Maths

Fundamental Theorem of Arithmetic

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.

Used to compute HCF and LCM; proving the irrationality of numbers like $\sqrt{2}, \sqrt{3},$ and $\sqrt{5}$; exploring the nature (terminating/non-terminating repeating) of the decimal expansion of a rational number $p/q$ by looking at the prime factorisation of $q$.

HCF and LCM (Two Positive Integers $a$ and $b$)

$\text{HCF} (a, b) \times \text{LCM} (a, b) = a \times b$.

Used to find the LCM of two positive integers if their HCF is already known.

HCF (Prime Factorisation Method)

Product of the smallest power of each common prime factor in the numbers.

Used for computing the HCF of two positive integers.

LCM (Prime Factorisation Method)

Product of the greatest power of each prime factor, involved in the numbers.

Used for computing the LCM of two positive integers.

Divisibility Theorem

If $p$ is a prime number and $p$ divides $a^2$, then $p$ divides $a$, where $a$ is a positive integer.

A supporting theorem for proving the irrationality of numbers like $\sqrt{2}$.

Linear

$p(x) = ax + b$ ($a \neq 0$).

Zero $k = \frac{-b}{a}$.

Finding the single zero of a linear polynomial.

Quadratic (Zeroes $\alpha, \beta$)

$ax^2 + bx + c$ ($a \neq 0$).

Sum of zeroes: $\alpha + \beta = \frac{-b}{a}$

Relating the zeroes of the polynomial to its coefficients.

Product of zeroes: $\alpha\beta = \frac{c}{a}$

Cubic (Zeroes $\alpha, \beta, \gamma$)

$ax^3 + bx^2 + cx + d$ ($a \neq 0$).

Sum of zeroes: $\alpha + \beta + \gamma = \frac{-b}{a}$

Relating the zeroes of the polynomial to its coefficients.

Sum of products of zeroes taken two at a time: $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$

Product of zeroes: $\alpha\beta\gamma = \frac{-d}{a}$

$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

Intersecting lines

Exactly one solution (Unique solution) / Consistent pair of equations.

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Coincident lines

Infinitely many solutions / Dependent (Consistent) pair of equations.

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

Parallel lines

No solution / Inconsistent pair of equations.

Methods of Solution

Algebraic methods used for finding solutions include the Substitution Method and the Elimination Method.

Quadratic Formula (Roots $x$)

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Used to find the roots (solutions) of the quadratic equation. Must have $b^2 - 4ac \ge 0$.

Discriminant ($D$)

$D = b^2 - 4ac$.

Determines the nature of the roots.

Nature of Roots

Two distinct real roots if $b^2 - 4ac > 0$.

Used to check if real solutions exist for a given problem.

Two equal real roots if $b^2 - 4ac = 0$.

When $b^2 - 4ac = 0$, both roots are $x = -b/2a$.

No real roots if $b^2 - 4ac < 0$.

Indicates that no real solution exists for the equation.

Solution by Factorisation

Factorise $ax^2 + bx + c$ into a product of two linear factors and equate each factor to zero.

Used to find the roots/solutions of a quadratic equation.

Common Difference ($d$)

$d = a_{k+1} - a_k$.

The fixed number added to the preceding term to obtain the successive term.

$n$th Term (General Term) ($a_n$ or $l$)

$a_n = a + (n - 1)d$.

Finding the value of any specific term in the AP, where $a$ is the first term and $n$ is the term number.

Sum of First $n$ Terms ($S_n$)

$S = \frac{n}{2} [2a + (n - 1)d]$.

Finding the sum of the first $n$ terms of an AP.

Sum of First $n$ Terms (Alternate)

$S = \frac{n}{2} (a + a_n)$. $S = \frac{n}{2} (a + l)$.

Used when the first term ($a$) and the last term ($a_n$ or $l$) are known.

$n$th term from Sum

$a_n = S_n - S_{n-1}$.

Finding the $n$th term if the sum of $n$ terms and $(n-1)$ terms is known.

