EJU Mathematics Course 1 flashcards

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

Quadratic formula

For ax²+bx+c=0 with a≠0, the solutions are x = (−b ± √(b²−4ac))/(2a). Apply it whenever factoring is not obvious.

Discriminant rule

D = b²−4ac. If D>0 there are two distinct real roots, if D=0 one repeated real root, if D<0 no real roots.

Vertex of a parabola

For y = ax²+bx+c the vertex lies at x = −b/(2a). Its y-coordinate is c − b²/(4a), the maximum or minimum value.

Sum and product of roots

If α and β are roots of ax²+bx+c=0, then α+β = −b/a and αβ = c/a. These build a new quadratic from given roots.

Remainder theorem

Dividing polynomial P(x) by (x−k) leaves remainder P(k). If that remainder is zero, (x−k) is a factor of P(x).

Absolute value inequality

For a>0, |x|<a means −a<x<a, while |x|>a means x<−a or x>a. Split into two cases before solving.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

Extraneous roots check

Squaring both sides of an equation can introduce solutions that do not satisfy the original. Substitute every candidate back to verify.

Permutation formula

The number of ordered arrangements of r objects chosen from n distinct objects is nPr = n!/(n−r)!. Order matters in permutations.

Combination formula

The number of unordered selections of r objects from n is nCr = n!/(r!(n−r)!). Note nCr = nC(n−r) and nC0 = 1.

Addition rule of probability

P(A∪B) = P(A)+P(B)−P(A∩B). Subtract the overlap; for mutually exclusive events the intersection term is zero.

Conditional probability

P(B|A) = P(A∩B)/P(A), defined when P(A)>0. Events are independent exactly when P(A∩B) = P(A)P(B).

Complement probability

P(not A) = 1 − P(A). Counting the complement is usually faster for 'at least one' type events.

Expected value

For a discrete random variable, E(X) = Σ x·P(X=x). It is the long-run average value of the variable.

Variance shortcut formula

V(X) = E(X²) − {E(X)}². Compute the mean first, then square it and subtract from the mean of squares.

Mean of grouped data

x̄ = Σ(f·x)/Σf, where f is each frequency and x the corresponding value or class midpoint. The denominator is the total count.

Standard deviation

s = √((1/n)Σ(xᵢ−x̄)²). It measures spread around the mean; squaring deviations first prevents cancellation.

Median and quartiles

Sort the data ascending. The median divides it into halves; the medians of those halves are the first and third quartiles.

Sine rule

a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. Use it with a known side and its opposite angle.

Cosine rule

a² = b²+c²−2bc·cos A. Use it when two sides and the included angle, or all three sides, are known.

Triangle area by sine

S = ½ab·sin C, using two sides and the included angle. Any pair of sides with their included angle works.

Pythagorean identity

sin²θ + cos²θ = 1 for every angle θ. Combined with tan θ = sin θ/cos θ it converts between the three ratios.

Supplementary angle relations

sin(180°−θ) = sin θ and cos(180°−θ) = −cos θ. These reduce obtuse angles to acute reference angles.

Sector arc and area

With central angle θ in radians and radius r: arc length = rθ, sector area = ½r²θ. Convert degrees to radians first.

Sphere volume and area

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². Both formulas appear in solid geometry problems.

Logarithm definition

If a^p = M with a>0 and a≠1, then p = log_a M. Logarithms answer the question: to what power must a be raised?

Product rule for logarithms

log_a(MN) = log_a M + log_a N, and log_a(M/N) = log_a M − log_a N. Valid only when M and N are positive.

Inverse function

f⁻¹ undoes f: f(f⁻¹(x)) = x on the range of f. Its graph is the reflection of y=f(x) across the line y=x.

Distance and midpoint

Between two points the distance is √((x₂−x₁)²+(y₂−y₁)²) and the midpoint is ((x₁+x₂)/2,(y₁+y₂)/2).

Line slope

The slope through two points is (y₂−y₁)/(x₂−x₁). Parallel lines share a slope; perpendicular slopes multiply to −1.

Circle equation

(x−a)²+(y−b)² = r² describes a circle with centre (a,b) and radius r. Expand to read centre and radius from general form.

Simultaneous equations determinant

For ax+by=e and cx+dy=f, a unique solution exists when ad−bc ≠ 0. If ad−bc = 0, solutions are none or infinitely many.