LC Higher Maths flashcards

36 cards. Read the term and the definition here. The deck at the top of this page is optional.

Discriminant of a quadratic

For ax² + bx + c = 0, the value b² − 4ac. Positive gives two distinct real roots, zero gives one repeated real root, negative gives no real roots.

Factor theorem

A polynomial p(x) has (x − a) as a factor exactly when p(a) = 0. Dividing by (x − a) leaves remainder p(a), so a zero remainder confirms the factor.

Change of base formula

log_a x equals log_b x divided by log_b a, for any valid base b. It lets you evaluate logarithms on a calculator, which usually offers base 10 or base e only.

Solving simultaneous linear and quadratic

Isolate one variable in the linear equation, substitute that expression into the quadratic, and solve the resulting quadratic. Each solution pair must satisfy both original equations.

Proof by induction steps

Prove the statement for the base case, assume it holds for n = k, use that assumption to show it holds for n = k + 1, then conclude it holds for all integers from the base onward.

Composite function domain

For f(g(x)), x must lie in the domain of g, and g(x) must lie in the domain of f. Both conditions together give the domain of the composite.

Inverse function condition

A function has an inverse only if it is one-to-one, so each output comes from exactly one input. Its graph is the reflection of the original graph in the line y = x.

Even and odd functions

A function is even when f(−x) = f(x) for all x in its domain, giving symmetry about the y-axis. It is odd when f(−x) = −f(x), giving symmetry about the origin.

Completed square vertex form

Writing a quadratic as y = a(x − h)² + k displays the vertex at (h, k) and the axis of symmetry x = h. The sign of a tells whether the vertex is a maximum or minimum.

Horizontal stretch transformation

Replacing x by bx in y = f(x) stretches the graph horizontally by factor 1/b. Values of b greater than 1 compress the graph toward the y-axis.

Cosine rule

a² = b² + c² − 2bc cos A relates three sides to one angle. Use it with two sides and the included angle, or with all three sides to find an angle.

Sine rule ambiguous case

Given two sides and a non-included angle, the sine rule may produce two candidate angles, θ and 180° − θ. Check whether each candidate leaves a valid triangle.

Sector arc length and area

With angle θ in radians, arc length is rθ and sector area is ½r²θ. Convert degrees to radians by multiplying by π and dividing by 180 first.

Double angle identities

sin 2θ = 2 sin θ cos θ, and cos 2θ = cos²θ − sin²θ, which also equals 1 − 2sin²θ or 2cos²θ − 1. Choose the form matching the other terms present.

Circle equation expanded form

x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius √(g² + f² − c), provided the quantity under the root is positive.

Perpendicular line slopes

Two lines are perpendicular exactly when the product of their slopes is −1, so one slope is the negative reciprocal of the other. Vertical and horizontal lines are also perpendicular.

Perpendicular distance to a line

The distance from (x₁, y₁) to ax + by + c = 0 is |ax₁ + by₁ + c| divided by √(a² + b²). The absolute value ensures a non-negative distance.

Internal section formula

The point dividing the segment from (x₁, y₁) to (x₂, y₂) in ratio m:n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). For a midpoint take m = n.

Angle between two lines

For slopes m₁ and m₂, tan θ = |(m₁ − m₂)/(1 + m₁m₂)| gives the acute angle between the lines. If 1 + m₁m₂ = 0 the lines are perpendicular.

Conditional probability formula

P(A|B) equals P(A ∩ B) divided by P(B), provided P(B) is greater than zero. It restricts the sample space to outcomes where B has occurred.

Combinations notation

ⁿCᵣ = n! / (r!(n − r)!) counts the number of ways to choose r objects from n when order does not matter. It equals the binomial coefficient in expansions.

Bernoulli trial probability

For n independent trials each with success probability p, the probability of exactly r successes is ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ. This is the binomial distribution.

Expected value of discrete variable

E(X) = Σ x·P(X = x), the probability-weighted average of all possible values. Variance is E(X²) − (E(X))², and the standard deviation is its square root.

Sample standard deviation

s = √(Σ(x − x̄)² / (n − 1)), measuring typical spread about the mean. The population version divides by n instead of n − 1 and is never negative.

Interquartile range and outliers

IQR = Q₃ − Q₁, the spread of the middle half of the data. Values below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR are commonly flagged as outliers.

Least squares regression line

Slope b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² and intercept a = ȳ − b·x̄. The line passes through the mean point (x̄, ȳ) of the data.

Effect of adding a constant

Adding the same constant to every data value shifts the mean and median by that constant, but leaves the standard deviation, range, and interquartile range unchanged.

Compound interest formula

A = P(1 + r)ⁿ, where r is the decimal rate per period and n the number of periods. For depreciation, replace (1 + r) with (1 − r).

Similar figure scaling ratios

If corresponding lengths scale by factor k, then areas scale by k² and volumes by k³. This lets you compare surface areas and capacities of similar solids.

Sphere and cone volume

A sphere of radius r has volume 4/3 πr³. A cone with base radius r and perpendicular height h has volume ⅓πr²h, one third of the enclosing cylinder.

Differentiation from first principles

The derivative is the limit as h tends to zero of [f(x + h) − f(x)] divided by h. It represents the slope of the tangent at a point.

Chain rule

The derivative of f(g(x)) is f′(g(x)) multiplied by g′(x). Differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Second derivative test

At a stationary point where f′(x) = 0, a negative second derivative indicates a local maximum and a positive one a local minimum. A zero value needs a sign test.

Definite integral as area

∫ₐᵇ f(x) dx equals F(b) − F(a) for an antiderivative F. It gives the signed area between the curve and the x-axis, with regions below the axis counted negative.

Disproof by counterexample

A single valid case where a universal statement fails is enough to disprove it. State the case, show the statement is false for it, and conclude the statement is not generally true.

Optimisation procedure

Express the quantity as a function of one variable using the constraint, differentiate, set the derivative to zero, solve, and justify the maximum or minimum using the second derivative or an interval test.