Determinant

“A zero determinant means a transformation squashes space flat.”

A single number from a square matrix; if it's zero, the matrix has no inverse.

Memory trick: det = 0 → no inverse.

Probability basics

“From weather to cricket, uncertainty is measured on this 0-to-1 scale.”

Probability = favourable outcomes ÷ total outcomes, always between 0 and 1.

P=favourable/total

Memory trick: Never below 0 or above 1.

Independent vs mutually exclusive

“Mutually exclusive events are actually dependent — one rules the other out.”

Independent events don't affect each other; mutually exclusive events can't both happen.

Memory trick: Exclusive ≠ independent.

Conditional probability

“Medical tests are interpreted with conditional probability.”

P(A|B) = P(A∩B)/P(B) is the chance of A given B has occurred.

P(A|B)=P(A∩B)/P(B)

Memory trick: Given B, rescale by P(B).

Bayes' theorem

“Spam filters use Bayes to keep learning what junk looks like.”

Updates a probability when new evidence arrives, flipping P(A|B) into P(B|A).

Memory trick: Reverse the condition correctly.

Mean, median, mode

“A few billionaires can drag the mean income far above the typical (median) person.”

Mean is the average, median the middle value, mode the most frequent.

Memory trick: Skewed data → trust the median.

Standard deviation

“Two classes can share an average yet differ wildly in consistency.”

A measure of how spread out data is around the mean.

Memory trick: Bigger SD = more spread.

Set

“Your phone's contact list is a set — no duplicates allowed.”

A well-defined collection of distinct objects.

Memory trick: Elements are unique.

Union and intersection

“Union is 'either'; intersection is 'both' — like overlapping friend circles.”

Union (A∪B) gathers all elements of either set; intersection (A∩B) keeps only the common ones.

Memory trick: ∪ = OR, ∩ = AND.

Subset

“A team is a subset of the whole class.”

A is a subset of B if every element of A is in B.

A⊆B

Memory trick: All of A sits inside B.

De Morgan's laws

“Negating an 'or' flips it to an 'and' — a rule programmers use daily.”

(A∪B)' = A'∩B' and (A∩B)' = A'∪B'.

Memory trick: Complement flips ∪ and ∩.

Power set

“Three toppings give 8 possible pizzas — that's the power set.”

The set of all subsets of a set; a set with n elements has 2ⁿ subsets.

|P(A)|=2ⁿ

Memory trick: n elements → 2ⁿ subsets.

Types of relations

“'Is the same age as' is an equivalence relation — it sorts people into age groups.”

A relation can be reflexive, symmetric or transitive; all three make it an equivalence relation.

Memory trick: Reflexive+symmetric+transitive = equivalence.

Even and odd functions

“cos is even, sin is odd — their graphs mirror differently.”

Even functions satisfy f(−x)=f(x) (symmetric about y-axis); odd satisfy f(−x)=−f(x) (symmetric about origin).

Memory trick: Even = y-axis mirror; odd = origin flip.

Inverse function

“Logarithm undoes exponentiation, like subtraction undoes addition.”

Undoes a function; it exists only if the function is one-one and onto (bijective).

Memory trick: Only bijections are invertible.

Modulus function

“Distance can't be negative — that's the modulus in one idea.”

|x| gives the distance of x from zero, always non-negative.

|x|

Memory trick: |x| ≥ 0 always.

Greatest integer function

“[2.9] = 2 — it always rounds down to the nearest integer.”

[x] gives the greatest integer not exceeding x (the 'floor').

[x]

Memory trick: Floor: round down.

Exponential function

“Bacteria, viruses and compound interest all grow exponentially.”

f(x)=a^x (a>0) grows or decays by a constant factor.

a^x

Memory trick: Constant ratio per step.

Logarithm laws

“Logarithms turn hard multiplication into easy addition.”

log(mn)=log m+log n; log(m/n)=log m−log n; log(m^p)=p·log m.

Memory trick: Log turns × into +.

Standard trig values

“These few values unlock most trigonometry problems.”

sin30°=½, sin45°=1/√2, sin60°=√3/2, sin90°=1.

Memory trick: Learn 0,30,45,60,90 cold.

Double-angle formulas

“They let you shrink a doubled angle back to a single one.”

sin2θ=2sinθcosθ; cos2θ=cos²θ−sin²θ=1−2sin²θ.

sin2θ=2sinθcosθ

Memory trick: cos2θ has three handy forms.

Tan of an angle

“tan is the slope of the line at angle θ.”

tanθ = sinθ/cosθ; undefined at 90° where cos is zero.

tanθ=sinθ/cosθ

Memory trick: tan = sin/cos.

Trigonometric ratios of complementary angles

“'Co' in cosine literally means the complement of sine.”

sin(90°−θ)=cosθ and cos(90°−θ)=sinθ.

Memory trick: Swap sin↔cos at 90°−θ.

General solution of trig equations

“Trig equations have infinitely many solutions — these formulas list them all.”

sinθ=sinα → θ=nπ+(−1)ⁿα; cosθ=cosα → θ=2nπ±α.

Memory trick: Add the periodic family, not one answer.