DOST Study Guide
Simple Notes, Formulas & Sample Questions
Notes to Remember
1. Memorize the DOST Primer (~10% Appears on the Exam)
Around 10% of the actual exam comes directly from the DOST Primer (same question style, just different numbers). Study the Primer and memorize how every item is solved.
2. No Calculator Allowed (Heavy Calculation = Trick Question)
Calculators are strictly banned. If a question looks like it needs long, messy calculation, it is a trick question! Look for a shortcut:
- Cancel numbers out before multiplying.
- Check the units (for gas laws, always add $273$ to convert $^\circ\text{C}$ to Kelvin).
- Test Choice B or C directly in the problem instead of doing long algebra.
3. Manage Your Time (45 Seconds Per Question)
Each part of the exam is timed separately. Once time is up for Part 1, you cannot go back.
- Answer all easy questions first.
- If a question takes more than 45 seconds, skip it and come back later.
4. Never Leave Blanks (No Right-Minus-Wrong)
Wrong answers do not deduct points. In the last 2 minutes of each part, pick one letter (like B or C) and shade all remaining blanks.
⏱️ Tips from Previous Test-Takers
Part I: Creative Reasoning
~45 items
35–40 mins
Part II: Language
~35 items
25–30 mins
Part III: Science
~60 items
50–60 mins
Part IV: Math
~40 items
45–50 mins
Mechanical Reasoning
📖 Definition
Tests how simple machines, gears, pulleys, levers, springs, pendulums, fluids, and circuits work.
Seesaw / Lever Balance
$\text{Weight}_1 \times \text{Distance}_1 = \text{Weight}_2 \times \text{Distance}_2$
Heavier objects must sit closer to the center (fulcrum) to balance.
3 Classes of Levers (FLE 1-2-3 Shortcut)
Tell which part is in the middle:
Class 1 (Fulcrum in middle): Seesaw, scissors, pliers, crowbar.
Class 2 (Load in middle): Wheelbarrow, nutcracker, bottle opener (always multiplies force).
Class 3 (Effort in middle): Tweezers, fishing rod, human arm/bicep, broom (multiplies speed).
Pulleys
$\text{Effort } F = \frac{\text{Load Weight}}{n}$ (where $n =$ number of ropes holding the movable pulley).
A single fixed pulley only changes direction ($F = \text{Weight}$).
Inclined Plane (Ramp)
$\text{Effort } F = \text{Weight} \times \frac{\text{Height } (h)}{\text{Ramp Length } (L)}$
A longer, gentler ramp needs less force, but total work ($W = F \times d$) stays the same.
Wheel & Axle / Screw Jack
Wheel & Axle: $\text{MA} = \frac{\text{Radius of Wheel } (R)}{\text{Radius of Axle } (r)}$ (Bigger wheel = less force needed).
Screw Jack: $\text{MA} = \frac{2\pi L}{p}$ ($L =$ handle length, $p =$ thread pitch).
Gears & Belts
$\text{Teeth}_1 \times \text{Speed}_1 = \text{Teeth}_2 \times \text{Speed}_2$
Smallest gear spins the fastest.
Touching gears spin in opposite directions (Odd gears $1, 3, 5$ spin the same way; Even gears $2, 4$ spin the opposite way).
Open belt: pulleys spin the same way. Crossed belt: pulleys spin opposite ways.
Bicycle Chain & Sprockets: Chain connects front chainring (pedals) to rear wheel cog $\to$ both rotate in the same direction (unlike meshed gears).
High Gear (Cruising on flat ground): Large front chainring $+$ Small rear cog $\to$ multiplies speed (1 pedal turn spins rear wheel multiple times).
Low Gear (Uphill climbing): Small front chainring $+$ Large rear cog $\to$ multiplies torque (easiest to pedal up steep slopes).
Springs (Hooke's Law)
$F = k \cdot x$ (Force $=$ stiffness $k \times$ stretch $x$).
Parallel springs (side-by-side): $k_{\text{eq}} = k_1 + k_2$ (stiffer, stretches half as much).
Series springs (end-to-end): $k_{\text{eq}} = \frac{k_1 k_2}{k_1 + k_2}$ (stretches twice as much for identical springs).
Pendulum Swing
$T = 2\pi\sqrt{\frac{L}{g}}$
Swing time depends only on string length $L$ (and gravity). Changing the weight/mass of the bob does not change the swing time! Longer string = slower swing.
Hydraulic Press (Pascal's Law)
$\frac{F_1}{A_1} = \frac{F_2}{A_2}$
Pressure is equal everywhere in a closed liquid. A small force on a small piston lifts a heavy load on a big piston.
Water in a Moving Bus
Bus speeds up forward $\rightarrow$ water tilts backward (higher at the back).
Bus brakes / slows down $\rightarrow$ water tilts forward (higher in front).
Constant speed $\rightarrow$ water stays flat.
Bulb Circuits
Series: 1 path. If one bulb burns out, all bulbs go out.
Parallel: Separate branches. If one parallel bulb burns out, total resistance goes up $\rightarrow$ main line bulbs get dimmer, while the remaining parallel bulb gets brighter.
1. You are holding a glass of soda on a bus that is speeding up forward (to the right). What happens to the liquid surface?
Answer: Tilted upward toward the back of the bus — Because of inertia, the liquid stays behind as the bus moves forward, making it higher at the back and lower in front.
2. Bulbs W and X are in the main line, while Bulbs Y and Z are in parallel. If Bulb Y burns out, what happens to the brightness of Bulb Z, and Bulbs W and X?
Answer: Bulb Z gets brighter; Bulbs W and X get darker — When Y breaks, total resistance increases, so less current flows through W and X (darker), while Z gets a bigger share of voltage (brighter).
3. Three kites fly in a horizontal wind: Kite I is flat ($0^\circ$), Kite II is tilted at an angle, and Kite III is straight up ($90^\circ$). Which kite gets the most upward lift?
Answer: Kite II (tilted at an angle) — The angled surface pushes oncoming wind downward, which pushes the kite upward.
Spatial & Pattern Reasoning
📖 Definition
Tests how you fold flat nets into 3D cubes, count painted or hidden blocks, unfold punched paper, and spot visual patterns.
Folded Cube Net (Skip-One Rule)
On a flat cube net, faces separated by 1 square in a straight line ($1 \leftrightarrow 3$, $2 \leftrightarrow 4$, $5 \leftrightarrow 6$) are opposite faces.
Opposite faces can never touch or appear together on the same 3D view.
Painted Cube Cutting ($n \times n \times n$ Small Cubes)
When a big cube painted on the outside is cut into $n \times n \times n$ small cubes:
Total small cubes: $n^3$
3 faces painted (corners): Always $8$
2 faces painted (edges): $12(n - 2)$
1 face painted (face centers): $6(n - 2)^2$
0 faces painted (inside core): $(n - 2)^3$
2D Rotations
$90^\circ$ = quarter turn
$180^\circ$ = upside down (half turn)
$270^\circ$ clockwise = $90^\circ$ counterclockwise
$360^\circ$ = back to original position.
Mirror vs. Water Image
Side Mirror (Vertical): Flips Left $\leftrightarrow$ Right (top & bottom stay the same).
Water Image (Horizontal): Flips Top $\leftrightarrow$ Bottom (left & right stay the same).
Paper Folding & Hole Punching
Unfold the paper backward from the last fold to the first fold. Each fold line acts like a mirror that doubles the holes across the crease.
Spray-Painted Blocks on a Table
Count Top $+$ 4 Sides (do NOT count the bottom touching the table!):
$1 \times 4 \times 1$ flat row on table $\to$ $14$ painted faces
$2 \times 2 \times 2$ cube on table $\to$ $20$ painted faces
$1 \times 1 \times 5$ vertical tower on table $\to$ $21$ painted faces
$3 \times 3 \times 1$ flat layer on table $\to$ $21$ painted faces
3D Cross-Sections (Slicing Solids)
Cylinder: Sliced parallel to base $=$ Circle; sliced perpendicular to base $=$ Rectangle.
Square Pyramid: Sliced parallel to base $=$ Square.
Cube: Sliced diagonally through 3 corner vertices $=$ Equilateral Triangle.
Compass Directions & Tangrams
Walking $N\text{ km}$ North, then $E\text{ km}$ East, then $N\text{ km}$ South leaves you Directly East of start.
Two right isosceles triangles joined at their hypotenuse form 1 Square.
Counting 3D Stacked Blocks
Count column by column from top to bottom. If a block is on the 3rd floor, there are 2 hidden blocks supporting it underneath!
Shape Sequences
Track one element at a time:
1. Count sides or lines ($3, 4, 5, 6\dots$).
2. Track rotation ($45^\circ$ or $90^\circ$ per step).
3. Check shading (alternating black/white) or inside-outside swaps.
1. What figure comes next in the sequence below?
Answer: 7-sided shape (heptagon) with the blue dot at the top — The outer shape adds 1 side each step ($3 \to 4 \to 5 \to 6 \to 7$), and the blue dot turns $90^\circ$ clockwise each step (top $\to$ right $\to$ bottom $\to$ left $\to$ top).
2. A flat cube net has 4 squares in a vertical column labeled 1, 2, 3, 4 from top to bottom, with square 5 on the left and square 6 on the right. Which pairs of faces are opposite each other when folded?
Answer: (1 & 3), (2 & 4), and (5 & 6) — Using the Skip-One Rule, any two faces in this list are on opposite sides of the cube and can never touch.
Verbal Reasoning, Vocabulary & Grammar
📖 Definition
Tests word pair analogies, synonyms (same meaning), antonyms (opposite meaning), root words, and grammar rules.
Common Science Word Roots
Geo = Earth
Ferro = Iron ($\text{Fe}$)
Hydro = Water
Pyro = Fire
Photo = Light
Thermo = Heat
Chrono = Time
Morph = Shape
Cardio = Heart
Nephro = Kidney
Hepato = Liver
Pulmo = Lung
Neuro = Nerve
Osteo = Bone
Derm = Skin
Hema = Blood
Neither...nor / Either...or
The verb matches the closer subject:
"Neither the students nor the teacher was present."
"Neither the teacher nor the students were present."
Along with / Together with / As well as
Ignore the phrase inside commas! The verb matches the first subject:
"The principal, along with the teachers, is attending."
Everyone / Each / Somebody / Neither
Always singular! Takes a singular verb and singular pronoun:
"Everyone needs to bring his or her ID."
The number vs. A number & Collective Nouns
"The number of applicants is growing." (Singular)
"A number of applicants are waiting." (Plural = many)
Collective nouns & Book titles: "The student body/committee is" (singular as one unit); book titles like "The Chronicles of Narnia is" are always singular.
If-Clauses (Conditionals) & Correlatives
Unreal / Wish (Subjunctive): Always use were, never was ("If I were taller, I would join").
Past Regret: Had + past participle $\to$ would have + past participle ("Had I known, I would have left earlier").
Correlative Pair: Always pair No sooner ... than (never when) and protest against.
