Unit 1Set Language
Union and intersection
$A \cup B = \{x : x \in A \text{ or } x \in B\}$; $A \cap B = \{x : x \in A \text{ and } x \in B\}$.
Cardinality of a union
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Three sets
$n(A\cup B\cup C) = n(A)+n(B)+n(C) - n(A\cap B) - n(B\cap C) - n(A\cap C) + n(A\cap B\cap C)$
De Morgan's laws
$(A \cup B)' = A' \cap B'$; $(A \cap B)' = A' \cup B'$.
Number of subsets
A set with $n$ elements has $2^n$ subsets and $2^n - 1$ proper subsets.
Unit 2Real Numbers
Rational number
A number of the form $\dfrac{p}{q}$, where $p,q$ are integers and $q \neq 0$.
Laws of exponents
$a^m \times a^n = a^{m+n}$; $\dfrac{a^m}{a^n} = a^{m-n}$; $(a^m)^n = a^{mn}$; $a^0 = 1$.
Surds
$\sqrt{a}\,\sqrt{b} = \sqrt{ab}$; $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$.
Rationalising the denominator
Multiply numerator and denominator by a suitable factor, e.g. $\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$; for $\dfrac{1}{a+\sqrt{b}}$ multiply by the conjugate $a-\sqrt{b}$.
Unit 3Algebra
Square identities
$(a+b)^2 = a^2 + 2ab + b^2$; $(a-b)^2 = a^2 - 2ab + b^2$; $a^2 - b^2 = (a+b)(a-b)$.
Three-term square
$(a+b+c)^2 = a^2+b^2+c^2 + 2ab + 2bc + 2ca$
Product identity
$(x+a)(x+b) = x^2 + (a+b)x + ab$
Cube identities
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$; $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$; $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$.
Remainder theorem
When $p(x)$ is divided by $(x-a)$, the remainder is $p(a)$.
Factor theorem
$(x-a)$ is a factor of $p(x)$ if and only if $p(a) = 0$.
Unit 4Geometry
Angle sum of a triangle
The three interior angles of a triangle add up to $180^\circ$.
Exterior angle
An exterior angle of a triangle equals the sum of the two opposite interior angles.
Parallelogram
Opposite sides and angles are equal; the diagonals bisect each other.
Centroid
The centroid divides each median in the ratio $2:1$ from the vertex.
Angle in a circle
The angle subtended by an arc at the centre is twice the angle subtended at any point on the remaining circle.
Cyclic quadrilateral
Opposite angles of a cyclic quadrilateral are supplementary (sum to $180^\circ$).
Unit 5Coordinate Geometry
Distance between two points
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint
$\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$
Section formula
$\left(\dfrac{mx_2+nx_1}{m+n},\ \dfrac{my_2+ny_1}{m+n}\right)$
Area of a triangle
$\dfrac{1}{2}\,\big|\,x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\,\big|$
Collinearity
Three points are collinear if the area of the triangle they form is $0$.
Unit 6Trigonometry
The six ratios
$\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$, and their reciprocals $\mathrm{cosec}\,\theta$, $\sec\theta$, $\cot\theta$.
Pythagorean identities
$\sin^2\theta + \cos^2\theta = 1$; $1 + \tan^2\theta = \sec^2\theta$; $1 + \cot^2\theta = \mathrm{cosec}^2\theta$.
Quotient relations
$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$.
Standard angle values
$\sin$: $0,\ \tfrac12,\ \tfrac1{\sqrt2},\ \tfrac{\sqrt3}{2},\ 1$ (for $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$); $\cos$ is the reverse; $\tan$: $0,\ \tfrac1{\sqrt3},\ 1,\ \sqrt3,\ \infty$.
Unit 7Mensuration
Heron's formula
Area $= \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \dfrac{a+b+c}{2}$.
Sector of a circle
Area $= \dfrac{\theta}{360^\circ}\times \pi r^2$; arc length $= \dfrac{\theta}{360^\circ}\times 2\pi r$.
Cylinder
CSA $= 2\pi rh$; TSA $= 2\pi r(h+r)$; Volume $= \pi r^2 h$.
Cone
CSA $= \pi r l$; TSA $= \pi r(l+r)$; Volume $= \dfrac{1}{3}\pi r^2 h$; $l = \sqrt{r^2+h^2}$.
Sphere and hemisphere
Sphere: SA $= 4\pi r^2$, Volume $= \dfrac{4}{3}\pi r^3$. Hemisphere: CSA $= 2\pi r^2$, TSA $= 3\pi r^2$, Volume $= \dfrac{2}{3}\pi r^3$.
Unit 8Statistics
Mean
$\bar{x} = \dfrac{\sum x}{n}$; for a frequency table $\bar{x} = \dfrac{\sum fx}{\sum f}$.
Median
The middle value of data arranged in order; for an even count, the average of the two middle values.
Mode
The value that occurs most frequently in the data.
Range
Range $=$ largest value $-$ smallest value.
Unit 9Probability
Probability of an event
$P(E) = \dfrac{n(E)}{n(S)} = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$
Range of probability
$0 \le P(E) \le 1$. $P(E) = 0$ for an impossible event, $P(E) = 1$ for a sure event.
Complement
$P(\bar{E}) = 1 - P(E)$
Frequently asked questions
- What are the common algebraic identities in Class 9 Maths?
- $(a+b)^2 = a^2+2ab+b^2$, $(a-b)^2 = a^2-2ab+b^2$, $a^2-b^2=(a+b)(a-b)$, $(a+b)^3 = a^3+3a^2b+3ab^2+b^3$.
- What is Heron's formula?
- Area $= \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \dfrac{a+b+c}{2}$ is the semi-perimeter.
- What is De Morgan's law for sets?
- $(A\cup B)' = A'\cap B'$ and $(A\cap B)' = A'\cup B'$.
- What is the distance formula in coordinate geometry?
- The distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
These formulas follow the Tamil Nadu Samacheer Kalvi Class 9 Maths textbook. For worked examples and book-back answers, open each unit's full solutions linked above.