Sum of First $n$ Positive Integers

$S_n = \frac{n(n+1)}{2}$.

Finding the sum of consecutive integers starting from 1.

Arithmetic Mean

If $a, b, c$ are in AP, then $b = \frac{a+c}{2}$.

Finding the middle term when three terms are in AP.

Basic Proportionality Theorem (Theorem 6.1 / Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. (i.e., $\frac{AD}{DB} = \frac{AE}{EC}$).

Used to prove segments are proportional if a line parallel to the base intersects the other sides.

Converse of BPT (Theorem 6.2)

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Used to prove lines are parallel.

AAA Similarity Criterion (Theorem 6.3)

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (proportional) and hence the two triangles are similar. (Also known as AA Similarity if two angles are equal).

Establishing similarity between triangles when angles are known.

SSS Similarity Criterion (Theorem 6.4)

If in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

Establishing similarity between triangles when all corresponding side ratios are equal.

SAS Similarity Criterion (Theorem 6.5)

If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

Establishing similarity when one angle and the ratio of the two included sides are known.

Distance Formula

Distance $PQ$ between $P(x_1, y_1)$ and $Q(x_2, y_2)$: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Finding the distance between any two points in a plane.

Distance from Origin

Distance of $P(x, y)$ from origin $(0, 0)$: $OP = \sqrt{x^2 + y^2}$.

Finding the distance of a point from the origin.

Section Formula (Internal Division)

Coordinates $(x, y)$ of point $P$ dividing segment $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in ratio $m_1 : m_2$: $(x, y) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right)$.

Finding the coordinates of a point that divides a line segment into a given ratio.

Section Formula (Ratio $k:1$)

$\left( \frac{k x_2 + x_1}{k+1}, \frac{k y_2 + y_1}{k+1} \right)$.

Simplified form of the section formula when the ratio is expressed as $k:1$.

Mid-point Formula

Coordinates of the mid-point $P$ (ratio $1:1$): $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.

Finding the coordinates of the mid-point of a line segment.

Sine (sin)

$\sin A = \frac{\text{Side opposite to } A}{\text{Hypotenuse}} = \frac{BC}{AC}$.

Expresses the relationship between an acute angle and the lengths of the sides of a right triangle.

Cosine (cos)

$\cos A = \frac{\text{Side adjacent to } A}{\text{Hypotenuse}} = \frac{AB}{AC}$.

Tangent (tan)

$\tan A = \frac{\text{Side opposite to } A}{\text{Side adjacent to } A} = \frac{BC}{AB}$.

Cosecant (csc or cosec)

$\csc A = \frac{1}{\sin A} = \frac{\text{Hypotenuse}}{\text{Side opposite to } A}$.

Reciprocal ratios.

Secant (sec)

$\sec A = \frac{1}{\cos A} = \frac{\text{Hypotenuse}}{\text{Side adjacent to } A}$.

Cotangent (cot)

$\cot A = \frac{1}{\tan A} = \frac{\text{Side adjacent to } A}{\text{Side opposite to } A}$.

Relationship Ratios

$\tan A = \frac{\sin A}{\cos A}$ and $\cot A = \frac{\cos A}{\sin A}$.

Expressing tangent and cotangent in terms of sine and cosine.

Pythagorean Identity 1

$\sin^2 A + \cos^2 A = 1$ (True for $0^\circ \le A \le 90^\circ$).

A fundamental trigonometric identity; used for relating sine and cosine.

Pythagorean Identity 2

$1 + \tan^2 A = \sec^2 A$ (True for $0^\circ \le A < 90^\circ$).

Relating tangent and secant.

Pythagorean Identity 3

$1 + \cot^2 A = \csc^2 A$ (True for $0^\circ < A \le 90^\circ$).

Relating cotangent and cosecant.

Heights and Distances

Distances or heights are determined by using trigonometric ratios ($\sin, \cos, \tan$, etc.) in right triangles formed by the object, the observer's position, and the ground.