Dangling / Misplaced Modifiers
Put the action phrase right next to the person doing the action:
Wrong: "Walking down the street, the trees looked tall."
Right: "Walking down the street, Maria saw the tall trees."
Parallel Structure & Polite Replies
Words in a list must follow the same pattern: "carefully, accurately, and quickly".
"Do you mind closing the window?" $\to$ Polite agreement is "Not at all" (meaning "I do not mind").
Common Confused Words
some time (a period of time): "I need some time to finish the exam."
sometime (at an unspecified time): "Let's meet sometime next week."
sometimes (occasionally): "She sometimes studies at night."
affect (verb = to influence): "Lack of sleep will affect your score."
effect (noun = result): "The medicine had a fast effect."
fewer (countable items): "There are fewer mistakes on this paper."
less (uncountable amount): "Drink less coffee and more water."
between (used for 2 items): "Divide the prize between Ana and Ben."
among (used for 3 or more items): "Divide the prize among the five winners."
Figures of Speech & Reading Rules
Personification: Giving human actions to non-human things ("The wind whispered").
Metonymy: Replacing a name with a closely linked object ("The pen [writing] is mightier than the sword [force]").
Antithesis: Contrasting ideas in parallel form ("It was the best of times, it was the worst of times").
Understatement: Making something big sound small ("It's just a scratch" for a broken leg).
Reading Passages: Paraphrasing = restating in your own words; including a counterargument makes an author more credible and objective.
| Analogy Pair | Relationship |
|---|---|
| LIFEBELT : SEA :: PARACHUTE : AIR | Rescue gear : Where it is used |
| GEO : EARTH :: FERRO : IRON | Root word : Meaning ($\text{Fe} =$ Iron) |
| NEWTON : CALCULUS :: COPERNICUS : ASTRONOMY | Scientist : Field of study |
| ISAAC NEWTON : LAW OF GRAVITY :: ALEXANDER FLEMING : PENICILLIN | Scientist : Major discovery |
| GALILEO GALILEI : TELESCOPE :: BENJAMIN FRANKLIN : LIGHTNING ROD | Scientist : Famous invention |
| HAMMER : CARPENTER :: BRUSH : PAINTER | Primary tool : Worker |
| DOCTOR : STETHOSCOPE :: CARPENTER : HAMMER | Worker : Primary tool |
| SCALPEL : SURGEON :: STETHOSCOPE : PHYSICIAN | Tool : Medical specialist |
| CARDIOLOGIST : HEART :: NEPHROLOGIST : KIDNEY | Specialist : Organ (nephro- = kidney) |
| ARCHITECT : BLUEPRINT :: COMPOSER : SCORE | Creator : Design plan |
| HYPOTHESIS : EXPERIMENT :: BLUEPRINT : CONSTRUCTION | Plan : Execution |
| APHASIA : SPEECH :: AMNESIA : MEMORY | Condition : What is lost |
| FAMINE : FOOD :: DROUGHT : WATER | Severe shortage : Missing resource |
| INHALE : EXHALE :: SYSTOLE : DIASTOLE | Opposite body phases (heart contraction vs. relaxation) |
| BENIGN : MALIGNANT :: ACUTE : CHRONIC | Medical opposites (harmless vs. harmful; sudden vs. long-lasting) |
| ANTERIOR : POSTERIOR :: FRONT : HIND | Anatomical opposites (front vs. back) |
| AFFECTION : INFATUATION :: TIMIDITY : COWARDICE | Normal trait : Negative extreme |
| PARSIMONIOUS : FRUGALITY :: INTREPID : VALOR | Extreme trait : Noun (thrift vs. bravery) |
| OBSEQUIOUS : FLATTERY :: MENDACIOUS : DECEPTION | Trait : Action (excessive flattery vs. lying) |
| EPHEMERAL : PERMANENT :: CAPRICIOUS : PREDICTABLE | Opposites (short-lived vs. lasting; fickle vs. steady) |
| ARCHAIC : MODERN :: STAGNANT : FLOWING | Opposites (old vs. new; still vs. moving) |
| BIG : ENORMOUS :: SMALL : MINUSCULE | Normal size : Extreme size |
| HIDE : CONCEAL :: STOP : HALT | Synonyms (same meaning) |
| HOT : COLD :: TALL : SHORT | Opposites |
| TREE : FOREST :: SOLDIER : ARMY | Single member : Group |
| FISH : SCHOOL :: WOLF : PACK | Animal : Group name |
| WHALE : POD :: KITTEN : LITTER | Animal : Group name |
| BOOK : LIBRARY :: PAINTING : MUSEUM | Object : Place where it is kept |
| BAT : BALL :: BOW : ARROW | Launcher : Projectile |
| BOWED STRINGS : VIOLIN :: WOODWIND : FLUTE | Instrument family : Example |
| SPONGE : POROUS :: SKUNK : SMELLY | Object/animal : Defining trait |
| DO : DID :: COST : COST | Present tense : Past tense |
| Word | Synonym / Simple Meaning |
|---|---|
| Abolish | Eliminate (formally end) |
| Antiquated | Ancient (very old-fashioned) |
| Arduous | Difficult (hard and tiring) |
| Bamboozled | Duped (cheated or fooled) |
| Benevolent | Kind and charitable (generous) |
| Beset | Surrounded (hemmed in on all sides) |
| Big / Large | Enormous / Gigantic (very big) |
| Brief | Short |
| Coherent | Logical and consistent (clear and connected) |
| Consolidated | United (combined into one) |
| Diligent | Hardworking and careful |
| Evanescent | Fleeting (lasting a very short time) |
| Facilitate | Assist (make easier) |
| Frank | Candid (honest and direct) |
| Hide | Conceal |
| Interim | Interval (temporary period between events) |
| Intrepid | Fearless / Brave (showing valor) |
| Lenient | Forbearing (patient and merciful) |
| Mendacious | Deceitful / Dishonest (given to lying) |
| Meticulous | Thorough (very careful with small details) |
| Obsequious | Servile / Flattering (overly eager to please) |
| Obsolete | Outdated (no longer used) |
| Parsimonious | Frugal / Stingy (extremely thrifty) |
| Persistent | Continuing despite difficulty (never giving up) |
| Regressed | Deteriorated (returned to a worse state) |
| Rescind | Repeal (cancel a law or rule) |
| Scrupulous | Careful (honest and thorough) |
| Slim | Svelte (slender and graceful) |
| Small | Minuscule (extremely tiny) |
| Stop | Halt |
| Talkative | Loquacious (very chatty) |
| Transient | Transitory (temporary / passing quickly) |
| Ubiquitous | Everywhere (present everywhere at once) |
| Versatile | Able to adapt to many uses |
| Visible | Conspicuous (easily seen / noticeable) |
| Word | Antonym (Opposite) |
|---|---|
| Abolish (end) | Establish / Continue (start / keep) |
| Acute (sudden / short-term) | Chronic (long-lasting) |
| Anterior (front) | Posterior (back / hind) |
| Archaic (ancient / old) | Modern (new) |
| Belligerent (hostile / aggressive) | Peaceful |
| Benevolent (kind) | Cruel |
| Benign (harmless) | Malignant (harmful / cancerous) |
| Callow (immature / inexperienced) | Wise (having wisdom) |
| Capricious (fickle / changeable) | Predictable (steady) |
| Consolidated (united) | Divided |
| Ephemeral (short-lived) | Permanent (lasting) |
| Facilitate (help / make easy) | Hinder / Complicate (block) |
| Fastidious (very careful / fussy) | Careless |
| Front | Hind (back) |
| Hot | Cold |
| Inhale (breathe in) | Exhale (breathe out) |
| Optimal (best) | Worst |
| Prodigal (wasteful / extravagant) | Parsimonious / Frugal (saving / thrifty) |
| Pugnacious (eager to fight) | Conciliatory (peacemaking) |
| Regressed (got worse) | Progressed (improved) |
| Rigid (stiff) | Flexible (bendable) |
| Shy (timid / reserved) | Effusive (openly expressive / unrestrained) |
| Stagnant (still / not moving) | Flowing (moving) |
| Systole (heart contraction) | Diastole (heart relaxation) |
| Tall | Short |
| Timid (shy / fearful) | Bold / Fearless |
| Ubiquitous (everywhere) | Rare (hard to find) |
1. Find the error: "The old lady picked up the struggling puppy straightening her hat grimly and took it to the veterinarian."
Answer: "straightening her hat grimly" — It is placed right after "the struggling puppy", making it sound like the puppy is straightening the hat.
2. Fill in the blank: "Wait here, I will be back in ___________." (some time / sometime / sometimes)
Answer: some time — Two words (some time) means a length of time (a little while).
Physics
📖 Definition
Covers motion, forces, work & energy, momentum, fluids, heat, electricity, waves, and mirrors/lenses.
Speed & Motion (Kinematics)
Speed Formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
Round-Trip Average Speed: $v_{\text{avg}} = \frac{2v_1 v_2}{v_1 + v_2}$ (when traveling same distance at speeds $v_1$ and $v_2$).
The 3 Motion Equations:
$v_f = v_i + at$
$d = v_i t + \frac{1}{2}at^2$
$v_f^2 = v_i^2 + 2ad$
Free Fall (dropped from rest):
Fall height: $h = \frac{1}{2}gt^2$
Landing speed: $v_f = \sqrt{2gh}$ (use $g \approx 9.8\text{ or }10\text{ m/s}^2$)
Police Catch-Up Shortcut: $t = \frac{2v}{a}$ (from rest with acceleration $a$ chasing target at constant speed $v$).
Example Q: A car starts from rest and accelerates at $4\text{ m/s}^2$ for $5\text{ seconds}$. Find its final speed and distance.
✅ Solution: $v_f = 0 + (4)(5) = \mathbf{20\text{ m/s}}$. Distance $d = \frac{1}{2}(4)(5^2) = \frac{1}{2}(100) = \mathbf{50\text{ m}}$.
✅ Solution: $v_f = 0 + (4)(5) = \mathbf{20\text{ m/s}}$. Distance $d = \frac{1}{2}(4)(5^2) = \frac{1}{2}(100) = \mathbf{50\text{ m}}$.
Example Q (Free Fall): A coin is dropped from a bridge and hits the water in $3\text{ seconds}$ ($g = 9.8\text{ m/s}^2$). Find height.
✅ Solution: $h = \frac{1}{2}gt^2 = \frac{1}{2}(9.8)(3^2) = 4.9 \times 9 = \mathbf{44.1\text{ m}}$.
✅ Solution: $h = \frac{1}{2}gt^2 = \frac{1}{2}(9.8)(3^2) = 4.9 \times 9 = \mathbf{44.1\text{ m}}$.
Projectile Motion
Horizontal Speed: $v_x = v_0\cos\theta$ is constant (acceleration $a_x = 0$).
Fired Horizontally vs. Dropped Bullet: Both hit flat ground at the exact same time!
