Formula Sheet
Every key formula of the course on one page, grouped by chapter, with a link back to the section that develops it.
Chapter 1 · Real Numbers
Fundamental Theorem of Arithmetic§ 1.1 →
Every composite $n$: $\;n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ with primes $p_1\lt p_2\lt\cdots\lt p_k$, in exactly one way (apart from order)
Prime factorisation (example)§ 1.1 →
$360=2^3\times3^2\times5,\quad 4620=2^2\times3\times5\times7\times11$
Is $n$ prime?§ 1.1 →
Test only the primes $p$ with $p^2\le n$; if none divides $n$, then $n$ is prime
Ending in the digit 0§ 1.1 →
$N$ ends in $0$ $\iff$ $2$ and $5$ both appear in its prime factorisation (e.g. $6^n=2^n3^n$ never does)
HCF by prime factorisation§ 1.2 →
$\operatorname{HCF}=$ product of the smallest power of each prime common to all the numbers (no common prime: HCF $=1$)
LCM by prime factorisation§ 1.2 →
$\operatorname{LCM}=$ product of the greatest power of every prime that appears in any of the numbers
HCF × LCM (two numbers only)§ 1.2 →
$\operatorname{HCF}(a,b)\times\operatorname{LCM}(a,b)=a\times b$ for natural numbers $a,b$ — not true in general for three numbers
HCF or LCM?§ 1.2 →
Largest equal pieces / most identical groups → HCF; repeating events together / smallest common amount → LCM
Rational and irrational§ 1.3 →
$r$ rational $\iff r=\tfrac pq$ with integers $p,q$, $q\ne0$ (and we may take $p,q$ co-prime); irrational $=$ not rational
Prime dividing a square§ 1.3 →
$p$ prime, $a$ a positive integer: if $p$ divides $a^2$ then $p$ divides $a$ (false for composite $p$: $4$ divides $6^2$)
Square roots of primes§ 1.3 →
$\sqrt p$ is irrational for every prime $p$: $\sqrt2,\ \sqrt3,\ \sqrt5,\ \sqrt7,\dots$
Rational with irrational§ 1.3 →
$r$ rational, $s$ irrational: $r\pm s$ irrational; if $r\ne0$, $rs$ and $\tfrac sr$ irrational. Two irrationals can give a rational: $\sqrt2\times\sqrt8=4$
Proving $a+b\sqrt p$ irrational§ 1.3 →
Assume $a+b\sqrt p=\tfrac mn$ ($a,b$ rational, $b\ne0$) $\Rightarrow \sqrt p=\tfrac{m-an}{bn}$ is rational — contradiction
Chapter 2 · Polynomials
Types of polynomial§ 2.1 →
Linear $ax+b$, quadratic $ax^2+bx+c$, cubic $ax^3+bx^2+cx+d$ — always with $a\ne0$
Zero of a polynomial§ 2.1 →
$k$ is a zero of $p(x)$ $\iff$ $p(k)=0$ $\iff$ the graph of $y=p(x)$ passes through $(k, 0)$
Zero of a linear polynomial§ 2.1 →
$ax+b$ ($a\ne0$) has exactly one zero: $\;x=-\dfrac ba$
How many zeroes?§ 2.1 →
Degree $n$ ($n\ge1$): at most $n$ zeroes. Quadratic: $0$, $1$ or $2$ meeting points with the $x$-axis; cubic: $1$, $2$ or $3$
Zeroes and coefficients of a quadratic§ 2.2 →
Zeroes $\alpha,\beta$ of $ax^2+bx+c$ ($a\ne0$): $\quad\alpha+\beta=-\dfrac ba,\qquad\alpha\beta=\dfrac ca$
Quadratic from sum and product§ 2.2 →
