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 $\alpha$, $\beta$:
$$\alpha + \beta = -\frac{b}{a}$$$$\alpha\beta = \frac{c}{a}$$
For $ax^3 + bx^2 + cx + d$ with zeroes $\alpha$, $\beta$, $\gamma$:
$$\alpha + \beta + \gamma = -\frac{b}{a}$$$$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$$
$$\alpha\beta\gamma = -\frac{d}{a}$$
Quadratic with sum $S$ and product $P$ of zeroes: $k(x^2 - Sx + P)$
3. Pair of Linear Equations#
For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:
| Condition | Lines | Solutions |
|---|---|---|
| $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ | Intersecting | One (consistent) |
| $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ | Coincident | Infinitely many (dependent) |
| $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ | Parallel | None (inconsistent) |
4. Quadratic Equations#
For $ax^2 + bx + c = 0$, $a \neq 0$:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$$$D = b^2 - 4ac$$
$D > 0$: two distinct real roots · $D = 0$: two equal roots $\left(-\frac{b}{2a}\right)$ · $D < 0$: no real roots
5. Arithmetic Progressions#
$$d = a_{n+1} - a_n$$$$a_n = a + (n - 1)d$$
$$S_n = \frac{n}{2}\left[2a + (n - 1)d\right]$$
$$S_n = \frac{n}{2}(a + l)$$
$$a_n = S_n - S_{n-1}$$
- $n$th term from the end: $l - (n - 1)d$
- If $a$, $b$, $c$ are in AP: $b = \frac{a + c}{2}$
- $1 + 2 + \cdots + n = \frac{n(n + 1)}{2}$
6. Triangles#
- BPT: if $DE \parallel BC$ in $\triangle ABC$, then $\frac{AD}{DB} = \frac{AE}{EC}$. The converse is also true.
- Similarity criteria: AAA, AA, SSS, SAS (and RHS for right triangles).
- If $\triangle ABC \sim \triangle DEF$: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$
7. Coordinate Geometry#
$$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$$$OP = \sqrt{x^2 + y^2}$$
Section formula (ratio $m_1 : m_2$):
$$\left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2},\ \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)$$Mid-point:
$$\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$$8. Introduction to Trigonometry#
$$\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$$$$\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$$
$$\tan A = \frac{\text{opposite}}{\text{adjacent}}$$
$\cosec A = \frac{1}{\sin A}$, $\sec A = \frac{1}{\cos A}$, $\cot A = \frac{1}{\tan A}$, $\tan A = \frac{\sin A}{\cos A}$
| $0°$ | $30°$ | $45°$ | $60°$ | $90°$ | |
|---|---|---|---|---|---|
| $\sin$ | $0$ | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | $1$ |
| $\cos$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | $0$ |
| $\tan$ | $0$ | $\frac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | Not defined |
| $\cosec$ | Not defined | $2$ | $\sqrt{2}$ | $\frac{2}{\sqrt{3}}$ | $1$ |
| $\sec$ | $1$ | $\frac{2}{\sqrt{3}}$ | $\sqrt{2}$ | $2$ | Not defined |
| $\cot$ | Not defined | $\sqrt{3}$ | $1$ | $\frac{1}{\sqrt{3}}$ | $0$ |
$$1 + \tan^2 A = \sec^2 A$$
$$1 + \cot^2 A = \cosec^2 A$$
9. Applications of Trigonometry#
$$\tan\theta = \frac{\text{height}}{\text{horizontal distance}}$$- Angle of elevation from A to B = angle of depression from B to A.
- Subtract the observer’s eye height from the object’s height.
10. Circles#
- Tangent ⊥ radius at the point of contact.
- Tangents from an external point are equal: $PA = PB$.
- Tangents from inside: 0 · on the circle: 1 · outside: 2.
- Angle between two tangents + angle at centre = $180°$.
- Quadrilateral circumscribing a circle: $AB + CD = AD + BC$.
11. Areas Related to Circles#
$$\text{Length of arc} = \frac{\theta}{360°} \times 2\pi r$$$$\text{Area of sector} = \frac{\theta}{360°} \times \pi r^2$$
$$\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}$$
Triangle in the segment: $\frac{\sqrt{3}}{4}r^2$ for $60°$ · $\frac{1}{2}r^2$ for $90°$ · $\frac{\sqrt{3}}{4}r^2$ for $120°$
12. Surface Areas and Volumes#
| Solid | CSA | TSA | Volume |
|---|---|---|---|
| Cuboid | $2h(l + b)$ | $2(lb + bh + hl)$ | $lbh$ |
| Cube | $4a^2$ | $6a^2$ | $a^3$ |
| Cylinder | $2\pi rh$ | $2\pi r(r + h)$ | $\pi r^2 h$ |
| Cone | $\pi rl$ | $\pi r(r + l)$ | $\frac{1}{3}\pi r^2 h$ |
| Sphere | $4\pi r^2$ | $4\pi r^2$ | $\frac{4}{3}\pi r^3$ |
| Hemisphere | $2\pi r^2$ | $3\pi r^2$ | $\frac{2}{3}\pi r^3$ |
Cone: $l = \sqrt{r^2 + h^2}$. Combined solid: add only the visible surfaces; volumes simply add.
13. Statistics#
$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$$$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$$
$$\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h$$
$$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$
$$\text{Median} = l + \frac{\frac{n}{2} - cf}{f} \times h$$
$$3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$$
14. Probability#
$$P(E) = \frac{\text{favourable outcomes}}{\text{total outcomes}}$$$$P(E) + P(\text{not } E) = 1$$
- $0 \leq P(E) \leq 1$ · sure event: 1 · impossible event: 0
- Sum of the probabilities of all elementary events = 1
- Deck: 52 cards, 4 suits of 13, 26 red, 26 black, 12 face cards