Equation of a plane

“Give a wall one point and its perpendicular, and the wall is determined.”

A plane is fixed by a point on it and a normal vector.

Memory trick: Point + normal defines it.

Angle between a line and a plane

“Sunlight hitting a slope makes this very angle.”

Found from the line's direction and the plane's normal.

Memory trick: Use direction vs normal.

Orthogonal matrix

“Rotations are orthogonal — they turn shapes without distorting them.”

A matrix whose transpose is its inverse (AAᵀ = I); it preserves lengths.

Memory trick: Aᵀ = A⁻¹.

Singular matrix

“It collapses space, so it can't be undone.”

A square matrix with zero determinant and no inverse.

Memory trick: det = 0 → singular.

Properties of determinants

“A few row rules can simplify a determinant enormously.”

Swapping two rows changes the sign; a repeated row makes the determinant zero.

Memory trick: Repeated row → zero.

Consistency of linear equations

“Determinants reveal whether equations agree, clash, or overlap.”

A system is consistent if it has at least one solution, judged by determinants/rank.

Memory trick: Check via determinant/rank.

Total probability theorem

“Add up every path that leads to the outcome.”

The probability of an event is the sum over all ways it can occur.

Memory trick: Sum over all cases.

Random variable

“The number rolled on a die is a random variable.”

A variable whose value depends on the outcome of a random experiment.

Memory trick: Maps outcomes to numbers.

Variance of a distribution

“Two bets with equal expected value can differ wildly in risk (variance).”

Measures the spread of a random variable around its mean.

Memory trick: Spread around the mean.

Complement rule

“Sometimes 'at least one' is easiest as 1 minus 'none'.”

P(not A) = 1 − P(A).

P(A')=1−P(A)

Memory trick: Subtract from 1.

Coefficient of variation

“It fairly compares consistency between things measured in different units.”

Standard deviation as a percentage of the mean, for comparing spread across data sets.

Memory trick: SD relative to mean.

Correlation

“Height and weight are positively correlated — taller tends to be heavier.”

Measures how two variables move together, from −1 to +1.

Memory trick: +1 together, −1 opposite.

Contrapositive

“Proving the contrapositive is often easier than the original.”

'If p then q' is logically equivalent to 'if not q then not p'.

Memory trick: Flip and negate both.

Converse and inverse

“A true statement can have a false converse — don't assume otherwise.”

The converse of 'if p then q' is 'if q then p'; they aren't logically equivalent.

Memory trick: Converse ≠ original.

Triangle inequality

“You can't build a triangle from sticks 1, 2 and 10.”

The sum of any two sides of a triangle exceeds the third.

Memory trick: Two sides beat the third.

AM-GM-HM order

“Three means, always in the same pecking order.”

For positive numbers, AM ≥ GM ≥ HM, equal only when all are equal.

Memory trick: AM ≥ GM ≥ HM.

Factorial

“It counts the ways to arrange n distinct objects.”

n! is the product of all positive integers up to n, with 0! defined as 1.

n! = n×(n−1)!

Memory trick: 0! = 1 by definition.

Sum of an arithmetic progression

“Gauss's pairing trick generalised to any AP.”

The sum of n terms is n/2 (first + last) = n/2[2a+(n−1)d].

Sₙ=n/2(a+l)

Memory trick: n/2 × (first + last).

Symmetry of binomial coefficients

“Choosing who's on the team is the same as choosing who's left out.”

nCr = nC(n−r).

nCr=nC(n−r)

Memory trick: Choose r = leave out n−r.

Distance between parallel lines

“The constant gap between two rails that never meet.”

For ax+by+c₁=0 and ax+by+c₂=0, distance = |c₁−c₂|/√(a²+b²).

Memory trick: Difference of constants over √(a²+b²).

Derivative of tan x

“A staple derivative worth memorising outright.”

d/dx(tan x) = sec²x.

d/dx(tanx)=sec²x

Memory trick: tan → sec².

Binomial theorem for any index

“It stretches the binomial idea beyond whole-number powers.”

(1+x)ⁿ for fractional or negative n gives an infinite series valid for |x|<1.

Memory trick: Infinite series when |x|<1.

Discriminant — common mistake

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

A frequent error is thinking D<0 means 'no solution'. In reality, D = b² − 4ac tells the nature of the roots before you even solve: D>0 gives two real distinct roots, D=0 gives equal roots, D<0 gives complex roots.

Memory trick: D<0 still has two roots — they're complex conjugates.

Sum and product of roots — common mistake

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

A frequent error is dropping the minus sign in −b/a. In reality, for ax²+bx+c=0, the roots add to −b/a and multiply to c/a (Vieta's relations).

Memory trick: sum = −b/a, product = c/a — handy for building equations from roots.