139 formulas, theorems, identities and laws across the whole Compulsory Part — each with the conditions it actually needs, and a drill for the topics it powers.
The exam gives you none of these
Neither HKDSE Mathematics paper prints a formula sheet — both go from the instructions page straight into the questions, which we checked against the real papers. Everything here has to be memorised. The statements are standard mathematics, written and independently checked by us; no exam or textbook material is reproduced.
Each digit satisfies 0≤di<n0 less than or equal to d sub i less than n.
A rational number can be written as qpp over q with integers p and q=0q not equal to 0; its decimal terminates or recurs. An irrational number, such as 2square root of 2 or πpi, cannot.
% change=originalnew−original×100%
final=original(1+100a)(1+100b)
A decrease uses a negative rate. Two changes never simply add.
A=P(1+100r)n
r is the rate per period and n the number of periods — halve the rate and double n for half-yearly compounding.
M, N > 0. There is no law for log(M+N)log (M plus N).
logbN=logablogaN
loga1=0,logaa=1,alogax=x
y=axy equals a to the power x passes through (0, 1) and has the x-axis as an asymptote; y=logaxy equals log sub a x passes through (1, 0) and has the y-axis as an asymptote. The two are reflections of each other in the line y = x.
(a−b)(a+b)≡a2−b2
(a±b)2≡a2±2ab+b2
a3±b3≡(a±b)(a2∓ab+b2)
Undo the operations acting on the target variable in reverse order, applying each step to the whole of both sides. Collect every term containing the new subject on one side and factorise before dividing.
Solve by substitution or elimination. Two lines meet in one point unless they are parallel (no solution) or identical (infinitely many).
f(x)÷(x−a) leaves remainder f(a)
For a divisor bx - a, the remainder is f(ba)f ( a over b ).
(x−a) is a factor of f(x)⟺f(a)=0
f(x)=Q(x)D(x)+R(x)
The degree of R(x) is less than the degree of the divisor D(x).
H.C.F.×L.C.M.=product of the two expressions
Holds for two quantities, not for three or more.
ba±dc=bdad±bc,ba÷dc=ba×cd
Factorise first, then cancel common factors — never cancel a term that is part of a sum.
x=2a−b±b2−4ac
For ax2+bx+c=0ax squared plus bx plus c equals 0 with a=0a not equal to 0.
Δ=b2−4ac
Δ>0greater than 0 two distinct real roots; Δ=0equals 0 one double real root; Δ<0less than 0 no real roots.
α+β=−ab,αβ=ac
x2−(α+β)x+αβ=0
y=a(x−h)2+k,h=−2ab,k=c−4ab2
The vertex is (h, k): a minimum when a > 0, a maximum when a < 0.
x=−2ab=2α+β
The second form uses the two x-intercepts, when they exist.
The y-intercept is c. The x-intercepts are the roots, so their number is decided by the discriminant.
f(a) is the value of f at x = a — substitute a for every occurrence of x. f(a+b) is not f(a)+f(b) in general.
yyyyyy=f(x)+k=f(x+h)=−f(x)=f(−x)=kf(x)=f(kx)up kleft hreflect in the x-axisreflect in the y-axisstretch ×k verticallystretch ×k1 horizontally
Horizontal changes act in the opposite sense to the sign inside the bracket.
The domain is the set of permitted inputs. Exclude values that make a denominator zero or put a negative number under an even root.
Adding or subtracting the same quantity keeps the direction. Multiplying or dividing by a positive number keeps it; by a negative number reverses it.
With roots α<βalpha less than beta and a>0:ax2+bx+c<0a greater than 0: ax squared plus bx plus c less than 0 holds between the roots, and > 0 outside them. Sketching the parabola is safer than memorising the direction.
A linear objective function over a bounded feasible polygon attains its maximum and minimum at a vertex, so testing the vertices suffices.
Tn=T1+(n−1)d
Sn=2n(2T1+(n−1)d)
Equivalently Sn=2n(T1+Tn)S sub n equals n over 2 (T sub 1 plus T sub n ) when the last term is known.
Tn=T1rn−1
Sn=1−rT1(1−rn)
r=1r not equal to 1. Use r−1T1(rn−1)T sub 1 (r to the power n minus 1) over r minus 1 when r > 1 to keep both parts positive.
S∞=1−rT1(∣r∣<1)
Only converges for |r| < 1; otherwise the sum does not exist.
Tn+1−Tn=d(arithmetic),TnTn+1=r(geometric)
The difference or ratio must be the same for every n, not just the first pair.
y=kx
k=0k not equal to 0. The graph is a straight line through the origin.
y=xk
k=0,x=0k not equal to 0, x not equal to 0. Plotting y against x11 over x gives a straight line.
z=kxy(joint),y=k1+k2x(partial)
A partial variation needs two conditions to fix both constants.
