Concept library

Draw a horizontal number line (the x-axis) and a vertical number line (the y-axis) that cross at their zeros. The crossing point is the origin O, written (0, 0)…

The axes cut the plane into four quadrants, numbered anticlockwise starting from the top right. The signs of the coordinates tell you the quadrant immediately, without…

To find the distance between A(x1, y1) and B(x2, y2), draw a horizontal line from A and a vertical line from B. They meet at C(x2, y1), making a right-angled triangle…

The midpoint of a segment sits exactly halfway, so each of its coordinates is the average of the end-point coordinates.

Once points have coordinates, you can prove geometric facts with arithmetic. Collinearity: three points A, B, C lie on one line exactly when the two shorter distances…

A polynomial in x is a sum of terms of the form (number) × x^n, where every power n is a whole number (0, 1, 2, 3, …). For example, 4x^3 - 2x + 7 is a polynomial with…

A linear polynomial p(x) = ax + b (a ≠ 0) changes by exactly a every time x increases by 1. To evaluate it, substitute a number: if p(x) = 3x - 6, then p(4) = 3(4) - 6 =…

A pattern is linear when the difference between consecutive terms is constant. Look at matchstick squares in a row: 4, 7, 10, 13, … sticks. Each new square adds 3 sticks.

Linear growth means a quantity increases by the same fixed amount in each unit of time: V = (start) + (rate) × t, with a positive rate.

A relationship y = ax + b links two quantities so that y changes by a whenever x increases by 1. Plotting its pairs (x, y) always gives a straight line.

The natural numbers mathbbN = \1, 2, 3, …\ come from counting. Adding zero gives the whole numbers \0, 1, 2, …\. Zero (śūnya) was a major Indian contribution: it lets…

A rational number can be written as p/q where p and q are integers and q ≠ 0. Every integer is rational (-4 = (-4)/(1)), and so is every fraction and every terminating…

Between any two different rational numbers there is always another one — for example their average (a + b)/(2). Repeat the idea and you find infinitely many. This is…

An irrational number cannot be written as p/q with integers p, q. Examples: √(2), √(3), √(5), π. Their decimals go on forever without a repeating block.

Rational and irrational numbers together form the real numbers mathbbR. Every real number is a point on the number line, and every point on the line is a real number —…

Divide p by q and the decimal either terminates (3/8 = 0.375) or repeats forever (2/11 = 0.181818… = 0.18). It must repeat because only q different remainders are…

An equation such as 2x + 1 = 7 is true only for some values (x = 3). An identity such as (a + b)^2 = a^2 + 2ab + b^2 is true for all values. You can test a claimed…

A square of side a + b + c splits into three squares and six rectangles (two each of ab, bc, ca), which gives (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.

Factorising reverses expanding: write an expression as a product. Always start by taking out any common factor, then look for a pattern.

To factorise ax^2 + bx + c, find two numbers whose product is ac and whose sum is b. Use them to split bx into two terms, then group in pairs.

Multiplying (a + b)^2 by (a + b) once more gives (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, which can be written compactly as a^3 + b^3 + 3ab(a + b). Picture a cube of side…

A rational expression is a fraction of polynomials, like (x^2 - 9)/(x^2 + 5x + 6). To simplify, factorise top and bottom fully, then cancel factors that are common to…

A circle is the set of all points in a plane at a fixed distance (the radius r) from a fixed point (the centre O). A chord joins two points on the circle; a chord…

Through one point you can draw infinitely many circles; through two points, still infinitely many (their centres all lie on the perpendicular bisector of the segment…

The perpendicular from the centre to a chord bisects it. Proof: join O to both ends A, B of the chord and drop the perpendicular OM. Triangles OMA and OMB are…

The angle an arc subtends at the centre is twice the angle it subtends at any point on the remaining part of the circle. Proof idea: join the point P on the circle…

Points that lie on one circle are concyclic. A quadrilateral whose four vertices lie on a circle is cyclic.

The perimeter of a polygon is the sum of its side lengths. For a circle the perimeter is called the circumference C.

An arc with central angle θ (in degrees) is the fraction (θ)/(360) of the full circle, so its length is that fraction of the circumference.