Fall time: $t = \sqrt{\frac{2h}{g}}$ (depends only on vertical height $h$)
Horizontal reach: $d = v_0 \times t$
At the Highest Point (Peak):
Vertical speed: $v_y = 0$
Total speed: $v = v_x$ (only horizontal speed remains)
Acceleration: still $9.8\text{ m/s}^2$ downward (gravity never stops!)
Flight Symmetry: Same height $=$ Same speed (speed going up equals speed coming down).
Launch Angles & Range:
Maximum range: $45^\circ$
Equal range: Angles adding up to $90^\circ$ ($30^\circ$ & $60^\circ$) land at the exact same distance.
Example Q: A dart is thrown horizontally at $15\text{ m/s}$ from a height of $20\text{ m}$ ($g = 10\text{ m/s}^2$). How far does it land?
✅ Solution: Fall time $t = \sqrt{\frac{2(20)}{10}} = \sqrt{4} = 2\text{ s}$. Horizontal reach $d = v_x \times t = 15 \times 2 = \mathbf{30\text{ m}}$.
✅ Solution: Fall time $t = \sqrt{\frac{2(20)}{10}} = \sqrt{4} = 2\text{ s}$. Horizontal reach $d = v_x \times t = 15 \times 2 = \mathbf{30\text{ m}}$.
Forces, Elevator Scale & Circular Motion
Weight: $W = mg$
Newton's 2nd Law: $F_{\text{net}} = ma$
Friction: $f = \mu N$
Elevator Scale (Apparent Weight):
Accelerating upward: $N = m(g + a)$ (you feel heavier)
Accelerating downward: $N = m(g - a)$ (you feel lighter)
Constant speed: $N = mg$ (normal weight)
Connected Blocks: $a = \frac{F_{\text{net}}}{m_{\text{total}}}$
Atwood Machine (2 hanging masses): $a = \frac{(m_1 - m_2)g}{m_1 + m_2}$
Centripetal Force (turning in a circle): $F_c = \frac{mv^2}{r}$ (always points toward the center).
Example Q: A $60\text{ kg}$ student stands on a scale in an elevator accelerating upward at $2\text{ m/s}^2$ ($g = 9.8\text{ m/s}^2$). What does the scale read?
✅ Solution: $N = m(g + a) = 60(9.8 + 2) = 60(11.8) = \mathbf{708\text{ N}}$ (heavier than normal $588\text{ N}$).
✅ Solution: $N = m(g + a) = 60(9.8 + 2) = 60(11.8) = \mathbf{708\text{ N}}$ (heavier than normal $588\text{ N}$).
Work, Energy & Power
Work: $W = Fd\cos\theta$ (Carrying a bag horizontally at constant speed does $0\text{ J}$ of work against gravity).
Potential Energy: $PE = mgh$
Spring Energy: $PE_s = \frac{1}{2}kx^2$
Kinetic Energy: $KE = \frac{1}{2}mv^2$ (Doubling speed multiplies $KE$ and braking distance by $4$!)
Power: $P = \frac{W}{t} = Fv$ (measured in Watts, $\text{J/s}$).
Example Q: A $1000\text{ kg}$ car doubles its speed from $10\text{ m/s}$ to $20\text{ m/s}$. By how much does its kinetic energy increase?
✅ Solution: $KE_1 = \frac{1}{2}(1000)(10^2) = 50,000\text{ J}$. $KE_2 = \frac{1}{2}(1000)(20^2) = 200,000\text{ J}$. $KE$ increases by a factor of $2^2 = 4$ ($\mathbf{150,000\text{ J}}$ gain).
✅ Solution: $KE_1 = \frac{1}{2}(1000)(10^2) = 50,000\text{ J}$. $KE_2 = \frac{1}{2}(1000)(20^2) = 200,000\text{ J}$. $KE$ increases by a factor of $2^2 = 4$ ($\mathbf{150,000\text{ J}}$ gain).
Momentum & Impulse
Momentum: $p = mv$
Impulse: $J = F\Delta t = m(v_f - v_i)$ (bouncing off a wall gives a bigger impulse than sticking to it).
Collisions: $m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$ (Total momentum is always conserved).
Example Q: A $0.2\text{ kg}$ ball moving at $+15\text{ m/s}$ strikes a wall and bounces back at $-15\text{ m/s}$ in $0.05\text{ s}$. Find average force.
✅ Solution: $F = \frac{\Delta p}{\Delta t} = \frac{0.2(-15 - 15)}{0.05} = \frac{-6.0}{0.05} = \mathbf{-120\text{ N}}$ ($120\text{ N}$ away from wall).
✅ Solution: $F = \frac{\Delta p}{\Delta t} = \frac{0.2(-15 - 15)}{0.05} = \frac{-6.0}{0.05} = \mathbf{-120\text{ N}}$ ($120\text{ N}$ away from wall).
Fluids, Density & Buoyancy
Density: $\rho = \frac{m}{V}$ (water density $\rho = 1\text{ g/cm}^3 = 1000\text{ kg/m}^3$)
Pressure: $P = \frac{F}{A}$
Water Depth Pressure: $P = \rho g h$ (depends only on depth $h$, not container width).
Buoyant Force (Archimedes): $F_b = \rho_{\text{fluid}} V_{\text{submerged}} g =$ weight of displaced liquid.
Scrap-Iron Barge in a Lock: Throwing iron overboard causes water level to fall (floating iron displaces its heavy weight; submerged iron only displaces its small volume).
Pipe Flow (Bernoulli & Continuity): $A_1 v_1 = A_2 v_2$ (narrower pipe $\to$ faster flow and lower pressure).
Example Q: A metal cube of volume $0.002\text{ m}^3$ is completely submerged in water ($\rho = 1000\text{ kg/m}^3, g = 9.8\text{ m/s}^2$). Find buoyant force.
✅ Solution: $F_b = \rho V g = (1000)(0.002)(9.8) = \mathbf{19.6\text{ N}}$.
✅ Solution: $F_b = \rho V g = (1000)(0.002)(9.8) = \mathbf{19.6\text{ N}}$.
Heat & Thermodynamics
Temperature Change: $Q = mc\Delta T$
Phase Change: $Q = mL$ (During melting or boiling, temperature stays constant!)
High Altitude (Mountain / Baguio): Lower air pressure makes water boil at a lower temperature ($<100^\circ\text{C}$).
Thermal Expansion: $\Delta L = \alpha L_0 \Delta T$ (a hole inside a heated metal plate also gets larger).
Laws of Thermodynamics:
1st Law: Energy Conservation ($\Delta U = Q - W$)
2nd Law: Heat flows hot $\to$ cold; Entropy (disorder) always increases
Absolute Zero: $0\text{ K} = -273.15^\circ\text{C}$ (molecules stop moving)
Example Q: How much heat is needed to warm $200\text{ g}$ of water from $25^\circ\text{C}$ to $75^\circ\text{C}$? ($c = 4.184\text{ J/(g}\cdot^\circ\text{C)}$)
✅ Solution: $Q = mc\Delta T = 200 \times 4.184 \times (75 - 25) = 200 \times 4.184 \times 50 = \mathbf{41,840\text{ J}}$ ($41.84\text{ kJ}$).
✅ Solution: $Q = mc\Delta T = 200 \times 4.184 \times (75 - 25) = 200 \times 4.184 \times 50 = \mathbf{41,840\text{ J}}$ ($41.84\text{ kJ}$).
Electricity, Magnetism & Circuits
Coulomb's Law: $F = \frac{kq_1 q_2}{r^2}$ (Doubling the distance drops force to $\frac{1}{4}$).
Ohm's Law: $V = IR$
Electrical Power: $P = IV = I^2 R = \frac{V^2}{R}$
Series Resistors: $R_{\text{total}} = R_1 + R_2$ (same current $I$).
Parallel Resistors: $R_{\text{total}} = \frac{R_1 R_2}{R_1 + R_2}$ (same voltage $V$; adding branches lowers total resistance).
Generator vs. Motor:
Generator: Converts Mechanical $\to$ Electrical energy (Faraday's Law).
Motor: Converts Electrical $\to$ Mechanical energy.
Lenz's Law: Induced current opposes its cause (follows Conservation of Energy).
Example Q: Two resistors ($6\,\Omega$ and $3\,\Omega$) are connected in parallel across a $12\text{ V}$ battery. Find total current.
✅ Solution: Equivalent resistance $R_p = \frac{6 \times 3}{6 + 3} = \frac{18}{9} = 2\,\Omega$. Circuit current $I = \frac{V}{R_p} = \frac{12}{2} = \mathbf{6\text{ A}}$.
✅ Solution: Equivalent resistance $R_p = \frac{6 \times 3}{6 + 3} = \frac{18}{9} = 2\,\Omega$. Circuit current $I = \frac{V}{R_p} = \frac{12}{2} = \mathbf{6\text{ A}}$.
Waves, Sound & Light
Wave Speed: $v = f\lambda$
Period & Frequency: $T = \frac{1}{f}$
Index of Refraction: $n = \frac{c}{v}$
Snell's Law of Refraction: $n_1 \sin\theta_1 = n_2 \sin\theta_2$
Entering denser / slower medium (e.g. Air $\to$ Water): Light bends toward the normal.
Entering less dense / faster medium (e.g. Water $\to$ Air): Light bends away from the normal.
Sound Waves (Longitudinal): Travels fastest in Solids > Liquids > Gases (cannot travel through vacuum).
Light Waves (Transverse EM): Travels fastest in Vacuum ($3 \times 10^8\text{ m/s}$).
EM Spectrum (Low to High Energy/Frequency): Radio $\to$ Microwave $\to$ Infrared $\to$ Visible (ROYGBIV) $\to$ UV $\to$ X-ray $\to$ Gamma.
Example Q: An audio speaker produces sound at $170\text{ Hz}$ in air where speed is $340\text{ m/s}$. Find wavelength.
✅ Solution: $\lambda = \frac{v}{f} = \frac{340}{170} = \mathbf{2.0\text{ m}}$.
✅ Solution: $\lambda = \frac{v}{f} = \frac{340}{170} = \mathbf{2.0\text{ m}}$.
Mirrors, Lenses & Eye Defects
Mirror / Lens Formula: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \implies d_i = \frac{f \cdot d_o}{d_o - f}$
Magnification: $M = -\frac{d_i}{d_o}$
Convex Mirror & Concave Lens: Always produce a Virtual, Upright, and Smaller image.
Two $90^\circ$ Flat Mirrors: Second reflection angle $= 90^\circ - \theta_1$.
Eye Defects:
Nearsighted (Myopia): Fixed with a Concave (diverging) lens.
Farsighted (Hyperopia): Fixed with a Convex (converging) lens.
Example Q: An object is placed $30\text{ cm}$ in front of a converging lens with focal length $f = 10\text{ cm}$. Find image distance $d_i$.
✅ Solution: $d_i = \frac{f \cdot d_o}{d_o - f} = \frac{10 \times 30}{30 - 10} = \frac{300}{20} = \mathbf{15\text{ cm}}$ (real, inverted image).