Zeroes with sum $S$ and product $P$: $\;k\left[x^2-Sx+P\right]$, any $k\ne0$
Expressions in $\alpha$ and $\beta$§ 2.2 →
$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta,\quad(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta,\quad\dfrac1\alpha+\dfrac1\beta=\dfrac{\alpha+\beta}{\alpha\beta}$
Sum of cubes of the zeroes§ 2.2 →
$\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)$
Chapter 3 · Pair of Linear Equations in Two Variables
Pair of linear equations (general form)§ 3.1 →
$a_1x+b_1y+c_1=0,\ \ a_2x+b_2y+c_2=0$ ($a_1,b_1$ not both $0$; $a_2,b_2$ not both $0$)
Intersecting lines — exactly one solution§ 3.1 →
$\dfrac{a_1}{a_2}\ne\dfrac{b_1}{b_2}$ ⇒ lines intersect, unique solution, consistent ($a_2,b_2,c_2\ne0$)
Coincident lines — infinitely many solutions§ 3.1 →
$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$ ⇒ lines coincide, infinitely many solutions, dependent ($a_2,b_2,c_2\ne0$)
Parallel lines — no solution§ 3.1 →
$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne\dfrac{c_1}{c_2}$ ⇒ lines parallel, no solution, inconsistent ($a_2,b_2,c_2\ne0$)
Substitution method§ 3.2 →
From one equation write $y$ (or $x$) in terms of the other variable; substitute into the other equation; solve; back-substitute; check in both equations. Example: $y=7-x$ into $x-y=1$ gives $x-(7-x)=1$, so $x=4$, $y=3$.
What substitution tells you§ 3.2 →
One root (e.g. $7x=13$) ⇒ unique solution · false statement (e.g. $12=9$) ⇒ no solution · true statement (e.g. $-4=-4$) ⇒ infinitely many solutions
Clearing fractions and decimals§ 3.2 →
Multiply every term of an equation by the same non-zero number (e.g. the LCM of the denominators, or $10$); the solutions do not change.
Elimination method§ 3.3 →
Multiply the equations (every term, by non-zero numbers) so one variable's coefficients match; same signs → subtract, opposite signs → add; solve; substitute back; check. Example: $(x+y)+(x-y)=7+1\Rightarrow 2x=8$.
Operations that keep the solutions§ 3.3 →
Multiplying an equation by $k\ne0$, or replacing it by its sum/difference with a multiple of the other equation, gives a pair with the same solutions.
Two-digit numbers§ 3.3 →
Number $10x+y$, reversed $10y+x$ (tens digit $x$, units digit $y$); difference $9(y-x)$, sum $11(x+y)$
Upstream and downstream§ 3.3 →
Downstream speed $u+v$, upstream speed $u-v$ ($u$ = speed in still water, $v$ = stream, $u\gt v$)
Chapter 4 · Quadratic Equations
Quadratic equation (standard form)§ 4.1 →
$ax^2+bx+c=0$, with $a,b,c$ real and $a\ne0$ — simplify first: if the $x^2$ terms cancel, it is not quadratic
Root of a quadratic equation§ 4.1 →
$\alpha$ is a root of $ax^2+bx+c=0$ $\iff$ $a\alpha^2+b\alpha+c=0$ (the roots are the zeroes of $ax^2+bx+c$)
At most two real roots§ 4.1 →
