Unit 1Relations and Functions
Cartesian product
$A \times B = \{(a,b) : a \in A,\ b \in B\}$
Number of elements
$n(A \times B) = n(A) \times n(B)$
Number of relations from A to B
$2^{\,n(A)\times n(B)}$ (each ordered pair is either in the relation or not)
Types of functions
one-one (injective), onto (surjective), one-one and onto (bijective), into, and many-one.
Special functions
Identity $f(x)=x$; constant $f(x)=c$; linear $f(x)=ax+b$; quadratic $f(x)=ax^2+bx+c$.
Unit 2Numbers and Sequences
Euclid's division lemma
$a = bq + r,\quad 0 \le r \lt b$
HCF and LCM relation
$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$
AP — nth term
$a_n = a + (n-1)d$
AP — sum of n terms
$S_n = \dfrac{n}{2}\big[2a + (n-1)d\big] = \dfrac{n}{2}(a + l)$
GP — nth term
$a_n = a\,r^{\,n-1}$
GP — sum of n terms
$S_n = \dfrac{a(r^n - 1)}{r - 1},\ r \neq 1$
Special sums
$\displaystyle\sum n = \dfrac{n(n+1)}{2}$, $\displaystyle\sum n^2 = \dfrac{n(n+1)(2n+1)}{6}$, $\displaystyle\sum n^3 = \left[\dfrac{n(n+1)}{2}\right]^2$
Unit 3Algebra
GCD–LCM of polynomials
$\text{LCM} \times \text{GCD} = f(x) \times g(x)$
Quadratic formula
$x = \dfrac{-b \pm \sqrt{\,b^2 - 4ac\,}}{2a}$ for $ax^2+bx+c=0$
Discriminant
$\Delta = b^2 - 4ac$. $\Delta\gt0$: real, distinct roots; $\Delta=0$: real, equal roots; $\Delta\lt0$: no real roots.
Roots and coefficients
Sum $\alpha+\beta = -\dfrac{b}{a}$, Product $\alpha\beta = \dfrac{c}{a}$
Matrices
Two matrices can be multiplied only when columns of the first = rows of the second; $(A^T)^T = A$.
Unit 4Geometry
Basic Proportionality (Thales)
A line parallel to one side of a triangle divides the other two sides in the same ratio: $\dfrac{AD}{DB} = \dfrac{AE}{EC}$.
Similar triangles
Criteria: AA, SSS and SAS. Areas of similar triangles are in the ratio of the squares of corresponding sides.
Pythagoras theorem
$\text{(hypotenuse)}^2 = \text{(side)}^2 + \text{(side)}^2$
Angle bisector theorem
The internal bisector of an angle divides the opposite side in the ratio of the other two sides.
Tangents to a circle
A tangent is perpendicular to the radius at the point of contact; the two tangents from an external point are equal in length.
Unit 5Coordinate Geometry
Distance between two points
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Section formula
$\left(\dfrac{mx_2+nx_1}{m+n},\ \dfrac{my_2+ny_1}{m+n}\right)$
Midpoint
$\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$
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 $0$.
Slope of a line
$m = \dfrac{y_2-y_1}{x_2-x_1} = \tan\theta$
Equations of a line
Slope-intercept $y = mx + c$; point-slope $y - y_1 = m(x - x_1)$.
Parallel / perpendicular
Parallel: $m_1 = m_2$; Perpendicular: $m_1 \times m_2 = -1$.
Unit 6Trigonometry
Pythagorean identities
$\sin^2\theta + \cos^2\theta = 1$
$1 + \tan^2\theta = \sec^2\theta$
$1 + \cot^2\theta = \mathrm{cosec}^2\theta$
Reciprocal ratios
$\mathrm{cosec}\,\theta = \dfrac{1}{\sin\theta}$, $\sec\theta = \dfrac{1}{\cos\theta}$, $\cot\theta = \dfrac{1}{\tan\theta}$, $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
Heights and distances
Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal.
Unit 7Mensuration
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 $= \dfrac{1}{3}\pi r^2 h$; slant height $l = \sqrt{r^2 + h^2}$.
Sphere
Surface area $= 4\pi r^2$; Volume $= \dfrac{4}{3}\pi r^3$.
Hemisphere
CSA $= 2\pi r^2$; TSA $= 3\pi r^2$; Volume $= \dfrac{2}{3}\pi r^3$.
Frustum of a cone
Volume $= \dfrac{1}{3}\pi h\,(R^2 + Rr + r^2)$, where $R$ and $r$ are the two radii.
Unit 8Statistics and Probability
Range and coefficient of range
Range $= L - S$; Coefficient of range $= \dfrac{L - S}{L + S}$ ($L$ = largest, $S$ = smallest).
Standard deviation
$\sigma = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n}}$ (variance $= \sigma^2$).
Coefficient of variation
$\mathrm{C.V.} = \dfrac{\sigma}{\bar{x}} \times 100$
Probability of an event
$P(E) = \dfrac{n(E)}{n(S)},\quad 0 \le P(E) \le 1$
Complement
$P(\bar{E}) = 1 - P(E)$
Addition rule
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Frequently asked questions
- What is the quadratic formula in Class 10 Maths?
- For $ax^2+bx+c=0$ ($a\neq0$), the roots are $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$. The term $b^2-4ac$ is the discriminant.
- What is the formula for the nth term and sum of an AP?
- nth term $a_n = a+(n-1)d$; sum $S_n = \dfrac{n}{2}[2a+(n-1)d] = \dfrac{n}{2}(a+l)$.
- What are the three basic trigonometric identities?
- $\sin^2\theta+\cos^2\theta=1$, $1+\tan^2\theta=\sec^2\theta$, $1+\cot^2\theta=\mathrm{cosec}^2\theta$.
- How many relations can be formed from set A to set B?
- If $n(A)=p$ and $n(B)=q$, the number of relations is $2^{pq}$.
These formulas follow the Tamil Nadu Samacheer Kalvi Class 10 Maths textbook. For worked examples and book-back answers, open each unit's full solutions linked above.