✅ Solution: $d_i = \frac{f \cdot d_o}{d_o - f} = \frac{10 \times 30}{30 - 10} = \frac{300}{20} = \mathbf{15\text{ cm}}$ (real, inverted image).
1. A ball is thrown along a curved path through point X (going up), Z (the peak), and P (going down at the exact same height as X). How does its speed at X compare to P?
Answer: Speed at X equals speed at P — Points at the same height along a projectile path always have the exact same speed.
2. A car passes a sign at a constant $15\text{ m/s}$. A police motorcycle starts from rest at $a = 3\text{ m/s}^2$. How long until the officer catches the car?
Answer: $10\text{ seconds}$ — Use the shortcut $t = \frac{2v}{a} = \frac{2 \times 15}{3} = 10\text{ s}$.
3. An object is placed $50\text{ cm}$ in front of a concave mirror with focal length $f = 15\text{ cm}$. Find the image distance $d_i$.
Answer: $21.43\text{ cm}$ — Multiply over subtract: $d_i = \frac{15 \times 50}{50 - 15} = \frac{750}{35} = \frac{150}{7} \approx 21.43\text{ cm}$.
4. Two flat mirrors meet at a $90^\circ$ corner. A light ray hits Mirror 1 at $55^\circ$ and bounces to Mirror 2. What angle does it leave Mirror 2?
Answer: $35^\circ$ — Just subtract from $90^\circ$: $90^\circ - 55^\circ = 35^\circ$.
Chemistry
📖 Definition
Covers atoms, periodic trends, bonding, molecule shapes, moles, solutions, gas laws, acids/bases, redox, and half-life.
Atoms, Ions & Shells
Atomic Number ($Z$): $\text{Protons} = \text{Electrons}$ (in a neutral atom).
Mass Number ($A$): $\text{Protons} + \text{Neutrons} \implies \text{Neutrons} = A - Z$.
Isotopes: Same protons ($Z$), but different neutrons ($A$).
Max Electrons Per Shell ($n$): $2n^2$ ($2, 8, 18, 32$). Subshells: $s=2, p=6, d=10, f=14$.
Periodic Table Trends
Quick Memory Rule (The 2 Opposite Corners):
Top-Right Corner (Fluorine $\text{F}$): Pulls electrons the hardest.
Bottom-Left Corner (Francium $\text{Fr}$): Has the biggest atom size.
Pulling Electrons (Electronegativity & Ionization Energy):
What it means: How strongly an atom grabs electrons and refuses to give them up.
Trend: Increases moving Right ($\to$) and Up ($\uparrow$) toward Fluorine.
Atom Size (Atomic Radius):
What it means: How physically big or fat the atom is.
Trend: Increases moving Left ($\leftarrow$) and Down ($\downarrow$) toward Francium (atoms gain more electron shells).
Metals vs. Non-Metals:
Left & Bottom: Metals (good conductors, lose electrons easily).
Top-Right: Non-metals (gases, hold onto electrons tightly).
Dobereiner's Triads:
In a set of 3 similar elements (like $\text{Li, Na, K}$), the middle atomic mass is the average of the other two: $\frac{7 + 39}{2} = 23$ (Sodium $\text{Na}$).
Bonding, Lattice Energy & IMFs
Ionic Bond: Metal $+$ Nonmetal (electrons transferred).
Covalent Bond: Nonmetal $+$ Nonmetal (electrons shared).
Criss-Cross Formula: Swap ion charges to subscripts ($\text{Al}^{3+} + \text{SO}_4^{2-} \to \text{Al}_2(\text{SO}_4)_3$).
Lattice Energy: Higher charges & smaller ions $=$ stronger bond ($\text{MgO}$ is higher than $\text{NaCl}$ or $\text{CaO}$).
Intermolecular Forces (Strongest $\to$ Weakest):
Ion-Dipole (strongest)
Hydrogen Bond ($\text{H}$ bonded with $\text{F, O, N}$, e.g. Alcohols $-\text{OH}$)
Dipole-Dipole
London Dispersion (weakest)
Rule: Stronger force = higher boiling point.
Molecule Shapes (VSEPR) & MO Theory
$\text{CO}_2, \text{BeCl}_2 \to$ Linear ($180^\circ$)
$\text{BF}_3, \text{SO}_3 \to$ Trigonal Planar ($120^\circ$)
$\text{CH}_4, \text{CCl}_4 \to$ Tetrahedral ($109.5^\circ$, $sp^3$)
$\text{NH}_3 \to$ Trigonal Pyramidal ($107^\circ$, 1 lone pair)
$\text{H}_2\text{O} \to$ Bent ($104.5^\circ$, 2 lone pairs)
$\text{O}_2$ Molecule: Bond order $= 2$ and paramagnetic (has 2 unpaired electrons).
Purple $\text{MnO}_4^-$ Color: Caused by Ligand-to-Metal Charge Transfer (LMCT).
Semiconductors: Silicon doped with Group 13 (Gallium $\text{Ga}$) $=$ p-type; Group 15 (Phosphorus) $=$ n-type.
Moles, Solutions & Colligative Properties
Moles ($n$): $\text{Moles} = \frac{\text{Mass }(g)}{\text{Molar Mass }(g/\text{mol})}$
STP Standard: $1\text{ mol} = 6.022 \times 10^{23}\text{ particles} = 22.4\text{ L of gas}$ ($0^\circ\text{C}, 1\text{ atm}$).
Molarity ($M$): $\frac{\text{moles of solute}}{\text{Liters of solution}}$
Molality ($m$): $\frac{\text{moles of solute}}{\text{kg of solvent}}$
Dilution Formula: $M_1 V_1 = M_2 V_2$
Van 't Hoff Factor ($i$):
$\text{NaCl} \to i = 2$
$\text{CaCl}_2 \to i = 3$
$\text{Al}_2(\text{SO}_4)_3 \to i = 5$
Rule: More ions ($i$) = lowest freezing point and highest boiling point.
Osmotic Pressure: $\Pi = iMRT$
Gas Laws (Always use Kelvin!)
Temperature Conversion: $K = ^\circ\text{C} + 273$ (always do this first!)
Combined Gas Law: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$
Boyle's Law ($T$ constant): $P_1 V_1 = P_2 V_2$
Charles's Law ($P$ constant): $\frac{V_1}{T_1} = \frac{V_2}{T_2}$
Gay-Lussac's Law ($V$ constant): $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ (heating gas in a rigid aerosol can increases pressure).
Ideal Gas Law: $PV = nRT$ ($R = 0.0821\text{ L}\cdot\text{atm/mol}\cdot\text{K}$)
Dalton's Partial Pressure: $P_{\text{total}} = P_1 + P_2 + \dots$
Graham's Law: Lighter gases (smaller molar mass like $\text{H}_2, \text{He}$) diffuse and move fastest.
Equilibrium (Le Chatelier), Acids & Buffers
Le Chatelier's Principle:
Adding heat shifts an exothermic reaction left (and endothermic right).
Increasing pressure shifts toward the side with fewer moles of gas.
A catalyst lowers activation energy ($E_a$) to speed up reaction, but does not shift $K_{\text{eq}}$.
pH & pOH: $\text{pH} = -\log[\text{H}^+]$
$\text{pH} + \text{pOH} = 14$
$\text{pH} < 7$: Acid (turns blue litmus red; donates $\text{H}^+$).
$\text{pH} = 7$: Neutral.
$\text{pH} > 7$: Base (turns red litmus blue; accepts $\text{H}^+$).
Conjugate Acid-Base Pairs (Brønsted-Lowry):
Acid donates $\text{H}^+ \to$ becomes conjugate base (e.g. $\text{HCl} \to \text{Cl}^-$).
Base accepts $\text{H}^+ \to$ becomes conjugate acid (e.g. $\text{NH}_3 \to \text{NH}_4^+$).
Buffer Solution: Weak acid $+$ conjugate base salt (e.g. $\text{CH}_3\text{COOH} + \text{CH}_3\text{COONa}$); resists changes in pH.
Redox & Electrochemistry
OIL RIG:
Oxidation Is Loss of electrons (charge goes up).
Reduction Is Gain of electrons (charge goes down).
AN OX / RED CAT:
Oxidation always happens at the Anode.
Reduction always happens at the Cathode.
Organic Chemistry & Functional Groups Cheat Sheet
Alcohol ($-\text{OH}$): Suffix -ol (e.g. Ethanol, rubbing alcohol). Forms strong hydrogen bonds.
Aldehyde ($-\text{CHO}$): Carbonyl at end of chain, suffix -al (e.g. Formaldehyde / Methanal).
Ketone ($-\text{CO}-$): Carbonyl inside carbon chain, suffix -one (e.g. Acetone / Propanone nail polish remover).
Carboxylic Acid ($-\text{COOH}$): Suffix -oic acid (e.g. Acetic acid in vinegar). Turns litmus red.
Ester ($-\text{COO}-$): Sweet, fruity fragrance in perfumes and fruits. Formed by Carboxylic Acid $+$ Alcohol (Esterification).
Ether ($-\text{R-O-R'}$): Central oxygen bridge connecting two carbon chains (e.g. Diethyl ether anesthetic).
Amine ($-\text{NH}_2$): Fishy odor, basic; contains Nitrogen (e.g. Amino acid building blocks).
Amide ($-\text{CONH}_2$): Forms the peptide bond linking amino acids in proteins.
Thermodynamics, Nuclear & Organic Rules
Exothermic ($\Delta H < 0$): Releases heat (feels hot; burning, freezing, condensation).
Endothermic ($\Delta H > 0$): Absorbs heat (feels cold; melting, boiling, evaporating).
Spontaneity: $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ < 0$. At water's triple point, degrees of freedom $F = 0$.
Half-Life: $\text{Remaining} = \text{Initial} \times \left(\frac{1}{2}\right)^n$, where $n = \frac{\text{Total Time}}{\text{Half-Life}}$.
Beta ($\beta^-$) Decay: Neutron turns into a proton ($Z$ goes up by $1$, mass $A$ stays same).
Organic Shortcuts:
Markovnikov: Propene $+ \text{HCl} \to$ 2-chloropropane.
Oxidation: Primary alcohol $+ \text{PCC} \to$ Aldehyde.
$\text{S}_\text{N}2$ Substitution: Inverts 3D configuration (backside attack).
Lab Safety:
Always add Acid into Water (AA), never water into acid!
Never mix bleach with acid ($\text{Cl}_2$ gas) or with ammonia (chloramine gas).
Treat $\text{HF}$ skin burns with calcium gluconate gel.
Absorbance $A = 1.0 \implies 10\%$ transmittance.
1. A $25\text{ L}$ gas at $1\text{ atm}$ and $35^\circ\text{C}$ is compressed to $20\text{ L}$ at $2\text{ atm}$. What is the new temperature in $^\circ\text{C}$?
