STPM Mathematics 954 flashcards

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

Discriminant

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

Remainder Theorem

When polynomial P(x) is divided by (x−a), the remainder equals P(a). This holds for any polynomial and any real constant a.

Factor Theorem

The linear expression (x−a) is a factor of polynomial P(x) if and only if P(a)=0. Used to find unknown coefficients and factorise cubics.

Inverse function condition

A function f has an inverse f⁻¹ only if it is one-to-one, meaning each output comes from exactly one input. The graph of f⁻¹ is the reflection of f in y=x.

Logarithm quotient law

For positive x and y, log_a(x/y)=log_a x−log_a y. Combined with the product and power laws it allows expansion or compression of logarithmic expressions.

Modulus inequality rule

The inequality |x|<a, for a>0, is equivalent to −a<x<a. The inequality |x|>a gives x<−a or x>a.

Chain rule

If y=f(u) and u=g(x), then dy/dx=(dy/du)(du/dx). It differentiates composite functions and underlies integration by substitution.

Quotient rule

For differentiable u and v with v≠0, d/dx(u/v)=(u'v−uv')/v². Remember the numerator order: derivative of top times bottom minus top times derivative of bottom.

Second derivative test

At a stationary point where dy/dx=0, d²y/dx²>0 indicates a local minimum and d²y/dx²<0 indicates a local maximum. If it equals zero, further investigation is needed.

Integration by substitution

Replace u=g(x) so du=g'(x)dx, converting ∫f(g(x))g'(x)dx into ∫f(u)du. For definite integrals, also change the limits to values of u.

Volume of revolution

Rotating y=f(x) about the x-axis between x=a and x=b gives volume V=π∫_a^b y² dx. About the y-axis use V=π∫ x² dy.

Trapezium rule

Approximates ∫_a^b f(x)dx by (h/2)[y₀+2(y₁+…+y_(n−1))+y_n], where h=(b−a)/n and y_i are ordinates at equally spaced points.

Dot product

For vectors a and b, a·b=a₁b₁+a₂b₂+a₃b₃=|a||b|cos θ. The dot product is zero exactly when the two vectors are perpendicular.

Cross product magnitude

The vector product a×b has magnitude |a||b|sin θ and direction perpendicular to both a and b by the right-hand rule. It is zero when a and b are parallel.

Vector line equation

A straight line is r=a+tb, where a is the position vector of a point on the line, b is a direction vector, and t is a real parameter.

Plane equation

A plane can be written r·n=d, where n is a normal vector and d is a constant. In Cartesian form this is ax+by+cz=d with normal (a,b,c).

Angle between line and plane

If b is the line direction and n the plane normal, the angle θ satisfies sin θ=(b·n)/(|b||n|). The complementary angle uses cos with the normal.

Sample variance formula

For data x₁,…,x_n with mean x̄, sample variance is s²=Σ(x−x̄)²/(n−1). The divisor n−1 gives an unbiased estimator of population variance.

Interquartile range

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

Pearson correlation coefficient

r=Σ(x−x̄)(y−ȳ)/√[Σ(x−x̄)²Σ(y−ȳ)²]. It measures strength and direction of linear association and always lies between −1 and 1.

Least squares regression line

The line y=a+bx minimising squared vertical deviations has b=Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and a=ȳ−bx̄. It always passes through the mean point (x̄,ȳ).

Spearman rank correlation

r_s=1−6Σd²/[n(n²−1)], where d is the difference between the ranks of each pair. It assesses monotonic association and ranges from −1 to 1.

Conditional probability

P(A|B)=P(A∩B)/P(B) for P(B)>0. Rearranged, it gives the multiplication rule P(A∩B)=P(B)P(A|B) used for dependent events.

Independence of events

Events A and B are independent when P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A). Knowing one event occurs does not change the probability of the other.

Binomial distribution

X~B(n,p) counts successes in n independent trials with constant success probability p. P(X=r)=C(n,r)p^r(1−p)^(n−r), with mean np and variance np(1−p).

Poisson distribution

X~Po(λ) models counts of rare events with mean λ. P(X=r)=e^(−λ)λ^r/r! and mean equals variance equals λ. It approximates binomial when n is large and p small.

Standard normal variable

If X~N(μ,σ²), then Z=(X−μ)/σ follows N(0,1). Probabilities are found from Φ(z)=P(Z≤z), with P(Z>z)=1−Φ(z) by symmetry.

Normal approximation to binomial

When np>5 and n(1−p)>5, B(n,p) is approximately normal with mean np and variance np(1−p). Apply a continuity correction of ±0.5 to discrete values.

Null and alternative hypotheses

H₀ states no change or no difference; H₁ states the effect being tested, one- or two-tailed. The decision compares the test statistic with a critical value at significance level α.

Type I and Type II errors

A Type I error rejects a true H₀, with probability α. A Type II error fails to reject a false H₀, with probability β. Power equals 1−β.

One-sample t-test

Tests a mean when σ is unknown using t=(x̄−μ₀)/(s/√n) with n−1 degrees of freedom. It assumes the underlying population is approximately normal.

Chi-squared goodness-of-fit

Compares observed and expected frequencies by χ²=Σ(O−E)²/E. Degrees of freedom are k−1−m, where k is the number of categories and m the number of estimated parameters.