Class 10 Maths Notes (Latest CBSE Syllabus) on Padhaiwise

Class 10 Maths Notes Chapter 1 — Real Numbers

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 1 — Real Numbers#

Key terms#

Prime number — a natural number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, …).
Composite number — a natural number greater than 1 with a factor other than 1 and itself (4, 6, 9, …).
1 is neither prime nor composite.
HCF — the largest number that divides each of the given numbers exactly.
LCM — the smallest number that is a multiple of each of the given numbers.
Coprime numbers — two numbers whose HCF is 1 (for example, 8 and 15).
Rational number — can be written as $\frac{p}{q}$, where $p$, $q$ are integers and $q \neq 0$.
Irrational number — cannot be written as $\frac{p}{q}$. Examples: $\sqrt{2}$, $\sqrt{3}$, $\sqrt{15}$, $\pi$, $0.10110111\ldots$
Rational and irrational numbers together make up the real numbers.

Fundamental Theorem of Arithmetic#

Theorem 1.1: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Class 10 Maths Notes Chapter 2 — Polynomials

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 2 — Polynomials#

Key terms#

Polynomial in $x$ — an expression $a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$, where the powers are whole numbers and the coefficients are real numbers.
Degree — the highest power of $x$ in $p(x)$.
Zero of $p(x)$ — a real number $k$ such that $p(k) = 0$.

Type	Degree	General form
Linear	1	$ax + b$, $a \neq 0$
Quadratic	2	$ax^2 + bx + c$, $a \neq 0$
Cubic	3	$ax^3 + bx^2 + cx + d$, $a \neq 0$

The zero of the linear polynomial $ax + b$ is $-\frac{b}{a}$.
A polynomial of degree $n$ has at most $n$ zeroes. A quadratic has at most 2, a cubic at most 3.

Geometrical meaning of zeroes#

The zeroes of $p(x)$ are exactly the x-coordinates of the points where the graph of $y = p(x)$ meets the x-axis.

Class 10 Maths Notes Chapter 3 — Pair of Linear Equations in Two Variables

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 3 — Pair of Linear Equations in Two Variables#

Key terms#

Linear equation in two variables — $ax + by + c = 0$, where $a$, $b$, $c$ are real numbers and $a$, $b$ are not both zero. Its graph is a straight line, and it has infinitely many solutions.
Pair of linear equations — two such equations in the same variables: $$a_1x + b_1y + c_1 = 0$$ $$a_2x + b_2y + c_2 = 0$$
Solution of the pair — a pair of values $(x, y)$ that satisfies both equations.
Consistent — has at least one solution. Inconsistent — has no solution.
Dependent — the two equations represent the same line (infinitely many solutions).

Graphical method and consistency#

Draw both lines on the same axes. Their position tells you the number of solutions, and so do the ratios of the coefficients:

Class 10 Maths Notes Chapter 4 — Quadratic Equations

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 4 — Quadratic Equations#

Key terms#

Quadratic equation — an equation $ax^2 + bx + c = 0$, where $a$, $b$, $c$ are real numbers and $a \neq 0$. This is its standard form.
Root — a real number $\alpha$ is a root if $a\alpha^2 + b\alpha + c = 0$.
The roots of $ax^2 + bx + c = 0$ are the same as the zeroes of the polynomial $ax^2 + bx + c$.
A quadratic equation has at most two roots.
To check whether an equation is quadratic, expand and simplify it first. It is quadratic only if the $x^2$ term remains with a non-zero coefficient (and no higher power remains).

Solving by factorisation#

Write the equation in standard form $ax^2 + bx + c = 0$.
Split the middle term: find two numbers whose sum is $b$ and whose product is $ac$.
Factorise by grouping into two linear factors.
Put each factor equal to zero and solve.

Example: $2x^2 - 5x + 3 = 0$. We need two numbers with sum −5 and product $2 \times 3 = 6$. They are −2 and −3.

Class 10 Maths Notes Chapter 5 — Arithmetic Progressions

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 5 — Arithmetic Progressions#

Key terms#

Arithmetic progression (AP) — a list of numbers in which each term, except the first, is got by adding a fixed number $d$ to the term before it.
First term — $a$ (also written $a_1$).
Common difference — $d = a_{k+1} - a_k$. It can be positive, negative or zero.
General form — $a,\ a + d,\ a + 2d,\ a + 3d,\ \ldots$
Finite AP has a last term, written $l$. An infinite AP has no last term.