Area counts unit squares. A rectangle has area l × b. A parallelogram can be cut and slid into a rectangle with the same base and the perpendicular height, so its area…

When you know all three sides a, b, c of a triangle but not its height, use Heron's formula. First compute the semi-perimeter s = (a + b + c)/(2).

Brahmagupta (628 CE) gave the area of a cyclic quadrilateral (all four vertices on a circle) with sides a, b, c, d using its semi-perimeter s = (a + b + c + d)/(2).

Cut a circle into many thin sectors and arrange them alternately up and down: they form a shape close to a rectangle of length π r (half the circumference) and height r…

The probability of an event is a number from 0 to 1 that measures how likely it is. 0 means impossible (rolling a 7 on a normal die), 1 means certain (the sun rising…

Experimental (empirical) probability comes from what actually happened: P(E) = (number of trials in which E happened)/(total number of trials). It is also called the…

An experiment is any action with uncertain results. Each possible result is an outcome, and the list of all outcomes is the sample space S. An event is a set of outcomes…

A tree diagram shows a multi-stage experiment: the first set of branches for stage 1, and from the end of each, branches for stage 2. Write the probability on every…

A sequence is an ordered list of numbers a1, a2, a3, …; the subscript is the position n and an is the term at that position. Position and value are different things —…

An arithmetic progression (AP) has a constant common difference d = an+1 - an. Starting from the first term a, each step adds d, so after n - 1 steps: an = a + (n - 1)d.

To add 1 + 2 + … + n, write the sum twice — forwards and backwards — and add the columns: every column gives n + 1, and there are n columns. So twice the sum is n(n + 1).

A geometric progression (GP) has a constant non-zero common ratio r = (an+1)/(an). Each step multiplies by r, so an = a r^n-1.

A fractal is a shape built by repeating the same step at smaller and smaller scales. Start with an equilateral triangle, join the midpoints of its sides and remove the…

A proposition is a sentence that is definitely true or false, not both. '7 is a prime number' is a proposition (true). '12 is odd' is a proposition (false). 'Close the…

The converse of 'If X, then Y' is 'If Y, then X' — the hypothesis and conclusion swap places.

A claim about all cases is disproved by one case where it fails — a counterexample. To disprove 'Every prime is odd', the single example 2 is enough.

Sometimes a statement and its converse are both true. Then X and Y are equivalent, written X Leftrightarrow Y and read 'X if and only if Y'.

Baudhāyana–Pythagoras theorem: if a triangle has a right angle, then the square on the hypotenuse equals the sum of the squares on the other two sides (a^2 + b^2 = c^2)…

The mean (average) of a set of values is their total divided by how many there are. To combine two groups, go back to totals: total of all values ÷ total number of…

In a weighted mean, each value counts according to its weight — how important it is or how often it occurs. Multiply each value by its weight, add, then divide by the…

The concentration of a mixture is (amount of ingredient) ÷ (total amount). When you mix two solutions, find the amount of ingredient in each, add them, and divide by the…

Adding a new value above the mean raises it; adding one below the mean lowers it; adding a value equal to the mean leaves it unchanged. Removing values works the other…

A clustered bar chart puts bars side by side to compare categories. A stacked bar chart divides each bar into coloured parts, so you can see both the total (whole bar…

An algorithm is a finite sequence of clear, unambiguous steps that takes an input, and after a finite number of steps produces an output — the same output every time for…

A divisor of n divides it with no remainder. The greatest common divisor gcd(a, b) (also called HCF) is the largest number that divides both. The least common multiple…

Euclid (around 300 BCE) noticed: any number that divides both a and b also divides a - b, and vice versa. So gcd(a, b) = gcd(a - b, b) when a > b.

Repeated subtraction of b from a is just division: subtracting b as many times as possible leaves the remainder. So gcd(a, b) = gcd(b, r) where a = bq + r, 0 ≤ r < b.

Algorithms often store intermediate results in a list (a simple data structure) — for example the divisors found so far, or the primes already known.

A quadrilateral ABCD has four vertices, four sides (AB, BC, CD, DA) and two diagonals (AC and BD). Sides that share a vertex are adjacent; sides that do not are opposite.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Draw diagonal AC of parallelogram ABCD: alternate angles give ∠ BAC = ∠ DCA and ∠ BCA = ∠…

Most properties have true converses (Chapter 9), and each gives a test. A quadrilateral is a parallelogram if: both pairs of opposite sides are equal; or both pairs of…

Midpoint theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it. If E and F are the midpoints of AB and AC…

A median joins a vertex to the midpoint of the opposite side. A triangle's three medians always pass through one point, the centroid G — its balance point.