Answer: $220^\circ\text{C}$ — Convert $T_1 = 35 + 273 = 308\text{ K}$. Solve $T_2 = 308 \times \frac{2 \times 20}{1 \times 25} = 493\text{ K}$. Convert back to Celsius: $493 - 273 = 220^\circ\text{C}$.
2. Element Davelium (Dv) has 2 valence electrons. Element Rulenium (Ru*) has 7 valence electrons. What formula forms when they combine?
Answer: $\text{Dv}(\text{Ru}^*)_2$ — Dv has 2 valence electrons ($\text{Dv}^{2+}$) and Ru* needs 1 electron ($(\text{Ru}^*)^-$). Crossing charges gives $\text{Dv}(\text{Ru}^*)_2$.
3. Combine $\text{Al}^{3+}$ and $\text{SO}_4^{2-}$.
Answer: $\text{Al}_2(\text{SO}_4)_3$ — Cross the $3$ and the $2$ to get $\text{Al}_2(\text{SO}_4)_3$.
Biology
📖 Definition
Covers cells, osmosis, photosynthesis, cell division, DNA/RNA, genetics, human anatomy, and ecology.
Microscope & Cell Organelles
Microscope Total Magnification: $\text{Eyepiece} \times \text{Objective}$ (e.g. $10\times \cdot 40\times = 400\times$).
Use the fine adjustment knob only under high power.
Cell Membrane: Phospholipid bilayer (hydrophilic heads face water outside; hydrophobic fatty-acid tails face inside).
Key Organelles:
Mitochondria: Makes ATP energy (cellular respiration).
Ribosome: Builds proteins.
Golgi Body: Packages & ships proteins.
Rough ER: Produces proteins for export.
Lysosome: Digests cellular waste.
Chloroplast: Performs photosynthesis in plants.
Plant vs. Animal: Plant cells have a Cell Wall (cellulose), Chloroplasts, and a large central vacuole; animal cells do not.
Prokaryote vs. Eukaryote:
Prokaryote (Bacteria): No nucleus (Staphylococcus = clusters; Streptococcus = chains).
Eukaryote (Plants, Animals, Fungi): Has a true nucleus.
Osmosis (Water Movement)
Golden Rule: Water always moves toward more salt/solute!
Hypertonic (salty outside): Water leaves cell $\to$ cell shrinks/shrivels.
Hypotonic (pure water outside): Water enters cell $\to$ cell swells/bursts (plants become turgid).
Isotonic: Equal salt concentration $\to$ cell stays normal.
Photosynthesis, Respiration & Enzymes
Photosynthesis (Chloroplast): $6\text{CO}_2 + 6\text{H}_2\text{O} + \text{Light} \to \text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2$
Light reaction (Thylakoid): splits water, releases $\text{O}_2$.
Calvin cycle (Stroma): fixes carbon into glucose.
Cellular Respiration (Mitochondria):
Glycolysis (in cytoplasm, no $\text{O}_2$ needed): nets $2\text{ ATP}$ & $2\text{ NADH}$.
Krebs Cycle & ETC (in mitochondria): uses $\text{O}_2$ as final electron acceptor to produce $36\text{–}38\text{ ATP}$.
Enzymes & Lipids:
Lower $K_m =$ higher enzyme binding affinity.
Competitive inhibitors: bind active site (overcome by adding more substrate).
Cholesterol: precursor for steroid hormones (testosterone, estrogen).
Mitosis vs. Meiosis (Stages: PMAT)
Cell Cycle (Interphase $+$ M Phase):
$G_1$ Phase: Cell growth, active metabolism, and protein synthesis.
$S$ Phase (Synthesis): DNA replication (chromosomes duplicate into sister chromatids).
$G_2$ Phase: Final growth and safety checkpoints before division.
M Phase (Mitosis): Nuclear and cytoplasmic division (PMAT).
PMAT Stages:
Prophase: chromosomes condense.
Metaphase: chromosomes line up in the Middle.
Anaphase: chromatids pull Apart to opposite poles.
Telophase: Two new nuclei form.
Mitosis: Body cells $\to$ 2 identical diploid ($2n = 46$) cells.
Meiosis: Sex cells $\to$ 4 unique haploid ($n = 23$) cells (crossing-over occurs in Prophase I).
Sperm vs. Egg: Spermatogenesis is continuous ($4$ sperm); Oogenesis pauses and makes $1$ ovum $+$ polar bodies.
DNA & RNA Base Pairing
DNA (Chargaff's Rule): $\mathbf{A = T}$ ($2$ H-bonds) and $\mathbf{G \equiv C}$ ($3$ H-bonds).
More $\text{G-C}$ pairs $=$ higher DNA melting temperature.
DNA Helicase: Unzips the double helix during replication.
RNA Pairing: Uses Uracil ($\text{U}$) instead of Thymine ($\text{A} \to \text{U}$, $\text{T} \to \text{A}$, $\text{C} \leftrightarrow \text{G}$).
tRNA: carries the matching anticodon and amino acid.
Central Dogma: $\text{DNA} \xrightarrow{\text{Transcription}} \text{mRNA} \xrightarrow{\text{Translation}} \text{Protein}$.
Genetics & Blood Types
Monohybrid Cross ($Aa \times Aa$): $3\text{ Dominant} : 1\text{ Recessive}$ ($25\%\text{ }AA, 50\%\text{ }Aa, 25\%\text{ }aa$).
Dihybrid Cross ($AaBb \times AaBb$): $9 : 3 : 3 : 1$
Test Cross ($Aa \times aa$): $1 : 1$
Blood Types:
Type O ($ii$): Universal RBC Donor.
Type AB ($I^A I^B$, codominant): Universal Recipient.
Rule: An AB parent can never have an O child.
X-Linked Recessive (Colorblindness / Hemophilia): Mothers pass traits to their sons ($XY$).
Mendel's Laws of Inheritance:
Law of Segregation: Two alleles for each trait separate during gamete formation.
Law of Independent Assortment: Alleles for different genes assort independently of each other.
Hardy-Weinberg Population Equilibrium:
Allele frequencies: $p + q = 1$ ($p = \text{dominant allele } A$, $q = \text{recessive allele } a$).
Genotype frequencies: $p^2 + 2pq + q^2 = 1$ ($p^2 = AA$, $2pq = Aa \text{ carriers}$, $q^2 = aa \text{ affected}$).
Shortcut: Always compute $q = \sqrt{q^2}$ first from the recessive count, then $p = 1 - q$, then carrier frequency $= 2pq$.
Human Anatomy, Immunity & Plants
Connective Tissue:
Ligament: Connects Bone to Bone.
Tendon: Connects Muscle to Bone.
Circulation:
Arteries: Carry blood Away from heart.
Veins: Return blood to heart (have one-way valves).
Pulmonary Vein: Only vein carrying oxygenated blood.
Key Organs:
Nephron: Kidney filtration unit.
Alveoli: Lung gas exchange.
Villi: Small intestine nutrient absorption.
Liver & Gallbladder: Liver makes bile; gallbladder stores it.
Pancreas: Produces insulin to lower blood sugar.
Cerebellum: Coordinates balance & posture.
Medulla: Controls breathing & heart rate.
Immunity:
Passive Natural: Mother's antibodies (breast milk/placenta).
Active Artificial: Vaccines.
Interferons: Antiviral proteins.
Histamine: Released by mast cells during allergies.
Pus: Dead neutrophils at infection site.
Plant Vessels & Roots:
Xylem: Carries water upward.
Phloem: Transports food/sugar throughout plant.
Monocot: Fibrous roots, parallel leaf veins (corn, grass).
Eudicot: Taproot, netted leaf veins (mango, munggo).
Taxonomy, Evolution & Ecology
Taxonomy Order: Domain $\to$ Kingdom $\to$ Phylum $\to$ Class $\to$ Order $\to$ Family $\to$ Genus $\to$ Species.
Memory trick: Dear King Philip Came Over For Good Soup.
Homologous vs. Analogous:
Homologous: Same bone structure, different use (human arm, bat wing, whale flipper).
Analogous: Different anatomy, same use (bird wing vs. butterfly wing).
10% Energy Rule: Only $10\%$ of energy transfers to next trophic level ($10{,}000\text{ J} \to 1{,}000\text{ J} \to 100\text{ J} \to 10\text{ J}$).
Crop Rotation (Corn & Munggo): Legumes (munggo) house root bacteria (Rhizobium) that restore nitrogen to soil.
1. A farmer plants Corn in Year 1, Munggo in Year 2, Corn in Year 3, and Munggo in Year 4. Why does he alternate Corn and Munggo?
Answer: To restore nitrogen in the soil — Munggo is a legume with root bacteria that put nitrogen back into the soil after corn uses it up.
2. Trace the path of sound vibrations from the outer ear to the inner ear.
Answer: Pinna $\to$ Tympanic membrane $\to$ Malleus $\to$ Incus $\to$ Stapes $\to$ Oval window $\to$ Cochlea — Remember MIS for the 3 middle-ear bones, ending at the Cochlea.
Earth & Space Science
📖 Definition
Covers plate tectonics, earthquakes, rocks & minerals, atmosphere layers, monsoons, seasons, eclipses, and planets.
3 Plate Boundaries & PH Geology
Convergent ($\to \leftarrow$ collide): Deep ocean trenches, volcanic arcs, folded mountains.
Divergent ($\leftarrow \to$ pull apart): Mid-ocean ridges, rift valleys, new seafloor crust.
Transform ($\uparrow \downarrow$ slide past): Strike-slip faults, shallow earthquakes (no volcanoes).
Philippine Geology Highlights:
Philippine Trench: East of the Philippines, where Philippine Sea Plate subducts westward.
Manila Trench: West of Luzon, where South China Sea crust subducts eastward.
Philippine Fault Zone: $1{,}200\text{ km}$ left-lateral strike-slip fault.
Zambales Ophiolite: Ancient oceanic crust uplifted onto Luzon.
Agency: PHIVOLCS (DOST agency monitoring volcanoes and earthquakes).
Earthquakes & Seismic Waves
Focus vs. Epicenter:
Focus (Hypocenter): Underground point where fault rock breaks.
Epicenter: Surface point directly above focus (requires 3 seismic stations to triangulate).
P-waves (Primary): Fastest wave; travels through solids, liquids, and gases.
S-waves (Secondary): Slower shear wave; travels only through solids (blocked by liquid outer core).
3 Types of Rocks, Folds & Mohs Scale
Igneous (cooled molten rock):
Underground cooling (slow, large crystals): Granite.
Surface cooling (fast, fine/glassy): Basalt, Pumice, Obsidian.
Sedimentary (compacted layers, contains fossils): Limestone, Sandstone, Shale, Coal (oldest layer at bottom).
Metamorphic (heat & pressure): Limestone $\to$ Marble; Shale $\to$ Slate; Sandstone $\to$ Quartzite.
Rock Folds & Weathering:
Syncline: Downward U-shaped fold (youngest rocks in center).
Anticline: Upward A-shaped fold (oldest rocks in center).
Angular Unconformity: Tilted rock layers eroded flat beneath horizontal layers.