Each term is the one before it plus the common difference $d$.

To check whether a list is an AP, find $a_2 - a_1$, $a_3 - a_2$, $a_4 - a_3$, … If all the differences are equal, it is an AP. One unequal difference is enough to show it is not.

Class 10 Maths Notes Chapter 6 — Triangles

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 6 — Triangles#

Similar figures#

Similar figures have the same shape but not necessarily the same size.
Congruent figures have the same shape and the same size.
All congruent figures are similar, but similar figures need not be congruent.
All circles are similar. All squares are similar. All equilateral triangles are similar.

Two polygons with the same number of sides are similar if both:

Class 10 Maths Notes Chapter 7 — Coordinate Geometry

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 7 — Coordinate Geometry#

Distance formula#

The distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$ is
$$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
The distance of $P(x, y)$ from the origin is
$$OP = \sqrt{x^2 + y^2}$$

The distance $PQ$ is the hypotenuse of a right triangle with legs $x_2 - x_1$ and $y_2 - y_1$, so Pythagoras gives the formula.

Class 10 Maths Notes Chapter 8 — Introduction to Trigonometry

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 8 — Introduction to Trigonometry#

Trigonometric ratios#

In $\triangle ABC$, right-angled at $B$, for the acute angle $A$:

Class 10 Maths Notes Chapter 9 — Some Applications of Trigonometry

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 9 — Some Applications of Trigonometry#

Key terms#

Line of sight — the line from the observer’s eye to the point on the object being viewed.
Horizontal — the line through the observer’s eye, parallel to the ground.
Angle of elevation — the angle between the line of sight and the horizontal when the object is above eye level (you raise your head).
Angle of depression — the angle between the line of sight and the horizontal when the object is below eye level (you lower your head).

The angle of depression from A to B equals the angle of elevation from B to A. They are alternate angles, because both horizontals are parallel.

Class 10 Maths Notes Chapter 10 — Circles

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 10 — Circles#

Key terms#

Secant — a line that meets a circle at two points.
Tangent — a line that meets a circle at exactly one point. That point is the point of contact.
A tangent is the special case of a secant when its two points of intersection come together.
Length of a tangent — the length of the segment from an external point to the point of contact.
A line and a circle can have 0, 1 or 2 common points: non-intersecting, tangent, or secant.
A circle has infinitely many tangents, one at each point. It can have at most two parallel tangents, at the ends of a diameter.

Tangent and radius#

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Class 10 Maths Notes Chapter 11 — Areas Related to Circles

Mon, 01 Jan 0001 00:00:00 +0000

Key terms#

Sector — the region between an arc and the two radii to its ends. The smaller one is the minor sector, the larger one the major sector.
Segment — the region between a chord and its arc. The smaller one is the minor segment, the larger one the major segment.
Angle of the sector — the angle $\theta$ that the two radii make at the centre.
For a minor sector of angle $\theta$, the major sector has angle $360° - \theta$.

OAB with the arc AB is the minor sector. The region between chord AB and arc AB is the minor segment.

Class 10 Maths Notes Chapter 12 — Surface Areas and Volumes

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 12 — Surface Areas and Volumes#

Formulas for basic solids#

The measurements used in the formulas: $r$ radius, $h$ height, $l$ slant height (cone).

Solid	Curved / lateral surface area	Total surface area	Volume
Cuboid ($l \times b \times h$)	$2h(l + b)$	$2(lb + bh + hl)$	$lbh$
Cube (edge $a$)	$4a^2$	$6a^2$	$a^3$
Cylinder ($r$, $h$)	$2\pi rh$	$2\pi r(r + h)$	$\pi r^2 h$
Cone ($r$, $h$, $l$)	$\pi rl$	$\pi r(r + l)$	$\frac{1}{3}\pi r^2 h$
Sphere ($r$)	$4\pi r^2$	$4\pi r^2$	$\frac{4}{3}\pi r^3$
Hemisphere ($r$)	$2\pi r^2$	$3\pi r^2$	$\frac{2}{3}\pi r^3$

Slant height of a cone:

Class 10 Maths Notes Chapter 13 — Statistics

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 13 — Statistics#

Key terms#

Class mark $x_i$ — the mid-point of a class: $$x_i = \frac{\text{upper limit} + \text{lower limit}}{2}$$
Class size $h$ — upper limit minus lower limit.
Cumulative frequency ($cf$) of a class — the sum of the frequencies of that class and all classes before it.
Modal class — the class with the highest frequency.
Median class — the first class whose cumulative frequency is greater than $\frac{n}{2}$.