If y∝xny x to the power n and x is multiplied by t, then y is multiplied by tnt to the power n. For y∝xn1,yy 1 over x to the power n, y is multiplied by tn11 over t to the power n.
i=−1,i2=−1
(a+bi)(c+di)=(ac−bd)+(ad+bc)i,a+bi1=a2+b2a−bi
Divide by multiplying numerator and denominator by the conjugate.
a+bi=c+di⟺a=c and b=d
a, b, c, d real. Complex roots of a real quadratic occur in conjugate pairs.
A=21bh
A∥=bh,Atrap=21(a+b)h
h is the perpendicular height, not a slant side.
C=2πr,A=πr2
l=360∘θ×2πr
A=360∘θ×πr2
V=πr2h
A=2πrh
Add 2πr22 pi r squared for the two ends to get the total surface area.
V=31πr2h
A=πrl,l=r2+h2
l is the slant height, not the vertical height.
V=34πr3
A=4πr2
Vprism=Abaseh,Vpyramid=31Abaseh
V=lwh,A=2(lw+lh+wh)
∠A+∠B+∠C=180∘
An exterior angle equals the sum of the two interior opposite angles.
With a transversal cutting parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior angles are supplementary.
(n−2)×180∘
For a regular polygon each interior angle is n(n−2)180∘(n minus 2)180 degrees over n.
360∘
True for every convex polygon, whatever the number of sides — each exterior angle of a regular n-gon is n360∘360 degrees over n.
at a point=360∘,on a straight line=180∘
Vertically opposite angles are equal.
A2A1=(l2l1)2,V2V1=(l2l1)3
Areas scale as the square of the length ratio, volumes as the cube.
SSS, SAS, ASA, AAS and RHS each prove congruence. SSA does not, except in the right-angled RHS case.
AAA (equiangular), three sides proportional, or two sides proportional with the included angle equal.
Centroid — medians, divides each 2:1 from the vertex. Orthocentre — altitudes. Circumcentre — perpendicular bisectors, equidistant from the vertices. Incentre — angle bisectors, equidistant from the sides.
a2+b2=c2
c is the hypotenuse. The converse also holds: if a2+b2=c2a squared plus b squared equals c squared the triangle is right-angled.
Leading zeros are never significant; trailing zeros after a decimal point are. Round only at the final step — rounding partway through compounds the error.
A line from the centre perpendicular to a chord bisects it, and the line from the centre to a chord's mid-point is perpendicular to it. Equal chords are equidistant from the centre.
∠at centre=2×∠at circumference
Both angles must stand on the same arc.
∠in a semicircle=90∘
The chord must be a diameter.
Angles in the same segment, standing on the same arc, are equal. Equal arcs subtend equal angles, and equal angles stand on equal arcs.
∠A+∠C=180∘
Opposite angles are supplementary, and an exterior angle equals the interior opposite angle. The converse proves four points concyclic.
A tangent is perpendicular to the radius at the point of contact. The two tangents drawn from an external point are equal in length and subtend equal angles at the centre.
The angle between a tangent and a chord equals the angle in the alternate segment.
In the same circle, arcs are proportional to the angles they subtend at the centre. Equal arcs cut off equal chords, and equal chords cut off equal arcs.
d=(x2−x1)2+(y2−y1)2
(2x1+x2,2y1+y2)
(r+srx2+sx1,r+sry2+sy1)
For the point dividing AB internally in the ratio r : s — note the coordinates cross over.
m=x2−x1y2−y1
Also m=tanθm equals tangent theta, where θtheta is the angle the line makes with the positive x-axis.
m1=m2(parallel),m1m2=−1(perpendicular)
The perpendicular rule fails when one line is vertical.
y−y1=m(x−x1),y=mx+c,ax+by=1,Ax+By+C=0
For Ax+By+C=0 the slope is −BAnegative A over B.
d=A2+B2∣Ax0+By0+C∣
For the line Ax+By+C=0 and the point (x0,y0)(x sub 0, y sub 0 ).
(x−h)2+(y−k)2=r2
Centre (h, k), radius r.
x2+y2+Dx+Ey+F=0,centre(−2D,−2E),r=4D2+E2−F
A real circle needs 4D2+E2−F>0D squared plus E squared over 4 minus F greater than 0; it is a point when the expression is 0 and imaginary when negative.
Substitute the line into the circle and read the discriminant: Δ>0greater than 0 two intersections, Δ=0equals 0 tangent, Δ<0less than 0 no intersection.