Frost Wedging: Water enters cracks, freezes, expands, and splits rock.
Mohs Hardness ($1\text{ to }10$): 1 = Talc (softest), 7 = Quartz, 10 = Diamond (hardest).
5 Atmosphere Layers & Air Composition
Dry Air Composition: $78\%$ Nitrogen ($\text{N}_2$, chemically inert), $21\%$ Oxygen ($\text{O}_2$), $0.9\%$ Argon, $0.04\%$ $\text{CO}_2$.
The 5 Layers (Bottom to Top):
1. Troposphere: Weather, clouds, rain (temperature drops $10^\circ\text{C/km}$ for rising dry air).
2. Stratosphere: Contains Ozone Layer ($\text{O}_3$) absorbing UV rays; jet flight level.
3. Mesosphere: Coldest layer; meteors burn up here.
4. Thermosphere: Hottest layer; auroras and ionosphere.
5. Exosphere: Outermost boundary merging into outer space.
Monsoons, Breezes & Coriolis
Amihan (Northeast Monsoon, Nov–Feb): Cool, dry air from Siberia.
Habagat (Southwest Monsoon, Jun–Oct): Warm, moisture-laden air bringing heavy monsoon rains.
Coastal Breezes:
Daytime = Sea Breeze (cool breeze blows from sea to warm land).
Nighttime = Land Breeze (cool breeze blows from land to sea).
Coriolis Effect: Earth's rotation curves winds to the right in Northern Hemisphere and left in Southern Hemisphere.
Seasons, Eclipses & Tides
Seasons & Uneven Heating: Caused by Earth's spherical shape and $23.5^\circ$ axial tilt.
Equinox (Mar 21 / Sep 23): Sun directly over equator ($12\text{ hr}$ day and night everywhere).
Precession: Earth's rotational axis wobbles in a $26{,}000\text{-year}$ cycle.
Eclipses:
Solar Eclipse (S–M–E): Moon in middle (New Moon).
Lunar Eclipse (S–E–M): Earth in middle (Full Moon).
Ocean Tides:
Spring Tide: Highest high tides (New & Full Moon alignment).
Neap Tide: Weakest tidal range (Quarter Moons at $90^\circ$).
Planets, Sun & Scientific Method
Planetary Order: Mercury, Venus, Earth, Mars (Rocky)
Jupiter, Saturn, Uranus, Neptune (Gas/Ice Giants).
Venus: Hottest planet (runaway $\text{CO}_2$ greenhouse atmosphere), rotates retrograde.
Earth & Moon Origin:
Chemical Differentiation: Dense molten iron sank to Earth's core.
Giant Impact Hypothesis: Mars-sized body (Theia) collided with young Earth, forming Moon.
Cyanobacteria: Photosynthetic bacteria that generated Earth's first atmospheric $\text{O}_2$.
The Sun & Stars:
Core: Powered by nuclear fusion ($\text{H} \to \text{He}$).
Visible surface: Photosphere.
End state: Sun will finish life cycle as a White Dwarf.
Star temperatures: Blue (hottest) $\to$ White $\to$ Yellow (Sun) $\to$ Red (coolest).
Scientific Method Variables:
Independent Variable: Factor changed/manipulated by experimenter.
Dependent Variable: Factor measured as response/result.
Controlled Variables: Factors kept constant throughout trial.
1. What two factors cause the uneven heating of the Earth's surface by the Sun?
Answer: Earth's curved (spherical) shape and its $23.5^\circ$ tilted axis — Sunlight hits different latitudes at different angles. (Heat from Earth's core does not affect surface weather.)
Algebra, Logs, Sequences & Word Problems
📖 Definition
Covers exponents, factoring, quadratics, polynomials, logarithms, sequences, and word problem shortcuts.
Exponent & Radical Rules
$a^m \cdot a^n = a^{m+n}$
$\frac{a^m}{a^n} = a^{m-n}$
$(a^m)^n = a^{mn}$
$a^0 = 1$
$a^{-n} = \frac{1}{a^n}$
$a^{m/n} = \sqrt[n]{a^m}$
Example Q: Simplify $(8x^6)^{2/3}$ completely.
✅ Solution: $8^{2/3} \cdot (x^6)^{2/3} = (\sqrt[3]{8})^2 \cdot x^{6 \times \frac{2}{3}} = 2^2 \cdot x^4 = \mathbf{4x^4}$.
✅ Solution: $8^{2/3} \cdot (x^6)^{2/3} = (\sqrt[3]{8})^2 \cdot x^{6 \times \frac{2}{3}} = 2^2 \cdot x^4 = \mathbf{4x^4}$.
Fast Factoring Patterns
Difference of Two Squares: $a^2 - b^2 = (a - b)(a + b)$
Perfect Square Trinomial: $(a \pm b)^2 = a^2 \pm 2ab + b^2$
Sum & Difference of Cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
Example Q: Factor $27x^3 - 8$ using difference of cubes.
✅ Solution: Here $a = 3x$ and $b = 2$: $(a - b)(a^2 + ab + b^2) = \mathbf{(3x - 2)(9x^2 + 6x + 4)}$.
✅ Solution: Here $a = 3x$ and $b = 2$: $(a - b)(a^2 + ab + b^2) = \mathbf{(3x - 2)(9x^2 + 6x + 4)}$.
Functions, Composites & Inverses
Evaluating Functions: To find $f(3)$, substitute $x = 3$ into the function.
Composite Function $f(g(x))$: Substitute the entire function $g(x)$ into every $x$ in $f(x)$.
Example: If $f(x) = 2x + 1$ and $g(x) = x^2$, then $f(g(3)) = f(9) = 2(9) + 1 = 19$.
Finding Inverse Function $f^{-1}(x)$ (3-Step Shortcut):
1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$ ($x = \text{equation with } y$).
3. Solve for $y$. (Example: $f(x) = 3x - 5 \to x = 3y - 5 \implies f^{-1}(x) = \frac{x + 5}{3}$).
Domain Restrictions:
Denominators can never be zero: $\frac{1}{x - 4} \implies x \neq 4$.
Values under even radicals must be non-negative: $\sqrt{2x - 6} \implies 2x - 6 \ge 0 \implies x \ge 3$.
Example Q: If $f(x) = \frac{2x + 1}{x - 3}$, find its domain and inverse $f^{-1}(x)$.
✅ Solution: Domain: $x \neq 3$. Inverse: $x = \frac{2y + 1}{y - 3} \implies x(y - 3) = 2y + 1 \implies y(x - 2) = 3x + 1 \implies \mathbf{f^{-1}(x) = \frac{3x + 1}{x - 2}}$.
✅ Solution: Domain: $x \neq 3$. Inverse: $x = \frac{2y + 1}{y - 3} \implies x(y - 3) = 2y + 1 \implies y(x - 2) = 3x + 1 \implies \mathbf{f^{-1}(x) = \frac{3x + 1}{x - 2}}$.
Quadratics ($ax^2 + bx + c = 0$) & Vieta's
Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Parabola Vertex: $x = -\frac{b}{2a}$
Sum of Roots: $r_1 + r_2 = -\frac{b}{a}$
Product of Roots: $r_1 \cdot r_2 = \frac{c}{a}$
Sum of Squares Shortcut: $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2$
Reciprocal Sum Shortcut: $\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2} = -\frac{b}{c}$
Discriminant ($D = b^2 - 4ac$):
$D > 0$: 2 distinct real roots
$D = 0$: 1 repeated real root
$D < 0$: no real roots
Example Q: For $2x^2 - 8x + 5 = 0$, find $r_1^2 + r_2^2$ without solving for roots.
✅ Solution: $r_1 + r_2 = -\frac{-8}{2} = 4$; $r_1 r_2 = \frac{5}{2} = 2.5$. Then $r_1^2 + r_2^2 = 4^2 - 2(2.5) = 16 - 5 = \mathbf{11}$.
✅ Solution: $r_1 + r_2 = -\frac{-8}{2} = 4$; $r_1 r_2 = \frac{5}{2} = 2.5$. Then $r_1^2 + r_2^2 = 4^2 - 2(2.5) = 16 - 5 = \mathbf{11}$.
Remainder & Factor Theorems
Remainder Theorem: To find the remainder when $P(x)$ is divided by $(x - c)$, simply plug in $x = c$ to get $P(c)$!
Factor Theorem: If $P(c) = 0$, then $(x - c)$ is an exact factor.
Example Q: Find remainder when $P(x) = 2x^3 - 5x^2 + 3x - 7$ is divided by $(x - 2)$.
✅ Solution: Plug in $x = 2$: $P(2) = 2(8) - 5(4) + 3(2) - 7 = 16 - 20 + 6 - 7 = \mathbf{-5}$.
✅ Solution: Plug in $x = 2$: $P(2) = 2(8) - 5(4) + 3(2) - 7 = 16 - 20 + 6 - 7 = \mathbf{-5}$.
Logarithms
Definition: $\log_b(x) = y \iff b^y = x$
Base Rules: $\log_b(1) = 0$
$\log_b(b) = 1$
Power Fraction Shortcut: $\log_{b^m}(b^n) = \frac{n}{m}$ (Example: $\log_4(32) = \log_{2^2}(2^5) = \frac{5}{2}$)
Product Rule: $\log(xy) = \log x + \log y$
Quotient Rule: $\log\left(\frac{x}{y}\right) = \log x - \log y$
Exponent Rule: $\log(x^k) = k\log x$
Example Q: Solve for $x$: $\log_2(x) + \log_2(x - 2) = 3$.
✅ Solution: $\log_2[x(x - 2)] = 3 \implies x^2 - 2x = 2^3 = 8 \implies x^2 - 2x - 8 = 0 \implies (x - 4)(x + 2) = 0 \implies \mathbf{x = 4}$ ($x > 2$).
✅ Solution: $\log_2[x(x - 2)] = 3 \implies x^2 - 2x = 2^3 = 8 \implies x^2 - 2x - 8 = 0 \implies (x - 4)(x + 2) = 0 \implies \mathbf{x = 4}$ ($x > 2$).
Arithmetic & Odd Number Sums
$n$th Term: $a_n = a_1 + (n - 1)d$
Sum of $n$ Terms: $S_n = \frac{n}{2}(a_1 + a_n)$
Sum of First $n$ Odd Numbers: $1 + 3 + 5 + \dots = n^2$ (Example: sum of first 10 odds is $10^2 = 100$).
Example Q: Find sum of first $20$ terms of sequence $3, 7, 11, 15, \dots$
✅ Solution: $a_1 = 3, d = 4$. Term $a_{20} = 3 + 19(4) = 79$. Sum $S_{20} = \frac{20}{2}(3 + 79) = 10(82) = \mathbf{820}$.
✅ Solution: $a_1 = 3, d = 4$. Term $a_{20} = 3 + 19(4) = 79$. Sum $S_{20} = \frac{20}{2}(3 + 79) = 10(82) = \mathbf{820}$.
