axis and imaginary part along the vertical axis.

A non-zero complex number can be expressed using modulus r and argument θ as z=r(cosθ+i sinθ).

Complex numbers allow quadratic equations with negative discriminant to be expressed in terms of conjugate complex roots.

A quadratic equation is ax²+bx+c=0, where a≠0.

The roots are given by x=(−b±√(b²−4ac))/(2a).

D=b²−4ac determines the nature of roots: D>0 distinct real roots; D=0 equal real roots; D<0 non-real conjugate roots.

If roots are α and β, then α+β=−b/a and αβ=c/a.

Questions may ask for an equation whose roots are related to given roots. Express the new roots in terms of α and β and use the sum/product relations.

Quadratic inequalities are solved by finding critical roots, examining signs in intervals and selecting the interval(s) satisfying the inequality.

If one task can be done in m ways and a second independent task in n ways, the two tasks together can be done in mn ways.

n!=n(n−1)(n−2)…2·1 and 0!=1. Factorials count ordered arrangements in many problems.

A permutation is an arrangement where order matters: ⁿPᵣ=n!/(n−r)!.

A combination is a selection where order does not matter: ⁿCᵣ=n!/[r!(n−r)!].

ⁿPᵣ=ⁿCᵣ·r!. This shows that a permutation can be formed by first selecting objects and then arranging them.

Translate words carefully: 'arrange', 'rank', 'seat' usually suggest permutations; 'select', 'choose', 'committee' usually suggest combinations, subject to restrictions.

The binomial theorem expands powers of a sum: (a+b)ⁿ=Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏbᵏ.

The (r+1)th term is Tᵣ₊₁=ⁿCᵣ aⁿ⁻ʳbʳ.

The coefficients follow Pascal's triangle and are binomial coefficients. Symmetry gives {f('ⁿCᵣ=ⁿCₙ₋ᵣ')}.

For even n there is one middle term; for odd n there are two middle terms. Identify them from their term numbers.

Use the general term to find a particular term, coefficient or constant term. Substitute carefully and simplify powers systematically.

A sequence is an ordered list of numbers following a rule. A series is the sum of terms of a sequence.

In an AP, common difference d is constant: aₙ=a+(n−1)d and Sₙ=n/2[2a+(n−1)d].

In a GP, common ratio r is constant: aₙ=arⁿ⁻¹ and Sₙ=a(rⁿ−1)/(r−1) for r≠1.

When |r|<1, the infinite sum exists and S∞=a/(1−r).

For positive numbers a,b, arithmetic mean is AM=(a+b)/2 and geometric mean is GM=√ab; AM≥GM.

Summation notation provides a compact way to represent repeated addition. Learn standard sums of natural numbers, squares and cubes and apply them carefully.

The coordinate plane locates points using ordered pairs (x,y). The distance and midpoint formulae connect geometry with algebra.

For two points, m=(y₂−y₁)/(x₂−x₁). A vertical line has undefined slope.

Common forms include y=mx+c, y−y₁=m(x−x₁) and x/a+y/b=1. Choose the form that matches the information given.

Parallel non-vertical lines have equal slopes. Perpendicular non-vertical lines satisfy m₁m₂=−1.

For slopes m₁,m₂, tanθ=|(m₂−m₁)/(1+m₁m₂)| when the denominator is non-zero.

For ax+by+c=0 and point (x₁,y₁), distance is |ax₁+by₁+c|/√(a²+b²).

A circle is the locus of a point at a fixed distance from a fixed point. Standard form: x²+y²+2gx+2fy+c=0, with centre (−g,−f).

A parabola is the locus of a point equidistant from a fixed point (focus) and fixed line (directrix). Its standard forms include y²=4ax and x²=4ay.

An ellipse is the locus for which the sum of distances from two fixed foci is constant. Its standard forms depend on whether the major axis is horizontal or vertical.

A hyperbola is the locus for which the absolute difference of distances from two fixed foci is constant.

Focus, directrix, eccentricity, axis, vertex, latus rectum and centre describe important geometric features. Understand each before applying a formula.

Conic Section belongs to Section B in the ISC Class XI Mathematics syllabus; students attempt Section B or Section C as prescribed by their course structure.

A point in space is represented by (x,y,z), where three mutually perpendicular axes locate its position.

Distance between (x₁,y₁,z₁) and (x₂,y₂,z₂) is √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²].

If a point divides a segment joining two points internally in ratio m:n, each coordinate is obtained by the same weighted-average rule: P=((mx₂+nx₁)/(m+n), …).

Direction ratios are proportional numbers describing the direction of a line in space. Direction cosines are the cosines of its angles with the coordinate axes.

Use a coordinate sketch where possible. Keep x-, y- and z-coordinates in the correct order throughout the calculation.

A limit describes the value a function approaches as the independent variable approaches a particular value. It need not equal the function's value at that point.

Important standard results include lim(x→0) sinx/x=1 and lim(x→0)(1−cosx)/x²=1/2, with angles measured in radians.

A function is continuous at a point when its limiting value, left/right limits and function value agree appropriately.

The derivative measures instantaneous rate of change and slope of the tangent. From first principles, f′(x)=lim(h→0)[f(x+h)−f(x)]/h.

d(xⁿ)/dx=nxⁿ⁻¹ and d(c)/dx=0. Sum, product and quotient rules extend differentiation to more complex functions.

Derivatives help determine rates of change, slopes of tangents/normals and increasing/decreasing behaviour. Interpret the derivative physically or geometrically, not just algebraically.

Mean, median and mode describe a typical or central value. Select the measure according to the data and purpose.

For ungrouped observations, x̄=Σx/n; for frequency data, x̄=Σfx/Σf.

Range, variance and standard deviation measure how spread out observations are. A small standard deviation means observations are relatively close to the mean.

For equally likely outcomes, P(E)=favourable outcomes/total outcomes. Always ensure 0≤P(E)≤1.

For events A and B, P(A∪B)=P(A)+P(B)−P(A∩B). If they are mutually exclusive, the intersection term is zero.

For independent events, P(A∩B)=P(A)P(B). Do not use this relation simply because two events appear in the same question; establish independence first.

In the current ISC Class XI syllabus, Statistics & Probability is part of compulsory Section A. The official syllabus gives Section A 65 marks in the 2026 document.

A mathematical statement is a sentence that can be classified as true or false, but not both.

The negation of a statement changes its truth value. For compound statements, the wording of the negation must be handled logically rather than by simply adding 'not'.

AND, OR and NOT combine statements. Their truth values can be organised using truth tables.

A conditional statement has the form p→q, meaning 'if p, then q'. It is false only when p is true and q is false.

The converse of p→q is q→p. The contrapositive is ¬q→¬p and is logically equivalent to the original implication.

If p→q, p is sufficient for q and q is necessary for p. Distinguish these words carefully in proofs and reasoning questions.