Equidistant from two points — the perpendicular bisector. Equidistant from two intersecting lines — the pair of angle bisectors. Fixed distance from a point — a circle. Fixed distance from a line — a pair of parallel lines.
sinθ=hypopp,cosθ=hypadj,tanθ=adjopp
Right-angled triangles only.
sin30∘=21,sin45∘=21,sin60∘=23,tan45∘=1
tanθ=cosθsinθ
cosθ=0cosine theta not equal to 0.
sin2θ+cos2θ=1
All ratios are positive in the first quadrant; only sine in the second, only tangent in the third, only cosine in the fourth (ASTC).
Use for AAS/ASA, or SSA — the ambiguous case, where two triangles may satisfy the data.
c2=a2+b2−2abcosC
Use for SAS or SSS. Rearranged: cosC=2aba2+b2−c2cosine C equals a squared plus b squared minus c squared over 2ab.
Area=21absinC
C must be the angle between the two sides.
s=2a+b+c,Area=s(s−a)(s−b)(s−c)
Use when all three sides are known and no angle is.
Angles of elevation and depression are measured from the horizontal. A true bearing is measured clockwise from north as a three-figure angle; a compass bearing is given as N/S, an acute angle, then E/W.
The angle between a line and a plane is the angle between the line and its projection on the plane. The angle between two planes is measured along lines in each plane perpendicular to their line of intersection.
sinxsine x and cosxcosine x have period 360∘360 degrees and range [-1, 1]; tanxtangent x has period 180∘180 degrees and asymptotes at 90∘+180∘k90 degrees plus 180 degrees k.
The five-number summary: minimum, Q1Q sub 1, median, Q3Q sub 3, maximum. The box spans the interquartile range and the line inside it is the median, not the mean.
Read the median at the half-way height, and Q1Q sub 1 and Q3Q sub 3 at the quarter and three-quarter heights. Plot cumulative frequency against the upper class boundary.
A scatter diagram shows the direction and strength of a relationship between two variables. Correlation does not establish causation.
The class mark is the mean of the two class limits; the class boundaries lie half a unit outside the limits. Class width is the difference between the boundaries.
A truncated vertical axis exaggerates differences, an unrepresentative sample biases the conclusion, and the mean is pulled by outliers where the median is not.
xˉ=n∑xi,xˉ=∑fi∑fixi
The second form is for grouped data, where xix sub i is the class mark.
The median is the middle value once the data is sorted — the mean of the two middle values when n is even. The mode is the most frequent value; a set may have none or several.
xˉw=∑wi∑wixi
Combining two groups uses their sizes as the weights, not the plain mean of the two means.
Adding a constant c to every value raises the mean, median and mode by c but leaves the range, interquartile range and standard deviation unchanged. Multiplying every value by k multiplies all of them, including the measures of spread, by |k|.
range=max−min,IQR=Q3−Q1
The IQR ignores the extreme quarter at each end, so outliers do not affect it.
σ=N∑(xi−μ)2
For grouped data use σ=∑fi∑fi(xi−μ)2sigma equals square root of f sub i (x sub i minus mu ) squared over f sub i. The variance is σ2sigma squared.
z=σx−μ
A z-score has mean 0 and standard deviation 1, so it compares values from different distributions.
About 68% of the data lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three.
P(E)=total number of outcomesnumber of favourable outcomes
Requires equally likely outcomes. 0≤0 less than or equal to P(E) ≤1less than or equal to 1.
P(E′)=1−P(E)
Often far quicker than counting the event directly — especially for "at least one".
P(A∪B)=P(A)+P(B)−P(A∩B)
For mutually exclusive events P(A∩P(A B)=0, so the law reduces to P(A)+P(B).
P(A∩B)=P(A)P(B)
Only for independent events — drawing without replacement is not independent.
P(A∩B)=P(A)P(B∣A)
P(B∣P(B A) is the probability of B given that A has happened — the form to use for drawing without replacement.
E(X)=∑xiP(xi)
A game is fair when the expected gain is 0.
A tabulation suits two stages, a tree diagram suits several, and both guarantee every outcome is counted exactly once.
If tasks are alternatives, add the counts; if they are performed in sequence, multiply them.
Prn=(n−r)!n!
Order matters. n! =n(n−1)⋯2⋅1equals n(n minus 1) 2 times 1 and 0! = 1.
Crn=r!(n−r)!n!
Order does not matter. Crn=Cn−rnC to the power n sub r equals C to the power n sub n minus r.
(n−1)!
Seating n distinct objects round a circle, where only relative position counts.
p!q!⋯n!
Where p, q, … are the counts of each repeated object.
Compiled by Dsetopia against the EDB Mathematics curriculum. Statements are standard mathematical results, independently verified. Not affiliated with the HKEAA or the EDB.