Mean of grouped data#

Each class is represented by its class mark. Use any one method:

Class 10 Maths Notes Chapter 14 — Probability

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths Notes Chapter 14 — Probability#

Key terms#

Experiment — an action with well-defined outcomes, such as tossing a coin.
Outcome — a possible result of the experiment.
Equally likely outcomes — each outcome has the same chance of happening. A fair coin, a fair die and a well-shuffled deck give equally likely outcomes.
Event — a collection of outcomes, such as “getting an even number”.
Elementary event — an event with only one outcome.
Sure (certain) event — always happens. Impossible event — never happens.

Theoretical probability#

Class 10 Maths — All Formulas (Quick Revision)

Mon, 01 Jan 0001 00:00:00 +0000

Class 10 Maths — All Formulas#

Every formula in the latest NCERT Class 10 Maths book, chapter by chapter.

1. Real Numbers#

Every composite number = product of primes, in exactly one way (apart from order).
HCF = product of the smallest powers of the common primes.
LCM = product of the greatest powers of all the primes.
For two numbers only: $$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$$
If a prime $p$ divides $a^2$, then $p$ divides $a$.

2. Polynomials#

For $ax^2 + bx + c$ with zeroes $\alpha$, $\beta$:

Class 10 Maths Notes (Latest CBSE Syllabus) on Padhaiwise

Class 10 Maths Notes Chapter 1 — Real Numbers

Class 10 Maths Notes Chapter 1 — Real Numbers#

Key terms#

Fundamental Theorem of Arithmetic#

Class 10 Maths Notes Chapter 2 — Polynomials

Class 10 Maths Notes Chapter 2 — Polynomials#

Key terms#

Geometrical meaning of zeroes#

Class 10 Maths Notes Chapter 3 — Pair of Linear Equations in Two Variables

Class 10 Maths Notes Chapter 3 — Pair of Linear Equations in Two Variables#

Key terms#

Graphical method and consistency#

Class 10 Maths Notes Chapter 4 — Quadratic Equations

Class 10 Maths Notes Chapter 4 — Quadratic Equations#

Key terms#

Solving by factorisation#

Class 10 Maths Notes Chapter 5 — Arithmetic Progressions

Class 10 Maths Notes Chapter 5 — Arithmetic Progressions#

Key terms#

Class 10 Maths Notes Chapter 6 — Triangles

Class 10 Maths Notes Chapter 6 — Triangles#

Similar figures#

Class 10 Maths Notes Chapter 7 — Coordinate Geometry

Class 10 Maths Notes Chapter 7 — Coordinate Geometry#

Distance formula#

Class 10 Maths Notes Chapter 8 — Introduction to Trigonometry

Class 10 Maths Notes Chapter 8 — Introduction to Trigonometry#

Trigonometric ratios#

Class 10 Maths Notes Chapter 9 — Some Applications of Trigonometry

Class 10 Maths Notes Chapter 9 — Some Applications of Trigonometry#

Key terms#

Class 10 Maths Notes Chapter 10 — Circles

Class 10 Maths Notes Chapter 10 — Circles#

Key terms#

Tangent and radius#

Class 10 Maths Notes Chapter 11 — Areas Related to Circles

Class 10 Maths Notes Chapter 11 — Areas Related to Circles#

Key terms#

Class 10 Maths Notes Chapter 12 — Surface Areas and Volumes

Class 10 Maths Notes Chapter 12 — Surface Areas and Volumes#

Formulas for basic solids#

Class 10 Maths Notes Chapter 13 — Statistics

Class 10 Maths Notes Chapter 13 — Statistics#

Key terms#

Mean of grouped data#

Class 10 Maths Notes Chapter 14 — Probability

Class 10 Maths Notes Chapter 14 — Probability#

Key terms#

Theoretical probability#

Class 10 Maths — All Formulas (Quick Revision)

Class 10 Maths — All Formulas#

1. Real Numbers#

2. Polynomials#