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Welcome to Class 10 Mathematics

A self-study course for the whole Class 10 mathematics syllabus. Every idea is explained from scratch, with live labs, worked examples, board-style practice and full solutions.

1. Welcome

Class 10 is the year when school mathematics starts to fit together. Numbers, algebra, geometry and data stop being separate topics and begin to help each other. An arithmetic progression can turn into a quadratic equation. A tower's height comes from a right triangle. A coordinate grid can decide whether three points lie on one line.

This course follows the current NCERT (CBSE) syllabus chapter by chapter, in the same order as your textbook, and it is written for you working on your own. Nothing is skipped because "it is obvious". Every example is solved step by step. Every exercise has a full solution. Each section also has a ๐ŸŽฏ Exam corner with board-style questions, solved the way an examiner wants to see them.

๐Ÿ“ฆ Class 9 gave youโ€ฆ

rational and irrational numbers, polynomials such as $x^2-5x+6$, linear equations $ax+by+c=0$, triangles and circles, and areas and volumes of simple solids.

๐Ÿš€ Class 10 addsโ€ฆ

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ $$S_n=\frac n2\left[2a+(n-1)d\right]$$ $$\sin^2A+\cos^2A=1$$

2. A story that needs several chapters

A school is building seating for its annual sports day. The front row has $20$ seats, and every row behind it has $4$ more seats than the row in front: $20, 24, 28, 32, \dots$ A list of numbers that goes up by the same amount each time is an arithmetic progression (Chapter 5). The total number of seats in $n$ rows is given by a formula, $S_n=\tfrac n2\left[2a+(n-1)d\right]$.

Now turn the question round: how many rows are needed to seat $1160$ people? Setting $S_n=1160$ gives an equation with $n^2$ in it โ€” a quadratic equation (Chapter 4). Questions like this run through the whole course: one chapter sets up the problem and another solves it. Play with the lab, then read Example 1.

Interactive 1 โ€” Rows of Seats on Sports Day

Each bar is one row of seats, starting with $a$ seats in the front row and growing by $d$ seats per row. Press Play to add rows one at a time. The row lengths form an arithmetic progression, and the running total is $S_n$. Change $a$ and $d$ and see how quickly the total grows. It grows like $n^2$, which is why "how many rows?" leads to a quadratic equation.

204
rows $n$โ€”
seats in row $n$: $a+(n-1)d$โ€”
total seats $S_n$โ€”
โœ๏ธ Example 1 โ€” How many rows?
The front row has $20$ seats and each row has $4$ more than the one in front. How many rows are needed to seat exactly $1160$ people, and how many seats are in the last row?
AP (Ch. 5) $a=20$, $d=4$, so $S_n=\dfrac n2\left[2(20)+(n-1)4\right]=\dfrac n2\left(4n+36\right)=2n^2+18n$.
Quadratic (Ch. 4) $2n^2+18n=1160\ \Rightarrow\ n^2+9n-580=0$. We need two numbers with product $-580$ and sum $9$: $29$ and $-20$. So $(n+29)(n-20)=0$.
Choose $n=-29$ makes no sense for a number of rows, so $n=20$ rows.
Check Last row: $a_{20}=20+19\times4=96$ seats. Total: $S_{20}=\tfrac{20}{2}(20+96)=10\times116=1160$. โœ“ Set $a=20$, $d=4$ in the lab and watch the total reach $1160$ at row $20$.

3. The fourteen chapters at a glance

Numbers & algebra ยท Ch 1โ€“5โ†’Geometry & coordinates ยท Ch 6โ€“7โ†’Trigonometry ยท Ch 8โ€“9โ†’Circles & mensuration ยท Ch 10โ€“12โ†’Data & chance ยท Ch 13โ€“14
Chapter 1

Real Numbers

Primes as building blocks, HCF and LCM by prime factorisation, and proofs that $\sqrt2$ is irrational.

$360=2^3\times3^2\times5$
Open chapter โ†’ Chapter 2

Polynomials

Zeroes as the points where a graph meets the $x$-axis, and how the zeroes of a quadratic are tied to its coefficients.

$\alpha+\beta=-\dfrac ba,\ \alpha\beta=\dfrac ca$
Open chapter โ†’ Chapter 3

Pair of Linear Equations

Two lines that meet, run parallel or coincide; solving by graphs, substitution and elimination.

$\dfrac{a_1}{a_2}\ne\dfrac{b_1}{b_2}$
Open chapter โ†’ Chapter 4

Quadratic Equations

Modelling with $ax^2+bx+c=0$, factorisation, the quadratic formula and the discriminant.

