Class 9 — Notes 📘

A number that cannot be expressed in the form p/q where p and q are integers. Its decimal expansion is non-terminating and non-repeating. Examples: √2 = 1.41421..., √3 = 1.73205..., π = 3.14159...

Multiply numerator and denominator by √3: (1×√3)/(√3×√3) = √3/3

√4 = 2, which can be written as 2/1. So √4 is a rational number.

If a polynomial p(x) is divided by (x − a), the remainder is p(a). Example: if p(x) = x² + 3x + 2 is divided by (x − 1), remainder = p(1) = 1 + 3 + 2 = 6

Using (a+b)² = a²+2ab+b²: (2x)² + 2(2x)(3) + (3)² = 4x² + 12x + 9

An axiom is a general statement accepted as true in all branches of mathematics. A postulate is a statement specific to geometry accepted without proof. Both are assumed to be true.

Only ONE unique straight line can pass through two given distinct points. This is Euclid's first postulate.

Distance is the length of the actual path travelled by an object, regardless of direction (scalar). Displacement is the minimum straight-line distance from starting point to finishing point, along with direction (vector). Example: if you walk in a circle and return to start, distance > 0 but displacement = 0.

1st Law: An object remains at rest or in uniform motion unless an external unbalanced force acts. 2nd Law: F = ma; rate of change of momentum is proportional to applied force. 3rd Law: Every action has an equal and opposite reaction acting simultaneously on different objects.

a = (v−u)/t = (20−0)/5 = 4 m/s². s = ut + ½at² = 0 + ½×4×25 = 50 m

By Newton's 3rd law, the bullet exerts an equal and opposite force on the gun. Before firing, total momentum = 0. After firing, bullet moves forward, so gun moves backward to conserve total momentum (Law of Conservation of Momentum).

Energy can neither be created nor destroyed, only converted from one form to another. Total energy remains constant. Example: In free fall — at top, PE = mgh, KE = 0. At bottom, PE = 0, KE = mgh. At any point in between, PE + KE = mgh = constant.

KE = ½mv² = ½ × 2 × 25 = 25 J

1 kWh is the energy consumed when a device of power 1 kW works for 1 hour. 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J. It is used as the commercial unit of electricity (1 unit).

Work done = 0 when: (1) there is no displacement, or (2) force is perpendicular to displacement (θ = 90°). Example: A satellite moving in circular orbit — gravitational force is perpendicular to displacement, so work done by gravity = 0.

If the physical state of a conductor remains constant, the current (I) flowing through it is directly proportional to the potential difference (V) between its ends. V = IR, where R is resistance. Unit of resistance is Ohm (Ω).

1/Rp = 1/6 + 1/3 + 1/2 = 1/6 + 2/6 + 3/6 = 6/6 = 1. So Rp = 1 Ω. (Less than smallest individual resistance of 2Ω ✓)

In parallel: (1) Each appliance gets the same full voltage (220V). (2) If one appliance fails, others keep working. (3) Each appliance can be switched on/off independently. (4) Total current can be distributed as needed.

I = V/R = 230/1000 = 0.23 A

Proposed by Lavoisier (1785): During a chemical reaction, there is no rise or drop in the total mass. Total mass of reactants = total mass of products. Example: CaO (56g) + H₂O (18g) → Ca(OH)₂ (74g). Mass is conserved.

A mole is the quantity of a substance whose mass in grams equals its molecular mass in Daltons. It contains 6.022 × 10²³ molecules (Avogadro's number). Example: Molecular mass of CO₂ = 44u, so 1 mole CO₂ = 44 g and contains 6.022 × 10²³ molecules.

Molecular mass of CO₂ = 12 + 16×2 = 44 u. n = 66/44 = 1.5 mol. Number of molecules = 1.5 × 6.022 × 10²³ = 9.033 × 10²³ molecules.

The proportion by mass of constituent elements in different samples of the same compound is always fixed, regardless of the source. Example: In water (H₂O), H:O is always 1:8 by mass. So 9g water always has 1g H and 8g O.

Basic radical: Na⁺ (valency 1). Acidic radical: SO₄²⁻ (valency 2). Cross multiply: Na₂SO₄. Formula = Na₂SO₄

Sublimation is the direct change of a solid into gas without passing through the liquid state on heating. Examples: Dry ice (solid CO₂), Camphor, Ammonium chloride, Iodine.

During evaporation, particles with higher kinetic energy escape from the surface. They absorb heat from the remaining liquid, causing its temperature to fall. That is why sweating cools our body.

At 0°C, ice starts converting to water. During this process, temperature remains constant at 0°C even though heat is being absorbed (latent heat).

When a beam of light passes through a colloid, the particles scatter the light making the path of light visible. This is the Tyndall effect. Example: sunlight through fog, torch beam in smoky room.