Varignon's theorem: join the midpoints of the sides of any quadrilateral in order and you always get a parallelogram. Draw diagonal AC: by the midpoint theorem, the two…

A linear equation in two variables can be written as ax + by + c = 0, where a and b are not both zero. Examples: 2x + 3y = 12, y = 5x - 1, x = 4 (here b = 0).

Plot several solutions of a linear equation and they all lie on one straight line; every point on the line is a solution. To draw it, find two points (intercepts are…

A pair (system) of linear equations asks for (x, y) satisfying both. On a graph, that is a point on both lines.

Make one variable the subject of one equation, substitute that expression into the other equation, and solve the resulting one-variable equation. Then substitute back to…

Multiply one or both equations by suitable numbers so that one variable has equal (or opposite) coefficients, then subtract (or add) the equations to eliminate it.

Step 1: name the two unknowns clearly (with units). Step 2: translate each sentence into one equation. Step 3: solve by any method. Step 4: answer the question asked, in…

A cuboid with length l, breadth b and height h has 6 rectangular faces in 3 equal pairs. Unfold it into a net and add the areas: 2(lb + bh + hl). The four side faces…

Cut the curved surface of a cylinder (radius r, height h) straight down and unroll it: you get a rectangle with length equal to the base circumference 2π r and width h…

A cone has radius r, vertical height h (from the vertex straight down to the centre of the base) and slant height l (from the vertex to the edge of the base). These form…

A pyramid has a polygon base and triangular faces meeting at a vertex (apex). The Egyptian pyramids have square bases.

A sphere of radius r has surface area 4π r^2 — exactly four times the area of its largest circle. (Archimedes showed it equals the curved surface of the cylinder that…

Melting and recasting: when a solid is melted and reshaped, its volume stays the same (ignoring waste). Set old volume = new volume, or old volume = (number of pieces) ×…

In earlier classes you mostly described things in words: 'the ball moves fast', 'sugar disappears in water'. In Class 9 you start to measure and quantify: the ball moves…

An observation is what you detect with your senses or instruments: 'the solution turned milky', 'the thermometer read 31 °C', 'the plant leaned towards the window'.

Science begins with a testable question: 'Does the temperature of water affect how fast sugar dissolves?' A hypothesis is a possible answer that leads to a prediction…

Every measurement needs a number and a unit. Science uses the SI system: metre (m) for length, kilogram (kg) for mass, second (s) for time, kelvin (K) for temperature…

Many things in science are too small (atoms, cells), too large (Earth's atmosphere) or too complex (an ecosystem) to see all at once. Scientists use models — diagrams…

In 1665 Robert Hooke looked at a thin slice of cork through a simple microscope and saw tiny box-like compartments, which he called cells. He was actually seeing the…

Unicellular organisms (Amoeba, Paramecium, bacteria, yeast) are a single cell that does everything — feeding, moving, reproducing. Multicellular organisms have many…

Every cell is surrounded by a thin, flexible plasma membrane (cell membrane) made of lipids and proteins. It is selectively permeable: it lets some substances (water…

The nucleus is the control centre. It is enclosed by a double nuclear membrane with pores. Inside are chromosomes, made of DNA and proteins. DNA carries the instructions…

Organelles work together like departments of a factory. Mitochondria release energy from food by respiration and store it as ATP — they are the 'powerhouses'. Ribosomes…

Diffusion is the spreading of particles from a region of higher concentration to lower concentration because of their random motion. Oxygen diffuses into cells and…

A tissue is a group of cells, similar in structure (or working together), that perform a particular function. Blood, bone, xylem and the lining of the stomach are…

Meristematic tissue is made of actively dividing cells. The cells are small, thin-walled, packed closely without spaces, with a large nucleus, dense cytoplasm and few or…

Simple permanent tissues are made of one kind of cell. Parenchyma has living, thin-walled, loosely packed cells with spaces between them; it stores food and, when it…

Complex tissues contain more than one type of cell working together. Xylem and phloem together form vascular bundles, the plant's transport system.