Real roots of $ax^2+bx+c=0$ ($a\ne0$) $=$ $x$-coordinates where $y=ax^2+bx+c$ meets the $x$-axis: two, one or none
Modelling checklist§ 4.1 →
Name $x$ with its unit → express the other quantities in $x$ → write the condition → clear fractions → standard form → after solving, reject roots that make no sense (negative lengths, speeds, numbers of people)
Zero-product rule§ 4.2 →
$pq=0\iff p=0$ or $q=0$ — the other side must be $0$ before you use it
Roots from the factors§ 4.2 →
$(px+q)(rx+s)=0$ with $p,r\ne0$ $\Rightarrow$ $x=-\tfrac qp$ or $x=-\tfrac sr$
Splitting the middle term§ 4.2 →
For $ax^2+bx+c$ find $m,n$ with $m+n=b$ and $mn=ac$; write $bx=mx+nx$ and group. If $ac\gt0$, $m,n$ have the sign of $b$; if $ac\lt0$, opposite signs, the larger has the sign of $b$
Repeated root§ 4.2 →
$(px+q)^2=0$ $\Rightarrow$ $x=-\tfrac qp,\ -\tfrac qp$ (e.g. $9x^2-12x+4=(3x-2)^2$)
Quadratic formula§ 4.3 →
$ax^2+bx+c=0$ with $a\ne0$ and $b^2-4ac\ge0$ $\Rightarrow$ $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant§ 4.3 →
$D=b^2-4ac$ for $ax^2+bx+c=0$ ($a\ne0$)
Nature of roots§ 4.3 →
$D\gt0$: two distinct real roots · $D=0$: two equal real roots · $D\lt0$: no real roots
Equal roots§ 4.3 →
$D=0$ $\Rightarrow$ $x=-\tfrac{b}{2a},\ -\tfrac{b}{2a}$
Condition for real roots§ 4.3 →
Real roots $\iff D\ge0$ (equal roots included); keep $a\ne0$ when $a$ contains $k$
Chapter 5 · Arithmetic Progressions
Common difference of an AP§ 5.1 →
$d=a_2-a_1=a_3-a_2=\cdots=a_{k+1}-a_k$ (the same for every $k$)
General form of an AP§ 5.1 →
$a,\ a+d,\ a+2d,\ a+3d,\ \dots$
Three numbers in AP§ 5.1 →
$a,\ b,\ c$ in AP $\iff 2b=a+c$, i.e. $b=\tfrac{a+c}2$
Choosing unknown terms of an AP§ 5.1 →
Three terms: $a-d,\ a,\ a+d$ (sum $3a$); four terms: $a-3d,\ a-d,\ a+d,\ a+3d$ (sum $4a$, common difference $2d$)
The $n$th term of an AP§ 5.2 →
$a_n=a+(n-1)d,\quad n=1,2,3,\dots$
Last term of an AP with $m$ terms§ 5.2 →
$l=a+(m-1)d,\qquad m=\dfrac{l-a}{d}+1\ \ (d\ne0)$
Is $k$ a term? Which term?§ 5.2 →
Solve $a+(n-1)d=k$: $\;k$ is a term only if $n=\dfrac{k-a}{d}+1$ is a positive whole number ($d\ne0$)
The $n$th term from the end§ 5.2 →
$l-(n-1)d$ $\;=$ the $(m-n+1)$th term from the start
$d$ from two terms§ 5.2 →
$d=\dfrac{a_q-a_p}{q-p}\quad(p\ne q)$
Sum of the first $n$ terms of an AP§ 5.3 →
$S_n=\dfrac n2\left[2a+(n-1)d\right]$
Sum from the first and last terms§ 5.3 →
$S_n=\dfrac n2\,(a+l),\qquad l=a+(n-1)d$
Special sums§ 5.3 →
$1+2+\dots+n=\dfrac{n(n+1)}{2}$$1+3+\dots+(2n-1)=n^2$$2+4+\dots+2n=n(n+1)$
A term from the sums§ 5.3 →
$a_1=S_1,\qquad a_n=S_n-S_{n-1}\ \ (n\ge2)$
Number of terms from a sum§ 5.3 →
Solve $\tfrac n2\left[2a+(n-1)d\right]=S$ (a quadratic in $n$); keep only positive whole-number roots — with $d\lt0$ there can be two
Chapter 6 · Triangles
Similar polygons§ 6.1 →
