Rational Numbers (Q): numbers of form p/q where q β 0
Irrational Numbers: numbers that cannot be written as p/q. Example: β2, β3, Ο
Real Numbers (R): all rational + irrational numbers
Every rational number has either a terminating or repeating decimal
Irrational numbers have non-terminating, non-repeating decimals
Rationalization: converting irrational denominator to rational by multiplying with conjugate
β Digest Answers
Q. What is an irrational number? Give examples.
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: β2 = 1.41421..., β3 = 1.73205..., Ο = 3.14159...
Q. Rationalize: 1/β3
Multiply numerator and denominator by β3: (1Γβ3)/(β3Γβ3) = β3/3
Q. Is β4 rational or irrational?
β4 = 2, which can be written as 2/1. So β4 is a rational number.
π
Mathematics
Chapter 2 β Polynomials
π Key Points
Polynomial: algebraic expression with non-negative integer exponents
Degree: highest power of variable in polynomial
Monomial: one term. Binomial: two terms. Trinomial: three terms
Zero of polynomial: value of x where p(x) = 0
Factor Theorem: (xβa) is a factor of p(x) if p(a) = 0
Remainder Theorem: when p(x) divided by (xβa), remainder = p(a)
If a polynomial p(x) is divided by (x β a), the remainder is p(a). Example: if p(x) = xΒ² + 3x + 2 is divided by (x β 1), remainder = p(1) = 1 + 3 + 2 = 6
Line: extends in both directions infinitely, no width
Line segment: part of line with two endpoints
Ray: part of line with one endpoint, extends in one direction
Euclid's Postulate 1: A straight line can be drawn from any point to any other point
Euclid's Postulate 2: A terminated line can be produced indefinitely
Euclid's Postulate 3: A circle can be drawn with any centre and radius
Things equal to the same thing are equal to each other (Axiom)
β Digest Answers
Q. What is the difference between an axiom and a postulate?
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.
Q. How many lines can pass through two given points?
Only ONE unique straight line can pass through two given distinct points. This is Euclid's first postulate.
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Science
Chapter 1 β Matter in Our Surroundings
π Key Points
Matter: anything that has mass and occupies space
Three states of matter: Solid, Liquid, Gas
Solid: fixed shape and volume, particles closely packed
Liquid: fixed volume but no fixed shape, particles loosely packed
Gas: no fixed shape or volume, particles far apart
Melting point: temperature at which solid converts to liquid
Boiling point: temperature at which liquid converts to gas
Sublimation: direct conversion from solid to gas without liquid stage. Example: dry ice, camphor
Evaporation: conversion of liquid to gas at any temperature below boiling point
Latent heat of fusion: heat required to convert 1 kg of solid to liquid at melting point
β Digest Answers
Q. What is sublimation? Give two examples.
Sublimation is the direct change of a solid into gas without passing through the liquid state on heating. Examples: Dry ice (solid COβ), Camphor, Ammonium chloride, Iodine.
Q. Why does evaporation cause cooling?
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.
Q. Melting point of ice is 0Β°C β what does this mean?
At 0Β°C, ice starts converting to water. During this process, temperature remains constant at 0Β°C even though heat is being absorbed (latent heat).
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Science
Chapter 2 β Is Matter Around Us Pure?
π Key Points
Pure substance: made of single type of particles. Example: gold, water, salt
Mixture: two or more substances mixed but not chemically combined
Homogeneous mixture: uniform composition throughout. Example: salt water, air
Solution: homogeneous mixture of solute and solvent
Solute: substance that dissolves. Solvent: substance that dissolves solute
Colloid: mixture where particle size is between solution and suspension. Example: milk, fog
Suspension: heterogeneous mixture where particles settle down. Example: muddy water
Tyndall effect: scattering of light by colloid particles
β Digest Answers
Q. What is the Tyndall effect?
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.
Q. Difference between solution and suspension?
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).
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Science
Chapter 11 β Work and Energy
π Key Points
Work = Force Γ Displacement Γ cos ΞΈ. Unit: Joule (J)
Work is done only when force causes displacement in its direction
Positive work: force and displacement in same direction
Negative work: force and displacement in opposite directions
Kinetic Energy = Β½mvΒ². Unit: Joule
Potential Energy = mgh (gravitational). Unit: Joule
Law of Conservation of Energy: energy can neither be created nor destroyed, only converted from one form to another
Power = Work / Time. Unit: Watt (W). 1 W = 1 J/s
1 kilowatt-hour (kWh) = 3.6 Γ 10βΆ J (commercial unit of energy)
β Digest Answers
Q. State the law of conservation of energy.
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.
Q. A 5 kg ball falls from 10 m. Find its PE. (g = 10 m/sΒ²)
PE = mgh = 5 Γ 10 Γ 10 = 500 J
βοΈ
English Grammar
Tenses β Quick Reference
π Key Points
Simple Present: I eat / He eats β habitual action
Present Continuous: I am eating β action happening now
Present Perfect: I have eaten β completed with present effect
Simple Past: I ate β completed past action
Past Continuous: I was eating β ongoing past action
Past Perfect: I had eaten β completed before another past action
Simple Future: I will eat β future action
Future Perfect: I will have eaten β completed before a future time
β Digest Answers
Q. When do we use Present Perfect tense?
Present Perfect (has/have + past participle) is used for: 1) Actions completed recently with effect in present β "I have finished my homework." 2) Life experiences β "I have visited Delhi." 3) Actions with since/for β "She has lived here for 5 years."
Q. Difference between Simple Past and Past Perfect?
Simple Past: action completed in past β "I ate lunch." Past Perfect: action completed before another past action β "I had eaten lunch before he came." Past Perfect is used for the earlier of two past actions.
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