Angle of Elevation

The angle formed by the line of sight with the horizontal when the object viewed is above the horizontal level (raising the head).

Angle of Depression

The angle formed by the line of sight with the horizontal when the object viewed is below the horizontal level (lowering the head).

Tangent-Radius Perpendicularity (Theorem 10.1)

The tangent at any point $P$ of a circle is perpendicular to the radius $OP$ through the point of contact $P$.

Used to establish right angles in diagrams involving tangents and radii.

Length of Tangents (Theorem 10.2)

The lengths of tangents drawn from an external point $P$ to a circle are equal ($PQ = PR$).

Used to prove geometric equality and relationships involving tangents.

Area of a Sector

$\text{Area} = \frac{\theta}{360} \times \pi r^2$.

Finding the area of the portion enclosed by two radii and the corresponding arc.

Length of an Arc

$\text{Length} = \frac{\theta}{360} \times 2\pi r$.

Finding the length of the arc corresponding to the sector angle $\theta$.

Area of Minor Segment

$\text{Area} = \text{Area of the corresponding sector} - \text{Area of the corresponding triangle}$ ($\Delta OAB$).

Finding the area of the portion enclosed between a chord and the corresponding arc.

Area of Major Sector

$\text{Area} = \pi r^2 - \text{Area of the minor sector}$.

Finding the area of the remaining larger sector.

Area of Major Segment

$\text{Area} = \pi r^2 - \text{Area of the minor segment}$.

Finding the area of the remaining larger segment.

Volume of a Combination of Solids

The volume of the solid formed by joining two basic solids is the sum of the volumes of the constituents.

Calculating the total space occupied by a combined solid.

Surface Area of a Combination of Solids

Calculated by summing the Curved Surface Areas (CSA) of the individual parts, as the internal flat faces that are joined together disappear.

Calculating the total external surface area of a combined solid (e.g., a toy or a container).

Class Mark ($x_i$)

$x_i = \frac{\text{Upper class limit} + \text{Lower class limit}}{2}$.

Represents the mid-point of a class interval.

Mean (Direct Method)

$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.

Used when numerical values of $x_i$ and $f_i$ are sufficiently small.

Mean (Assumed Mean Method)

$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$, where $d_i = x_i - a$.

Used when $x_i$ and $f_i$ are numerically large.

Mean (Step-deviation Method)

$\bar{x} = a + h \left( \frac{\sum f_i u_i}{\sum f_i} \right)$, where $u_i = \frac{x_i - a}{h}$.

Most convenient if all deviations ($d_i$) have a common factor $h$.

Mode

$\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$.

Finding the value with the maximum frequency (modal class). ($l$: lower limit of modal class; $f_1$: modal frequency; $f_0$: preceding frequency; $f_2$: succeeding frequency; $h$: class size).

Median

$\text{Median} = l + \left( \frac{n/2 - cf}{f} \right) \times h$.

Finding the middle-most observation. ($l$: lower limit of median class; $n$: total observations; $cf$: preceding cumulative frequency; $f$: median frequency; $h$: class size).

Empirical Relationship

$3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean}$.

Relates the three measures of central tendency.

Theoretical Probability ($P(E)$)

$P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Number of all possible outcomes of the experiment}}$.

Calculating the probability of an event $E$ occurring, assuming equally likely outcomes.

Range of Probability

$0 \le P(E) \le 1$.

Probability value must be between 0 (impossible event) and 1 (sure event), inclusive.

Complementary Events ($E$ and $\bar{E}$)

$P(\bar{E}) = 1 - P(E)$ or $P(E) + P(\bar{E}) = 1$.

Calculating the probability that an event will not occur.

Sum of Elementary Events

The sum of the probabilities of all the elementary events of an experiment is 1.

Checks the completeness of probability distributions.

Impossible Event

$P(E) = 0$.

Describes an event that cannot happen.

Sure (Certain) Event

$P(E) = 1$.

Describes an event that is certain to happen.