Solution is a homogeneous mixture where particles are less than 1 nm and do not settle. Suspension is a heterogeneous mixture where particles are more than 100 nm and settle on standing. Example: salt water (solution), muddy water (suspension).

Energy can neither be created nor destroyed. It can only be transformed from one form to another. The total energy of an isolated system always remains constant. Example: in a pendulum, KE and PE keep converting into each other.

PE = mgh = 5 × 10 × 10 = 500 J

Present Perfect (has/have + past participle) is used for: 1) Actions completed recently with effect in present — "I have finished my homework." 2) Life experiences — "I have visited Delhi." 3) Actions with since/for — "She has lived here for 5 years."

Simple Past: action completed in past — "I ate lunch." Past Perfect: action completed before another past action — "I had eaten lunch before he came." Past Perfect is used for the earlier of two past actions.

Ohm's Law: If the physical state (length, area, temperature, material) of a conductor remains constant, the current (I) flowing through it is directly proportional to the potential difference (V) between its two ends. I ∝ V → V = IR → R = V/I

1 Ohm: The resistance of a conductor is 1 Ohm if 1 Ampere current flows through it when 1 Volt potential difference is applied between its ends. 1 Ω = 1V/1A

Series: Rs = R₁+R₂+R₃ | Same current through each | Voltage divides | Rs > each R | If one fails, circuit breaks | Used to increase resistance
Parallel: 1/Rp = 1/R₁+1/R₂+1/R₃ | Same voltage across each | Current divides | Rp < smallest R | If one fails, others work | Used to decrease resistance | Domestic appliances connected this way

L = 1000 m, r = 0.25×10⁻³ m, A = πr² = 3.14 × (0.25×10⁻³)² = 0.196×10⁻⁶ m²
R = ρL/A = (1.7×10⁻⁸ × 1000) / (0.196×10⁻⁶) = ≈ 87 Ω

1/Rp = 1/15 + 1/20 + 1/10 = 4/60 + 3/60 + 6/60 = 13/60
Rp = 60/13 = 4.615 Ω (less than smallest, 10Ω ✓)

(1) Same potential difference (220V) across each appliance. (2) Each appliance operates independently — if one fails, others continue to work. (3) Each appliance can be switched ON/OFF independently. (4) Each appliance gets same voltage required for its operation.

Fuse wire is made of an alloy with a specific low melting point. It is connected in series with the electric appliance. If for any reason the current increases excessively, the fuse wire heats up and melts, breaking the circuit and stopping current flow — thus protecting the appliance from damage. Domestic fuse ratings: 1A, 2A, 3A, 4A, 5A, 10A.

Law of Conservation of Matter (Lavoisier, 1785): During a chemical reaction, there is no rise or drop in the total mass. Total mass of reactants = Total mass of products. Ex: CaO (56g) + H₂O (18g) → Ca(OH)₂ (74g).

Law of Constant Proportion (Proust, 1794): The proportion by mass of constituent elements in different samples of the same compound is always fixed. Ex: In water (H₂O) from any source, H:O is always 1:8 by mass.

Mole: A mole is that quantity of a substance whose mass in grams equals the molecular mass of that substance in Daltons. Ex: Molecular mass of O₂ = 32u → 1 mole O₂ = 32g. Molecular mass of H₂O = 18u → 1 mole H₂O = 18g. n = mass in grams / molecular mass.

Avogadro's number (Nₐ) = 6.022×10²³. This is the number of molecules in one mole of any substance. Just like 1 dozen = 12, 1 mole = 6.022×10²³ molecules.

Molecular mass of H₂SO₄ = 2(1) + 32 + 4(16) = 2 + 32 + 64 = 98u
n = 49/98 = 0.5 mol
Number of molecules = 0.5 × 6.022×10²³ = 3.011×10²³ molecules

Sodium sulphate: Na⁺(1) + SO₄²⁻(2) → cross multiply → Na₂SO₄
Calcium carbonate: Ca²⁺(2) + CO₃²⁻(2) → CaCO₃
Aluminium hydroxide: Al³⁺(3) + OH⁻(1) → cross multiply → Al(OH)₃

Molecular mass of CO₂ = 12 + 2(16) = 44u
n = 66/44 = 1.5 mol
Molecules = 1.5 × 6.022×10²³ = 9.033×10²³ molecules

Valency = number of electrons an atom gives away or takes up while forming an ionic bond.
Na (2,8,1) → gives away 1 electron → Na⁺ (2,8) → valency = 1
Cl (2,8,7) → takes 1 electron → Cl⁻ (2,8,8) → valency = 1
Na⁺ + Cl⁻ → NaCl. Both ions get complete octet (8 electrons in outermost orbit). The force of attraction between opposite charges forms the ionic bond.