Geometric Sequence
$n$th Term: $a_n = a_1 \cdot r^{n-1}$
Finite Sum: $S_n = \frac{a_1(1 - r^n)}{1 - r}$
Infinite Sum ($|r| < 1$): $S_\infty = \frac{a_1}{1 - r}$
Example Q: Find the sum of the infinite geometric series $12 + 6 + 3 + 1.5 + \dots$
✅ Solution: $a_1 = 12$, $r = \frac{6}{12} = \frac{1}{2}$. Sum $S_\infty = \frac{12}{1 - 0.5} = \frac{12}{0.5} = \mathbf{24}$.
✅ Solution: $a_1 = 12$, $r = \frac{6}{12} = \frac{1}{2}$. Sum $S_\infty = \frac{12}{1 - 0.5} = \frac{12}{0.5} = \mathbf{24}$.
Number Theory, Binary & Sets
Divisibility by 9: Sum of digits is a multiple of $9$.
3-Digit Repdigits ($111, 222, \dots, 999$): Always divisible by $3, 37,$ and $111$ ($111 = 3 \times 37$).
Terminating Decimal: Reduced denominator has only prime factors $2$ and/or $5$.
Decimal to Binary: $0.6875 = \frac{11}{16} = \frac{1}{2} + \frac{1}{8} + \frac{1}{16} = \mathbf{.1011_2}$.
Venn Diagram (2 Sets): $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
Example Q: In a class of $50$ students, $32$ take Biology, $24$ Chemistry, and $10$ both. How many take neither?
✅ Solution: $n(B \cup C) = 32 + 24 - 10 = 46$. Neither $= 50 - 46 = \mathbf{4\text{ students}}$.
✅ Solution: $n(B \cup C) = 32 + 24 - 10 = 46$. Neither $= 50 - 46 = \mathbf{4\text{ students}}$.
Word Problem Shortcuts
Work Together (2 people): $T_{\text{together}} = \frac{t_1 \times t_2}{t_1 + t_2}$
Boat in Current: Downstream $= v_{\text{boat}} + v_{\text{water}}$; Upstream $= v_{\text{boat}} - v_{\text{water}}$.
Train Crossing a Bridge: $\text{Distance} = \text{Train Length} + \text{Bridge Length}$ (convert $\text{km/h} \times \frac{5}{18} = \text{m/s}$).
Bee Between Colliding Trains: Find collision time $t = \frac{D}{v_1 + v_2}$, then bee distance $d = v_{\text{bee}} \times t$.
Drying Crops: Water evaporates, but the solid matter weight never changes!
Successive Percent Changes: Multiply $(1 + a)(1 + b)$ (e.g. $+20\%$ then $-20\% \to 1.2 \times 0.8 = 0.96 = 4\%$ drop).
Simple vs. Compound Interest: Simple $I = Prt$
Compound $A = P(1 + r)^t$.
Example Q: A $150\text{ m}$ train at $72\text{ km/h}$ crosses a $350\text{ m}$ bridge. How many seconds does it take?
✅ Solution: Speed $= 72 \times \frac{5}{18} = 20\text{ m/s}$. Distance $= 150 + 350 = 500\text{ m}$. Time $t = \frac{500}{20} = \mathbf{25\text{ seconds}}$.
✅ Solution: Speed $= 72 \times \frac{5}{18} = 20\text{ m/s}$. Distance $= 150 + 350 = 500\text{ m}$. Time $t = \frac{500}{20} = \mathbf{25\text{ seconds}}$.
1. Find the exact value of $\log_4(32)$.
Answer: $\frac{5}{2}$ — Since $4 = 2^2$ and $32 = 2^5$, divide the exponents: $\frac{5}{2}$.
2. Divide $(3x^3 - 2x^2 + x - 5)$ by $(x + 2)$.
Answer: $3x^2 - 8x + 17 - \frac{39}{x+2}$ — Fast shortcut: plug $x = -2$ into the polynomial to get the remainder first: $3(-8) - 2(4) - 2 - 5 = -39$.
3. Math scores are $X = \{85, 90, 78, 92, 88\}$ and History scores are $Y = \{75, 80, 68, 82, 78\}$. What equation connects $X$ and $Y$?
Answer: $Y = X - 10$ — Every History score $Y$ is $10$ less than the Math score $X$.
Geometry, Conics & Trigonometry
📖 Definition
Covers triangles, polygons, 2D areas, 3D volumes, circle angles, lines, conic sections, and trigonometry.
Right Triangles, Triples & Inequality
Pythagorean Theorem: $a^2 + b^2 = c^2$
Common Triples: $(3,4,5)$, $(5,12,13)$, $(8,15,17)$, $(7,24,25)$
Triangle Inequality: Third side $z$ must satisfy $|x - y| < z < x + y$
Special Right Triangles:
$45^\circ\text{–}45^\circ\text{–}90^\circ$: sides $x : x : x\sqrt{2}$ (Square diagonal $= s\sqrt{2}$)
$30^\circ\text{–}60^\circ\text{–}90^\circ$: sides $x : x\sqrt{3} : 2x$
Equilateral Triangle Area: $A = \frac{s^2\sqrt{3}}{4}$
Example Q: In a $30^\circ\text{–}60^\circ\text{–}90^\circ$ triangle, hypotenuse is $12\text{ cm}$. Find both legs and area.
✅ Solution: Short leg $= \frac{12}{2} = 6\text{ cm}$; Long leg $= 6\sqrt{3}\text{ cm}$. Area $= \frac{1}{2}(6)(6\sqrt{3}) = \mathbf{18\sqrt{3}\text{ cm}^2}$.
✅ Solution: Short leg $= \frac{12}{2} = 6\text{ cm}$; Long leg $= 6\sqrt{3}\text{ cm}$. Area $= \frac{1}{2}(6)(6\sqrt{3}) = \mathbf{18\sqrt{3}\text{ cm}^2}$.
Polygons & Quadrilaterals
Sum of Interior Angles: $(n - 2) \times 180^\circ$
Sum of Exterior Angles: Always $360^\circ$ (Each regular exterior angle $= \frac{360^\circ}{n}$)
Number of Diagonals: $\frac{n(n - 3)}{2}$
Parallelograms from Grid Lines: $m$ parallel lines intersecting $n$ parallel lines form $\binom{m}{2} \times \binom{n}{2}$ parallelograms.
Quadrilateral Hierarchy: Every Square is a Rectangle and Rhombus, but not every Rhombus is a Square.
Example Q: Find the sum of interior angles and total diagonals in an octagon ($n = 8$).
✅ Solution: Interior sum $= (8 - 2) \times 180^\circ = \mathbf{1,080^\circ}$. Diagonals $= \frac{8(8 - 3)}{2} = \frac{40}{2} = \mathbf{20\text{ diagonals}}$.
✅ Solution: Interior sum $= (8 - 2) \times 180^\circ = \mathbf{1,080^\circ}$. Diagonals $= \frac{8(8 - 3)}{2} = \frac{40}{2} = \mathbf{20\text{ diagonals}}$.
2D Areas & 3D Volumes
Circle: Area $= \pi r^2$, Circumference $= 2\pi r$
Trapezoid: Area $= \frac{b_1 + b_2}{2}h$
Cube: Volume $= s^3$, Surface Area $= 6s^2$, Space Diagonal $= s\sqrt{3}$
Box from 3 Face Areas: Volume $V = \sqrt{A_1 \cdot A_2 \cdot A_3}$
Box Inscribed in Sphere: Sphere diameter $D = \sqrt{l^2 + w^2 + h^2}$
Cube to Inscribed Sphere Volume Ratio: Always $6 : \pi$
Cylinder: $V = \pi r^2 h$
Cone & Pyramid: $V = \frac{1}{3}\text{Base} \times h = \frac{1}{3}\pi r^2 h$
Sphere: Volume $= \frac{4}{3}\pi r^3$, Surface Area $= 4\pi r^2$
Example Q: A rectangular box has adjacent face areas of $12\text{ cm}^2$, $15\text{ cm}^2$, and $20\text{ cm}^2$. Find its volume.
✅ Solution: $V = \sqrt{A_1 \cdot A_2 \cdot A_3} = \sqrt{12 \times 15 \times 20} = \sqrt{3600} = \mathbf{60\text{ cm}^3}$.
✅ Solution: $V = \sqrt{A_1 \cdot A_2 \cdot A_3} = \sqrt{12 \times 15 \times 20} = \sqrt{3600} = \mathbf{60\text{ cm}^3}$.
Circle Angle Rules
Central Angle: Equals intercepted arc ($\text{Angle} = \text{Arc}$)
Inscribed Angle: Equals half intercepted arc ($\text{Angle} = \frac{1}{2}\text{Arc}$)
Angle inscribed in a semicircle is always $90^\circ$.
Two Chords Crossing Inside: $\text{Angle} = \frac{\text{Arc}_1 + \text{Arc}_2}{2}$ (chord lengths: $a \cdot b = c \cdot d$)
Two Secants/Tangents Outside: $\text{Angle} = \frac{\text{Far Arc} - \text{Near Arc}}{2}$
Example Q: Two chords intersect inside a circle. Opposite intercepted arcs are $70^\circ$ and $110^\circ$. Find interior angle.
✅ Solution: $\text{Angle} = \frac{70^\circ + 110^\circ}{2} = \frac{180^\circ}{2} = \mathbf{90^\circ}$.
✅ Solution: $\text{Angle} = \frac{70^\circ + 110^\circ}{2} = \frac{180^\circ}{2} = \mathbf{90^\circ}$.
Coordinate Geometry (Lines)
Distance Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Parallel Lines: Identical slopes ($m_1 = m_2$)
Perpendicular Lines: Negative reciprocal slopes ($m_1 \cdot m_2 = -1$)
Example Q: Find line perpendicular to $y = -\frac{2}{3}x + 4$ passing through $(4, 1)$.
✅ Solution: Slope $m_\perp = \frac{3}{2}$. Line: $y - 1 = \frac{3}{2}(x - 4) \implies y = \frac{3}{2}x - 6 + 1 \implies \mathbf{y = \frac{3}{2}x - 5}$.
✅ Solution: Slope $m_\perp = \frac{3}{2}$. Line: $y - 1 = \frac{3}{2}(x - 4) \implies y = \frac{3}{2}x - 6 + 1 \implies \mathbf{y = \frac{3}{2}x - 5}$.
4 Conic Sections
Circle: $(x-h)^2 + (y-k)^2 = r^2$ ($x^2$ and $y^2$ have equal positive coefficients).
Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ ($x^2$ and $y^2$ have unequal positive coefficients; $c^2 = a^2 - b^2$).
Hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ (one squared term is positive, one negative; $c^2 = a^2 + b^2$).
Parabola: $y = ax^2 + bx + c$ or $x = ay^2 + by + c$ (only one variable squared).
Example Q: Identify conic $4x^2 + 9y^2 - 16x + 54y + 61 = 0$ and find its center.