Epithelial tissue covers the body and lines organs and cavities. Its cells fit tightly with little space between them, resting on a basement membrane. Squamous…

Muscle tissue is made of long cells (muscle fibres) with contractile proteins that shorten to cause movement. Skeletal (striated) muscle is attached to bones and under…

An object is in motion if its position changes with time relative to a reference point (origin). Rest and motion are relative: a passenger in a moving bus is at rest…

Speed is distance travelled per unit time. Average speed = (total distance)/(total time). Its SI unit is m/s; km/h is common in daily life.

Acceleration is the rate of change of velocity: a = (v - u)/(t), where u is the initial velocity and v the final velocity after time t. Its SI unit is m/s² (metres per…

On a distance–time (or position–time) graph, the slope gives the speed (velocity). A straight sloping line = uniform motion; a steeper line = faster; a horizontal line =…

For motion in a straight line with uniform acceleration, three equations connect u (initial velocity), v (final velocity), a (acceleration), t (time) and s…

A pure substance has a fixed composition and fixed properties, such as a definite melting and boiling point. Elements (iron, oxygen) and compounds (water, sodium…

A solution is a homogeneous mixture. The substance present in the larger amount (usually the liquid) is the solvent; the substance dissolved in it is the solute…

A suspension is a heterogeneous mixture with particles big enough to see (larger than about 100 nm, often much larger). It looks cloudy, the particles settle on…

Every separation exploits a difference in a physical property between the components. Ask: do they differ in particle size, solubility, density, magnetism, boiling…

Simple distillation separates a liquid from dissolved solids (pure water from salt water) or two miscible liquids whose boiling points differ a lot (more than about 25…

Real mixtures usually need several methods in a sensible order. Plan by listing the components and their properties, then remove one component at a time.

A force is a push or a pull. It can start or stop motion, change speed, change direction, or change shape. Its SI unit is the newton (N), and it is a vector — direction…

Newton's first law: an object stays at rest, or keeps moving with uniform velocity in a straight line, unless an unbalanced external force acts on it.

Momentum p = mv (mass × velocity) measures the 'quantity of motion'. Its unit is kg m/s and it is a vector. A slow truck and a fast bullet can both have large momentum.

Newton's third law: whenever one object exerts a force on another, the second exerts an equal and opposite force on the first. Forces always come in action–reaction…

When two objects interact (collide, push apart) with no external force on them, the forces they exert on each other are equal and opposite and act for the same time, so…

Work is done when a force makes an object move (be displaced) in the direction of the force. For a constant force F along the displacement s: W = F × s. The SI unit is…

Power is the rate of doing work (or of transferring energy): P = W/t. Its SI unit is the watt (W): 1 W = 1 J/s. Larger units: 1 kW = 1000 W.

Energy is the capacity to do work or to cause change. It is measured in joules, like work.

Law of conservation of energy: energy can neither be created nor destroyed; it can only be transformed from one form to another or transferred between objects. The total…

A simple machine changes the size or direction of a force to make work easier. The force we apply is the effort; the force overcome is the load. Mechanical advantage MA…

Efficiency compares useful work (or energy) out with work (or energy) put in: η = (useful output)/(input) × 100%. For a machine, η = MA/VR × 100%.

Atoms contain three main particles. The electron (e^-), discovered by J.J. Thomson in 1897 using cathode rays, has a negative charge (−1) and a tiny mass — about 1/1836…

Once the electron was known, Thomson proposed (1904) that an atom is a sphere of positive charge with electrons embedded in it, like seeds in a watermelon (often called…

In 1911 Ernest Rutherford's team fired fast, positively charged alpha (α) particles at a very thin gold foil. Observations: (1) most α-particles passed straight through…

Niels Bohr (1913) proposed that electrons move only in certain allowed orbits, called shells or energy levels, labelled K, L, M, N (n = 1, 2, 3, 4) from the nucleus…

Atomic number (Z) = number of protons. It identifies the element: every carbon atom has 6 protons. In a neutral atom it also equals the number of electrons. Mass number…

Law of conservation of mass (Lavoisier): in a chemical reaction, mass is neither created nor destroyed — the total mass of the products equals the total mass of the…