Same number of sides, corresponding angles equal and corresponding sides in the same ratio (both conditions needed)
Similar triangles§ 6.1 →
$\triangle ABC\sim\triangle PQR\iff\angle A=\angle P,\ \angle B=\angle Q,\ \angle C=\angle R$ and $\dfrac{AB}{PQ}=\dfrac{BC}{QR}=\dfrac{CA}{RP}$
Scale factor§ 6.1 →
$k=\dfrac{\text{side of second figure}}{\text{corresponding side of first}}$; lengths and perimeters $\times k$, angles unchanged
Always similar§ 6.1 →
Any two circles · any two squares · any two equilateral triangles · regular polygons with equal numbers of sides
Basic Proportionality Theorem§ 6.2 →
$D$ on $AB$, $E$ on $AC$ (not vertices), $DE\parallel BC$ $\Rightarrow$ $\dfrac{AD}{DB}=\dfrac{AE}{EC}$
Converse of the BPT§ 6.2 →
$D$ on $AB$, $E$ on $AC$ and $\dfrac{AD}{DB}=\dfrac{AE}{EC}$ $\Rightarrow$ $DE\parallel BC$
Equivalent forms§ 6.2 →
$\dfrac{AD}{AB}=\dfrac{AE}{AC}$ · $\dfrac{DB}{AB}=\dfrac{EC}{AC}$ · $\dfrac{AB}{AD}=\dfrac{AC}{AE}$ (when $DE\parallel BC$)
Midpoints§ 6.2 →
$D,E$ midpoints of $AB,AC$ $\iff$ $DE\parallel BC$ with $E$ midpoint
Trapezium diagonals§ 6.2 →
$AB\parallel DC$, diagonals meet at $O$: $\dfrac{AO}{OC}=\dfrac{BO}{OD}$
AA criterion§ 6.3 →
$\angle A=\angle D$ and $\angle B=\angle E$ $\Rightarrow$ $\triangle ABC\sim\triangle DEF$
SSS criterion§ 6.3 →
$\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{CA}{FD}$ $\Rightarrow$ $\triangle ABC\sim\triangle DEF$ (match shortest with shortest)
SAS criterion§ 6.3 →
$\angle A=\angle D$ and $\dfrac{AB}{DE}=\dfrac{AC}{DF}$ (sides including the angle) $\Rightarrow$ $\triangle ABC\sim\triangle DEF$
Parallel line gives a similar triangle§ 6.3 →
$DE\parallel BC$ $\Rightarrow$ $\triangle ADE\sim\triangle ABC$, so $\dfrac{AD}{AB}=\dfrac{AE}{AC}=\dfrac{DE}{BC}$
Crossing ropes§ 6.3 →
Poles $a$, $b$, ropes crossing at height $h=\dfrac{ab}{a+b}$
Chapter 7 · Coordinate Geometry
Distance formula§ 7.1 →
$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$P(x_1,y_1),\ Q(x_2,y_2)$
Distance from the origin§ 7.1 →
$OP=\sqrt{x^2+y^2}$
Points on a horizontal or vertical line§ 7.1 →
same ordinate: $|x_2-x_1|$same abscissa: $|y_2-y_1|$
Collinearity test§ 7.1 →
$A,B,C$ collinear $\iff$ largest of $AB,BC,CA$ = sum of the other two
Shape tests with squared distances§ 7.1 →
right angle at $B$: $AB^2+BC^2=CA^2$ · rhombus: four equal sides · rectangle: parallelogram with equal diagonals · square: four equal sides and equal diagonals · $P$ equidistant from $A,B$: $PA^2=PB^2$
Section formula (internal division)§ 7.2 →
$P=\left(\dfrac{m_1x_2+m_2x_1}{m_1+m_2},\ \dfrac{m_1y_2+m_2y_1}{m_1+m_2}\right)$$AP:PB=m_1:m_2$, with $m_1,m_2\gt0$
Midpoint§ 7.2 →
$M=\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$
Trisection points§ 7.2 →