✅ Solution: Both squared terms positive & unequal $\to$ Ellipse. Complete squares: $4(x-2)^2 + 9(y+3)^2 = 36 \implies \frac{(x-2)^2}{9} + \frac{(y+3)^2}{4} = 1$. Center is $\mathbf{(2, -3)}$.
✅ Solution: Both squared terms positive & unequal $\to$ Ellipse. Complete squares: $4(x-2)^2 + 9(y+3)^2 = 36 \implies \frac{(x-2)^2}{9} + \frac{(y+3)^2}{4} = 1$. Center is $\mathbf{(2, -3)}$.
Trigonometry & Unit Circle
SOH-CAH-TOA: $\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$, $\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$, $\tan\theta = \frac{\text{Opp}}{\text{Adj}}$
Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$
Double-Angle Formulas:
$\sin(2\theta) = 2\sin\theta\cos\theta$
$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta$
Wave Graph ($y = A\sin(Bx - C)$):
Amplitude $= |A|$
Period $= \frac{2\pi}{|B|}$ (for $\tan$, Period $= \frac{\pi}{|B|}$)
Phase Shift $= \frac{C}{B}$
Special Angle Values:
$30^\circ$ ($\frac{\pi}{6}$): $\sin = \frac{1}{2}$, $\cos = \frac{\sqrt{3}}{2}$, $\tan = \frac{\sqrt{3}}{3}$
$45^\circ$ ($\frac{\pi}{4}$): $\sin = \frac{\sqrt{2}}{2}$, $\cos = \frac{\sqrt{2}}{2}$, $\tan = 1$
$60^\circ$ ($\frac{\pi}{3}$): $\sin = \frac{\sqrt{3}}{2}$, $\cos = \frac{1}{2}$, $\tan = \sqrt{3}$
Quadrant Signs $(\cos\theta, \sin\theta)$: Q1 $(+,+)$, Q2 $(-,+)$, Q3 $(-,-)$, Q4 $(+,-)$
Example Q: If $\sin\theta = \frac{3}{5}$ in Quadrant II, find $\cos\theta$ and $\sin(2\theta)$.
✅ Solution: In Q2, cosine is negative: $\cos\theta = -\frac{4}{5}$. Then $\sin(2\theta) = 2\sin\theta\cos\theta = 2\left(\frac{3}{5}\right)\left(-\frac{4}{5}\right) = \mathbf{-\frac{24}{25}}$.
✅ Solution: In Q2, cosine is negative: $\cos\theta = -\frac{4}{5}$. Then $\sin(2\theta) = 2\sin\theta\cos\theta = 2\left(\frac{3}{5}\right)\left(-\frac{4}{5}\right) = \mathbf{-\frac{24}{25}}$.
1. In how many points do $2(x+1)^2 + (y-2)^2 = 25$ and $4(x+1)^2 - 3(y-2)^2 = 15$ intersect?
Answer: $4$ points — An ellipse and a hyperbola with the same center $(-1, 2)$ cross each other in $4$ symmetrical points.
2. Chords $AC$ and $BD$ cross inside a circle at $M$. If $\text{Arc } CD = (x - 3)^\circ$ and $\text{Arc } AB = (3x + 11)^\circ$, find $\angle CMD$.
Answer: $2x + 4$ — Add the two arcs and divide by $2$: $\frac{(x - 3) + (3x + 11)}{2} = \frac{4x + 8}{2} = 2x + 4$.
3. Find the $(x, y)$ coordinates on the unit circle for $\theta = \frac{5\pi}{4}$.
Answer: $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ — $\frac{5\pi}{4}$ ($225^\circ$) is in Quadrant III, so both $x$ and $y$ are negative.
Probability, Statistics & Calculus
📖 Definition
Covers permutations, combinations, probability, statistics, limits, derivatives, and basic integrals.
Counting & Arrangements
Line of $n$ Items: $n!$ ($3! = 6, 4! = 24, 5! = 120, 6! = 720$)
Repeated Letters (e.g. LEVEL): $\frac{n!}{p!q!}$
Round Table (Circular): $(n - 1)!$
"Must Sit Together": Treat pair as 1 unit $\to (n - 1)! \times 2!$
"Refuse to Sit Together": $\text{Total } (n!) - \text{Together } ((n-1)! \times 2!)$
Permutation (Order matters): $_nP_r = \frac{n!}{(n - r)!}$
Combination (Order doesn't matter): $_nC_r = \frac{n!}{r!(n - r)!}$
Example Q: In how many distinct ways can the letters of the word SCIENCE be arranged?
✅ Solution: 7 letters with 2 C's and 2 E's: $\frac{7!}{2! \times 2!} = \frac{5040}{4} = \mathbf{1,260\text{ ways}}$.
✅ Solution: 7 letters with 2 C's and 2 E's: $\frac{7!}{2! \times 2!} = \frac{5040}{4} = \mathbf{1,260\text{ ways}}$.
Probability Rules
Basic Probability: $P(A) = \frac{\text{Wanted Outcomes}}{\text{Total Outcomes}}$
Complement Rule ("At least one"): $P(\text{at least } 1) = 1 - P(\text{none})$
OR (Addition Rule): $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
Example: Diamond or Ace in 52 cards $= \frac{13 + 4 - 1}{52} = \frac{16}{52} = \frac{4}{13}$
AND (Multiplication Rule): $P(A \text{ and } B) = P(A) \times P(B)$
Draw Without Replacement: Decrement total count by $1$ after first draw.
Example Q: Bag has $5$ red and $3$ blue marbles. Two marbles drawn without replacement. Find probability both are red.
✅ Solution: $P(\text{1st red}) = \frac{5}{8}$, $P(\text{2nd red}) = \frac{4}{7}$. $P(\text{both}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \mathbf{\frac{5}{14}}$.
✅ Solution: $P(\text{1st red}) = \frac{5}{8}$, $P(\text{2nd red}) = \frac{4}{7}$. $P(\text{both}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \mathbf{\frac{5}{14}}$.
Statistics & Wildlife Sampling
Mean (Average): $\frac{\text{Sum}}{n} \implies \text{Sum} = \text{Mean} \times n$
Median: Middle value in ordered list
Mode: Most frequent score
Range: $\text{Max} - \text{Min}$
Shift Rule: Adding constant $c$ to every score shifts Mean by $+c$, but Standard Deviation remains unchanged!
Capture-Recapture Wildlife Ratio:
$\frac{\text{Marked in 1st Catch } (M)}{\text{Total Population } (N)} = \frac{\text{Recaptured with Marks } (R)}{\text{2nd Catch Size } (C)}$
Example Q: Biologists tag $60$ fish in a lake ($M = 60$). Later catch of $80$ fish ($C = 80$) has $12$ tagged ($R = 12$). Estimate total fish ($N$).
✅ Solution: $\frac{60}{N} = \frac{12}{80} \implies N = \frac{60 \times 80}{12} = \frac{4800}{12} = \mathbf{400\text{ fish}}$.
✅ Solution: $\frac{60}{N} = \frac{12}{80} \implies N = \frac{60 \times 80}{12} = \frac{4800}{12} = \mathbf{400\text{ fish}}$.
Limits
First Step: Plug in $x$. If form is $\frac{0}{0}$, factor and cancel or apply L'Hôpital's Rule.
Trig Limit: $\lim_{x \to 0} \frac{\sin(ax)}{bx} = \frac{a}{b}$
Limit at Infinity ($\lim_{x \to \infty}$): If highest powers match, ratio is leading coefficients.
Example Q: Evaluate the trigonometric limit: $\lim_{x \to 0} \frac{\sin(6x)}{2x}$.
✅ Solution: Use the trig limit ratio shortcut $\lim_{x \to 0}\frac{\sin(ax)}{bx} = \frac{a}{b} = \frac{6}{2} = \mathbf{3}$.
✅ Solution: Use the trig limit ratio shortcut $\lim_{x \to 0}\frac{\sin(ax)}{bx} = \frac{a}{b} = \frac{6}{2} = \mathbf{3}$.
Derivatives (Slope / Rate)
Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
Constant: $\frac{d}{dx}(c) = 0$
Common Derivatives:
$\frac{d}{dx}(\sin x) = \cos x$
$\frac{d}{dx}(\cos x) = -\sin x$
$\frac{d}{dx}(e^x) = e^x$
$\frac{d}{dx}(\ln x) = \frac{1}{x}$
Product Rule: $(uv)' = u'v + uv'$
Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
Chain Rule: $[f(g(x))]' = f'(g(x)) \cdot g'(x)$
Circle Slope ($x^2 + y^2 = r^2$): $\frac{dy}{dx} = -\frac{x}{y}$
Physics Motion: Derivative of Position is Velocity; derivative of Velocity is Acceleration.
Max / Min Extrema: Set first derivative $f'(x) = 0$ and solve.
Example Q: Find the slope of tangent to $f(x) = 2x^3 - 4x + 5$ at $x = 2$.
✅ Solution: Derivative $f'(x) = 6x^2 - 4$. At $x = 2$: $f'(2) = 6(2^2) - 4 = 24 - 4 = \mathbf{20}$.
✅ Solution: Derivative $f'(x) = 6x^2 - 4$. At $x = 2$: $f'(2) = 6(2^2) - 4 = 24 - 4 = \mathbf{20}$.
Integrals & Area Between Curves
Power Rule: $\int x^n\,dx = \frac{x^{n+1}}{n + 1} + C$
Area Between Two Curves: $A = \int_a^b (y_{\text{top}} - y_{\text{bottom}})\,dx$
Common High-Frequency Areas:
Between $y = x^2$ and $y = x$: Area $= \mathbf{\frac{1}{6}}$
Between $y = x^2$ and $y = 2x - x^2$: Area $= \mathbf{\frac{1}{3}}$
Between $y = x^2$ and $y = 4$: Area $= \mathbf{\frac{32}{3}}$
Example Q: Evaluate definite integral: $\int_0^3 (3x^2 - 2x + 1)\,dx$.
✅ Solution: $[x^3 - x^2 + x]_0^3 = (3^3 - 3^2 + 3) - 0 = 27 - 9 + 3 = \mathbf{21}$.
✅ Solution: $[x^3 - x^2 + x]_0^3 = (3^3 - 3^2 + 3) - 0 = 27 - 9 + 3 = \mathbf{21}$.
1. Two chips are drawn at the same time from chips numbered $1$ to $10$. What is the probability that one is even and one is odd?
Answer: $\frac{5}{9}$ — After picking the first chip, $9$ chips remain and $5$ of them have the opposite parity (odd/even), giving $\frac{5}{9}$.
2. In how many ways can $5$ people sit in a row if $2$ of them refuse to sit next to each other?
Answer: $72$ ways — Total ways $= 5! = 120$. Ways sitting together $= 4! \times 2 = 48$. Subtract: $120 - 48 = 72$.
3. Find the limit: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.
Answer: $6$ — Factor $x^2 - 9 = (x - 3)(x + 3)$, cancel $(x - 3)$, and plug in $3 + 3 = 6$.