CBSE Class 10 Mathematics Standard (041)
Chapters: 7
1. Number Systems
Real Numbers
- Fundamental Theorem of Arithmetic – Every whole number bigger than 1 is either a prime or can be written as a product of primes in exactly one way (only the order can change). This is the Fundamental Theorem of Arithmetic. Using these prime "building blocks": HCF = product of the smallest powers of the common primes; LCM = product of the greatest powers of all primes. For two numbers, HCF × LCM = a × b.
- Proving √2, √3 and √5 are Irrational – A rational number can be written as p/q (q ≠ 0). An irrational number cannot; its decimal never ends and never repeats. To prove √2 is irrational we assume it is p/q in lowest terms, get p² = 2q², show both p and q are even, and reach a contradiction. The key fact: if a prime divides p², it divides p. Rational + or × irrational (non-zero) is irrational.
2. Algebra
Polynomials · Pair of linear equations in two variables · Quadratic equations · Arithmetic progressions
- Polynomials: Zeroes and Coefficients – A polynomial like ax² + bx + c has a degree (highest power). A zero is a value of x that makes it 0. On a graph, zeroes are the x-coordinates where the curve y = p(x) meets the x-axis. A polynomial of degree n has at most n zeroes. For ax² + bx + c with zeroes α, β: α + β = −b/a and αβ = c/a. A quadratic with given zeroes is k[x² − (α+β)x + αβ].
- Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
- Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
- Arithmetic Progressions – An arithmetic progression (AP) is a list of numbers where each term is made by adding the same fixed number d (the common difference) to the term before. With first term a: nth term aₙ = a + (n − 1)d; sum of the first n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
3. Coordinate Geometry
Lines in the plane
- Coordinate Geometry: Distance and Section Formula – Every point on a flat plane has an address (x, y). The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]; it is just Pythagoras on the x-gap and the y-gap. A point P that cuts AB inside in the ratio m : n is P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). When m = n, P is the midpoint.
4. Geometry
Triangles · Circles
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
- Tangent to a Circle – A tangent is a line that touches a circle at exactly one point. At that point it makes a 90° angle with the radius, and two tangents drawn from one outside point are always equal in length.
5. Trigonometry
Introduction to trigonometry · Heights and distances
- Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
- Trigonometric Identities – An identity is an equation that is true for every allowed angle. From Pythagoras on a right triangle with hypotenuse 1: sin²A + cos²A = 1. Dividing by cos²A gives 1 + tan²A = sec²A (A ≠ 90°). Dividing by sin²A gives 1 + cot²A = cosec²A (A ≠ 0°). Use them to find one ratio from another and to prove other statements.
- Heights and Distances – The line of sight joins your eye to the object. If the object is above you, the angle between the line of sight and the horizontal is the angle of elevation. If it is below you, that angle is the angle of depression. The object, the ground and the line of sight make a right triangle, so tan θ = height ÷ distance. The angle of depression from the top equals the angle of elevation from the bottom. Hard questions use two right triangles that share one side.
6. Mensuration
Areas related to circles · Surface areas and volumes
- Area of Sector and Segment of a Circle – A sector is a pizza slice of a circle: its area is θ/360 × πr². Its crust is an arc of length θ/360 × 2πr. A segment is the slice minus the triangle inside it: segment = sector − triangle.
- Surface Areas and Volumes: Combined Solids – Many real objects are two simple solids stuck together, like a cone on a cylinder. The surface area is only the outside skin you can touch, so the hidden joint is left out. The volume is the space inside, so you simply add the volumes of the parts.
7. Statistics and Probability
Statistics · Probability
- Statistics: Mean, Median and Mode of Grouped Data – When data is put into classes (like marks 20–30), we cannot see each value. We use the middle of each class to find the mean, the running total to find the median, and the tallest bar to find the mode. All three tell us the 'centre' of the data in different ways.
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.