An atom is the smallest particle of an element that takes part in a chemical reaction. Atoms are extremely small — a hydrogen atom has a radius of about 10^-10 m. Each…

An ion is a charged particle formed when an atom (or group of atoms) loses or gains electrons. Losing electrons gives a positive ion, a cation (Na⁺, Mg²⁺, mostly…

Valency is the combining capacity of an atom or ion, equal to the size of its charge for ions: Na⁺ 1, Ca²⁺ 2, Al³⁺ 3, Cl⁻ 1, O²⁻ 2, N³⁻ 3; NH₄⁺ 1, OH⁻ 1, NO₃⁻ 1, SO₄²⁻…

Atoms are too light to weigh in grams conveniently, so we use relative atomic mass on a scale where one atom of carbon-12 has a mass of exactly 12 atomic mass units (u)…

Sound is produced by vibrating objects: a guitar string, the skin of a tabla, your vocal cords. Touch a ringing bell gently and you can feel it vibrate.

The distance between two consecutive compressions (or rarefactions) is the wavelength λ (metre). The time for one complete oscillation is the time period T (second). The…

Pitch is how high or low a sound seems, and it depends mainly on frequency: a whistle or a bird's song has high frequency; a lion's roar or a drum has low frequency. A…

Sound reflects from hard surfaces like walls, cliffs and buildings, obeying the same laws as light: the angle of incidence equals the angle of reflection.

Humans hear frequencies from about 20 Hz to 20,000 Hz (20 kHz); the upper limit falls with age. Below 20 Hz is infrasound — produced by earthquakes, whales and…

The outer ear (pinna) collects sound and funnels it down the ear canal to the eardrum, a thin membrane. Compressions push the eardrum in; rarefactions let it move out —…

Reproduction is the production of new individuals of the same kind. It is not needed for an individual's survival, but it is essential for the continuity of the species.

Asexual reproduction involves a single parent and no gametes; offspring are almost identical to the parent (clones). It is fast and needs no partner.

The flower is the reproductive part of a flowering plant. Its parts, from outside in: sepals (protect the bud), petals (attract pollinators), stamens — the male part…

After landing on a suitable stigma, a pollen grain grows a pollen tube down through the style to an ovule. The male gamete travels down the tube and fuses with the egg…

Humans reproduce sexually. At puberty (about 10–14 years in girls, 12–16 in boys) hormones bring changes: growth spurts, body hair, voice changes in boys, breast…

Classification groups organisms by shared characteristics. It makes the huge diversity of life manageable, shows how organisms are related, and gives everyone a common…

Groups are nested from broad to narrow: Kingdom → Phylum (Division in plants) → Class → Order → Family → Genus → Species. The further down you go, the more features…

Robert Whittaker (1969) proposed five kingdoms. Monera: prokaryotic, unicellular; some autotrophic, some heterotrophic (bacteria, cyanobacteria). Protista: eukaryotic…

Plants are divided by three key questions: Is the body differentiated into root, stem and leaves? Does it have vascular tissue (xylem and phloem)? Does it produce seeds…

Animals are grouped by level of organisation (cells, tissues, organs), symmetry (none, radial, bilateral), body cavity (coelom), segmentation and the presence of a…

Organisms that share more fundamental features are usually more closely related — they descended from a more recent common ancestor. Modern classification tries to…

Scientists divide the Earth system into spheres: the atmosphere (the layer of air), the hydrosphere (oceans, rivers, lakes, groundwater), the geosphere or lithosphere…

Sunlight passes through the atmosphere and heats the ground and oceans. The warm surface then gives off energy as infrared radiation (heat). The ozone layer in the upper…

When air is heated it expands, becomes less dense and rises, leaving a region of low pressure near the ground. Cooler, denser air sinks and forms high pressure. Wind is…

Solar energy drives the water cycle. Water evaporates from oceans, lakes and soil, and plants release water vapour through transpiration. The vapour rises, cools and…

Carbon cycle: plants take in CO₂ from the air in photosynthesis and build it into sugars. Animals eat plants; respiration by living things, decomposition of dead matter…

Since the Industrial Revolution, burning coal, oil and gas has raised the CO₂ in the air from about 280 parts per million to over 420 ppm. Together with methane from…