ratios $1:2$ and $2:1$$\left(\dfrac{x_2+2x_1}{3},\ \dfrac{y_2+2y_1}{3}\right)$ and $\left(\dfrac{2x_2+x_1}{3},\ \dfrac{2y_2+y_1}{3}\right)$
Ratio in which an axis divides $AB$§ 7.2 →
$x$-axis: $k=-\dfrac{y_1}{y_2}$$y$-axis: $k=-\dfrac{x_1}{x_2}$ratio $k:1$; valid only if $k\gt0$
Missing endpoint§ 7.2 →
$M$ midpoint of $AB$: $B=2M-A$parallelogram $ABCD$: midpoint of $AC$ = midpoint of $BD$
Chapter 8 · Introduction to Trigonometry
Sine, cosine, tangent§ 8.1 →
$\sin A=\dfrac{\text{opp}}{\text{hyp}}$ · $\cos A=\dfrac{\text{adj}}{\text{hyp}}$ · $\tan A=\dfrac{\text{opp}}{\text{adj}}$ ($A$ acute, right angle elsewhere)
Reciprocal ratios§ 8.1 →
$\operatorname{cosec}A=\dfrac1{\sin A}$ · $\sec A=\dfrac1{\cos A}$ · $\cot A=\dfrac1{\tan A}$
Quotient form§ 8.1 →
$\tan A=\dfrac{\sin A}{\cos A}$, $\ \cot A=\dfrac{\cos A}{\sin A}$
Size of the ratios ($A$ acute)§ 8.1 →
$0\lt\sin A\lt1$, $0\lt\cos A\lt1$, $\sec A\gt1$, $\operatorname{cosec}A\gt1$
Same angle, same ratios§ 8.1 →
Right triangles with equal acute angle $A$ have equal $\sin A,\cos A,\tan A$
Table of standard angles§ 8.2 →
$\begin{array}{c|ccccc}A&0^\circ&30^\circ&45^\circ&60^\circ&90^\circ\\\hline\sin A&0&\tfrac12&\tfrac1{\sqrt2}&\tfrac{\sqrt3}2&1\\\cos A&1&\tfrac{\sqrt3}2&\tfrac1{\sqrt2}&\tfrac12&0\\\tan A&0&\tfrac1{\sqrt3}&1&\sqrt3&\text{n.d.}\end{array}$ (n.d. = not defined)
Reciprocal ratios of standard angles§ 8.2 →
$\operatorname{cosec}30^\circ=2,\ \sec30^\circ=\tfrac2{\sqrt3},\ \cot30^\circ=\sqrt3$ · $\operatorname{cosec}45^\circ=\sec45^\circ=\sqrt2,\ \cot45^\circ=1$ · $\operatorname{cosec}60^\circ=\tfrac2{\sqrt3},\ \sec60^\circ=2,\ \cot60^\circ=\tfrac1{\sqrt3}$
$45^\circ$–$45^\circ$–$90^\circ$ triangle§ 8.2 →
sides $1:1:\sqrt2$ (half of a square)
$30^\circ$–$60^\circ$–$90^\circ$ triangle§ 8.2 →
sides $1:\sqrt3:2$ (opposite $30^\circ$, opposite $60^\circ$, hypotenuse)
Not defined§ 8.2 →
$\tan90^\circ$, $\sec90^\circ$, $\operatorname{cosec}0^\circ$, $\cot0^\circ$
Fundamental identity§ 8.3 →
$\sin^2A+\cos^2A=1$ (every acute angle $A$)
Secant identity§ 8.3 →
$1+\tan^2A=\sec^2A$ (divide the fundamental identity by $\cos^2A$)
Cosecant identity§ 8.3 →
$1+\cot^2A=\operatorname{cosec}^2A$ (divide the fundamental identity by $\sin^2A$)
Solved-for forms§ 8.3 →
$\sin^2A=(1-\cos A)(1+\cos A)$, $\ \cos^2A=(1-\sin A)(1+\sin A)$
Product forms§ 8.3 →
$(\sec A-\tan A)(\sec A+\tan A)=1$, $\ (\operatorname{cosec}A-\cot A)(\operatorname{cosec}A+\cot A)=1$
Chapter 9 · Some Applications of Trigonometry
Elevation = depression§ 9.1 →
Angle of depression of $B$ from $A$ = angle of elevation of $A$ from $B$ (both measured from horizontal lines)
Right-triangle toolkit§ 9.1 →
$\sin\theta=\dfrac{h}{l}$ · $\cos\theta=\dfrac{x}{l}$ · $\tan\theta=\dfrac{h}{x}$$h$ vertical, $x$ horizontal, $l$ line of sight, $\theta$ measured from the horizontal
Tangent of standard angles§ 9.1 →
