14 chapters · Topics & Key Concepts
Euclid's division lemma (overview), Fundamental Theorem of Arithmetic, prime factorisation, HCF and LCM, proofs of irrationality of √2, √3, √5, decimal expansions
Every natural number can be uniquely factorised into primes. HCF and LCM can be found by prime factorisation. The product HCF × LCM = product of the two numbers. Numbers like √2 and √5 are irrational. Decimal expansions of rationals are either terminating or non-terminating recurring, depending on the prime factors of the denominator.
Geometrical meaning of zeroes of a polynomial, relationship between zeroes and coefficients of quadratic polynomials, formation of polynomial given zeroes
The zeroes of a polynomial are the x-coordinates where its graph cuts the x-axis. A quadratic polynomial has at most two zeroes. The coefficients determine the sum and product of zeroes. Polynomials can be formed when zeroes are given, connecting algebra and geometry.
Pair of linear equations, graphical method, algebraic methods (substitution, elimination, cross-multiplication), conditions for consistency and inconsistency, word problems
A pair of linear equations represents two straight lines. The lines may intersect, be parallel or coincide, giving unique solution, no solution, or infinitely many solutions. The relation between coefficients tells the nature of solutions without solving. Algebraic methods such as substitution and elimination give exact solutions.
Standard form of quadratic equation, solution by factorisation, quadratic formula, nature of roots, discriminant, word problems on speed, area, age, numbers
A quadratic equation has the form ax^2 + bx + c = 0. Solutions can be found by factorisation or the quadratic formula. The discriminant determines the nature of roots. Quadratic equations model many real-life problems including motion, area and finance.
Introduction to AP, general term, nth term, sum of first n terms, applications, word problems
An arithmetic progression is a sequence where consecutive terms differ by a constant called the common difference. APs model many real situations such as installments, savings, salary increments and seating arrangements.
Similar figures, similarity of triangles, criteria (AAA, SAS, SSS), Basic Proportionality Theorem (Thales theorem), areas of similar triangles, Pythagoras theorem and its converse
Similar figures have equal corresponding angles and proportional sides. The ratio of areas of similar triangles equals the square of the ratio of corresponding sides. Basic Proportionality Theorem and its converse are powerful tools. Pythagoras theorem and its converse connect right triangles with their side lengths.
Distance formula, section formula, mid-point formula, area of a triangle using coordinates, applications to geometry problems
Coordinate geometry uses algebra to study geometric figures. Distance, midpoint and division coordinates can be found using simple formulae. These allow proving geometric results such as collinearity and finding areas of triangles.
Trigonometric ratios of acute angles, ratios for specific angles (0°, 30°, 45°, 60°, 90°), trigonometric identities, complementary angles
Trigonometry is the study of relations between sides and angles of a triangle. The six trigonometric ratios are defined for an acute angle in a right triangle. Trigonometric identities are relations true for all values of the angle and are widely used to simplify expressions. Complementary angle relations are useful in finding unknown values.
Heights and distances, line of sight, angle of elevation, angle of depression, applications in towers, buildings, hills, ships
Heights and distances of objects that cannot be measured directly can be found using trigonometric ratios. The angle of elevation is from horizontal up to an object; angle of depression is from horizontal down to an object. The line of sight, horizontal and vertical form a right triangle.
Tangent to a circle, number of tangents from a point, length of tangent from external point, tangent perpendicular to radius, equal tangents from external point
A tangent touches a circle at exactly one point. From a point outside a circle, exactly two tangents can be drawn and they have equal lengths. The tangent at any point is perpendicular to the radius drawn to the point of contact. These properties solve many geometric problems involving circles.
Area of circle, area of sector, area of segment, perimeter of sector, length of arc, areas of combinations of plane figures
A sector is the region bounded by two radii and the corresponding arc; a segment is the region bounded by a chord and the arc. Many real-life shapes are combinations of circles, triangles and rectangles; their areas are found by addition or subtraction of basic shapes.
Surface area and volume of cuboid, cube, cylinder, cone, sphere, hemisphere; combinations of solids; conversion of one shape into another; frustum of a cone
Real-life objects are often combinations of two or more basic solids. Calculating their surface area and volume needs careful identification of the parts. When one solid is recast into another, the volume remains the same. The frustum of a cone (e.g., a bucket) is formed by cutting a cone with a plane parallel to its base.
Mean, median and mode of grouped data, cumulative frequency curves (ogives), finding median from ogive
Mean, median and mode are three measures of central tendency. For grouped data, formulas involving class intervals, frequencies and cumulative frequencies are used. Cumulative frequency curves (ogives) help find median graphically. The empirical relation connects the three measures.
Theoretical (classical) probability, simple problems using dice, coins, cards, balls in bags
Probability is a measure of the chance that an event will occur. In Class 10, theoretical probability is studied, assuming all outcomes are equally likely. Sum of probabilities of complementary events is 1. Probability is used in games, weather forecasting, insurance and decision-making.