$D=b^2-4ac$
Open chapter โ†’ Chapter 5

Arithmetic Progressions

Lists that grow by a fixed step: the $n$th term and the sum of the first $n$ terms.

$a_n=a+(n-1)d$
Open chapter โ†’ Chapter 6

Triangles

Similar figures, the Basic Proportionality Theorem and the criteria for similar triangles.

$\dfrac{AD}{DB}=\dfrac{AE}{EC}$
Open chapter โ†’ Chapter 7

Coordinate Geometry

Distances between points and the point dividing a segment in a given ratio.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Open chapter โ†’ Chapter 8

Introduction to Trigonometry

The six ratios of a right triangle, their values at special angles, and identities.

$\sin^2A+\cos^2A=1$
Open chapter โ†’ Chapter 9

Applications of Trigonometry

Angles of elevation and depression; heights of towers and widths of rivers.

$\tan60^\circ=\dfrac hx$
Open chapter โ†’ Chapter 10

Circles

Tangents: why they meet the radius at a right angle, and why two tangents from a point are equal.

$PA=PB$
Open chapter โ†’ Chapter 11

Areas Related to Circles

Arc length, area of a sector and area of a segment.

$A=\dfrac{\theta}{360^\circ}\pi r^2$
Open chapter โ†’ Chapter 12

Surface Areas and Volumes

Solids made by joining two simple ones: a cone on a hemisphere, a cylinder with hemispherical ends.

$V=\pi r^2h+\tfrac23\pi r^3$
Open chapter โ†’ Chapter 13

Statistics

Mean, mode and median of grouped data, and how they compare.

$\bar x=\dfrac{\sum f_ix_i}{\sum f_i}$
Open chapter โ†’ Chapter 14

Probability

Chances with coins, dice, cards and spinners, and the rule $P(E)+P(\overline E)=1$.

$P(E)=\dfrac{\text{favourable}}{\text{total}}$
Open chapter โ†’
๐Ÿ—บ๏ธ How the chapters connect
  1. Algebra feeds everything. Factorising in Chapters 2 and 4, and solving two equations at once in Chapter 3, come back in coordinate geometry, in progressions and in word problems all year.
  2. Chapter 5 meets Chapter 4. "How many terms give this sum?" is always a quadratic equation, as in Example 1.
  3. Similar triangles (Ch. 6) are why trigonometry works. All right triangles with the same angle are similar, so ratios like $\tfrac{\text{opposite}}{\text{hypotenuse}}$ depend only on the angle (Ch. 8). That lets us measure heights we can't reach (Ch. 9).
  4. Coordinates (Ch. 7) turn geometry into algebra. The distance formula is Pythagoras' theorem written with coordinates.
  5. Circles and solids (Ch. 10โ€“12) use $\pi$ and the area and volume ideas from Class 9, now for parts and combinations of shapes.
  6. Statistics and probability (Ch. 13โ€“14) deal with data and chance: summarising a table with one number, and measuring how likely an event is.
๐Ÿงญ How to use this course
  • Read with a pen and a notebook. Try each worked example yourself before opening its solution, then compare step by step.
  • Play with the labs. Each section has two interactives. Change the settings until you can predict what will happen before you press โ–ถ Play.
  • Do the exercises in order. They go from routine to challenging, and every part has a full solution under "๐Ÿ’ก Show solutions". Small check-labs let you test your own answers.
  • Practise for the board in the ๐ŸŽฏ Exam corner. Each problem is tagged with its marks (1, 2, 3, 5 or a 4-mark case study). Some carry a JEE Main tag for students who want a head start. The solutions show the steps that earn marks.
  • Use the quiz as a gate. Five questions end every section. If you score below 4, reread before moving on.
  • Look up words and formulas. Dotted words show their meaning when you tap them. The Glossary and the Formula Sheet collect everything in one place.

4. Quick check โ€” are you ready?

Eight quick questions on Class 9 skills this course uses all the time. Answer them all; for any you miss you will see what to revise.

Q1. Which of these numbers is irrational?
Q2. $(x+3)(x-5)=$
Q3. If $p(x)=x^2-3x+2$, then $p(-1)=$
Q4. The solution of $3x-7=11$ is
Q5. The point $(-3,4)$ lies in quadrant
Q6. A right triangle has legs $6$ cm and $8$ cm. Its hypotenuse is
Q7. Two angles of a triangle are $50^\circ$ and $65^\circ$. The third angle is
Q8. The volume of a cylinder with radius $7$ cm and height $10$ cm, taking $\pi=\tfrac{22}{7}$, is

When you are ready, start with Chapter 1 โ€” Real Numbers.

โ† Class 10 Mathematics โ€” ContentsChapter 1 โ€” Real Numbers โ†’