$\tan30^\circ=\tfrac1{\sqrt3}$, $\tan45^\circ=1$, $\tan60^\circ=\sqrt3$
Eye height§ 9.1 →
object height $=x\tan\theta+{}$eye height
Two points, same side§ 9.2 →
$h=\dfrac{d\tan\alpha\tan\beta}{\tan\beta-\tan\alpha}$, $PB=\dfrac{h}{\tan\beta}$, $QB=\dfrac{h}{\tan\alpha}$$d=PQ$; $\beta\gt\alpha$ are the elevations at the nearer and farther points
Two points, opposite sides§ 9.2 →
$h=\dfrac{d\tan\alpha\tan\beta}{\tan\alpha+\tan\beta}$$d=PQ$ with the foot of the tower between $P$ and $Q$
Pole on a support§ 9.2 →
$p=x(\tan\beta-\tan\alpha)$, $b=x\tan\alpha$$x$ distance from the foot; $\alpha$, $\beta$ elevations of bottom and top of the pole
Special case 30° and 60°§ 9.2 →
$h=\tfrac{\sqrt3}{2}\,d$, $PB=\tfrac d2$
Two boats from a height $H$§ 9.2 →
depressions $45^\circ$, $30^\circ$: gap $=H(\sqrt3\mp1)$ (same side $-$, opposite $+$)
Chapter 10 · Circles
Tangent and radius§ 10.1 →
The tangent at $P$ is perpendicular to the radius $OP$: $OP\perp$ tangent, $\angle OPT=90^\circ$ for $T$ on the tangent
Position of a line§ 10.1 →
$d\lt r$: secant (2 points) · $d=r$: tangent (1 point) · $d\gt r$: no common point ($d$ = perpendicular distance from the centre)
Right triangle at the contact point§ 10.1 →
$OT^2=r^2+PT^2$ ($T$ on the tangent at $P$)
Chord at distance $d$§ 10.1 →
chord $=2\sqrt{r^2-d^2}$ (for $d\lt r$)
How many tangents?§ 10.1 →
inside: 0 · on the circle: 1 · outside: 2
Equal tangents§ 10.2 →
Tangents $PA$, $PB$ from an external point $P$: $PA=PB$, and $PO$ bisects $\angle APB$ and $\angle AOB$
Length of a tangent§ 10.2 →
$PA=\sqrt{OP^2-r^2}$
Angle between tangents§ 10.2 →
$\angle APB+\angle AOB=180^\circ$
Quadrilateral around a circle§ 10.2 →
$ABCD$ circumscribes a circle $\Rightarrow$ $AB+CD=BC+DA$
Chord of contact§ 10.2 →
$AB=\dfrac{2\,r\,PA}{OP}$
Incircle of a right triangle§ 10.2 →
legs $p,q$, hypotenuse $h$: $r=\dfrac{p+q-h}{2}$
Chapter 11 · Areas Related to Circles
Length of an arc§ 11.1 →
$l=\dfrac{\theta}{360}\times2\pi r$$\theta$ = central angle in degrees, $0\lt\theta\lt360$; $r$ = radius
Area of a sector§ 11.1 →
$A=\dfrac{\theta}{360}\times\pi r^2$$\theta$ in degrees; the other sector of the circle has angle $360^\circ-\theta$
Sector from arc length§ 11.1 →
$A=\tfrac12\,l\,r$
Perimeter of a sector§ 11.1 →
$P=l+2r$
Sector of a ring§ 11.1 →
$A=\dfrac{\theta}{360}\,\pi\left(R^2-r^2\right)$$R\gt r$ are the radii of the two concentric arcs, same angle $\theta$
Clock hands§ 11.1 →
minute hand $6^\circ$/min; hour hand $0.5^\circ$/min
Area of a segment§ 11.2 →
minor segment $=$ sector $-$ triangle $OAB$; major segment $=\pi r^2-$ minor segment$O$ centre, $AB$ chord, $\angle AOB=\theta\lt180^\circ$
Segment, $\theta=60^\circ$§ 11.2 →
$r^2\left(\dfrac{\pi}{6}-\dfrac{\sqrt3}{4}\right)$, chord $=r$
Segment, $\theta=90^\circ$§ 11.2 →
$r^2\left(\dfrac{\pi}{4}-\dfrac12\right)$, chord $=r\sqrt2$
Segment, $\theta=120^\circ$§ 11.2 →
$r^2\left(\dfrac{\pi}{3}-\dfrac{\sqrt3}{4}\right)$, chord $=r\sqrt3$
Polygon inside a circle§ 11.2 →
$\pi r^2-\text{area of polygon}=n\times\text{one segment}$$n=3,4,6$ for central angles $120^\circ,90^\circ,60^\circ$
Chapter 12 · Surface Areas and Volumes
Surface areas of the basic solids§ 12.1 →
Cylinder: CSA $2\pi rh$, TSA $2\pi r(h+r)$. Cone: CSA $\pi rl$, TSA $\pi r(l+r)$Sphere: $4\pi r^2$. Hemisphere: CSA $2\pi r^2$, TSA $3\pi r^2$Cuboid: TSA $2(lb+bh+hl)$. Cube: TSA $6a^2$
Slant height of a cone§ 12.1 →
$l=\sqrt{r^2+h^2}$ ($r$ radius, $h$ height)
Two solids joined§ 12.1 →
$\text{TSA}=\text{TSA}_1+\text{TSA}_2-2A_c$, where $A_c$ is the contact areaAdd only the faces that stay on the outside
A hollow scooped out§ 12.1 →
$\text{TSA}=\text{TSA}_{\text{original}}-A_o+\text{(inside surface of the hollow)}$$A_o$ is the area of the opening
Capsule§ 12.1 →
$\text{TSA}=2\pi rh+4\pi r^2=2\pi rL$$L=h+2r$ is the total length
Cone on a hemisphere§ 12.1 →
$\text{TSA}=\pi rl+2\pi r^2$
Volumes of the basic solids§ 12.2 →
Cuboid $lbh$, cube $a^3$, cylinder $\pi r^2h$, cone $\tfrac13\pi r^2h$Sphere $\tfrac43\pi r^3$, hemisphere $\tfrac23\pi r^3$ ($h$ is the height, not the slant height)
Volume of a combination§ 12.2 →
Joined: $V=V_1+V_2$. Scooped out: $V=V_{\text{original}}-V_{\text{hollow}}$Volumes add or subtract with no loss at the joint (unlike surface areas)
Cone : hemisphere : cylinder§ 12.2 →
$1:2:3$ for radius $r$ and height $r$
Capsule§ 12.2 →
$V=\pi r^2h+\tfrac43\pi r^3=\pi r^2L-\tfrac23\pi r^3$
Cone with a hemisphere§ 12.2 →
$V=\tfrac13\pi r^2h+\tfrac23\pi r^3$
Capacity and mass§ 12.2 →
$1\ \text{L}=1000\ \text{cm}^3$, $1\ \text{m}^3=1000\ \text{L}$; mass $=$ volume $\times$ density
Chapter 13 · Statistics
Class mark and class size§ 13.1 →
$x_i=\dfrac{\text{upper}+\text{lower}}{2}$, $\ h=\text{upper}-\text{lower}$
Mean — direct method§ 13.1 →
$\bar x=\dfrac{\sum f_ix_i}{\sum f_i}$ ($x_i$ = class marks, $f_i$ = frequencies)
Mean — assumed-mean method§ 13.1 →
$\bar x=a+\dfrac{\sum f_id_i}{\sum f_i}$, $\ d_i=x_i-a$ ($a$ = any number)
Mean — step-deviation method§ 13.1 →
$\bar x=a+h\,\dfrac{\sum f_iu_i}{\sum f_i}$, $\ u_i=\dfrac{x_i-a}{h}$ (equal class size $h$)
Mode of grouped data§ 13.2 →
$\text{Mode}=l+\left(\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right)h$ — $l$ = lower limit of the modal class, $h$ = class size, $f_1$ = its frequency, $f_0,f_2$ = frequencies of the classes before and after (continuous classes)
Equal neighbours§ 13.2 →
If $f_0=f_2$: $\ \text{Mode}=l+\dfrac h2$ (the class mark)
First or last class§ 13.2 →
No neighbour: take $f_0=0$ (first class) or $f_2=0$ (last class)
Making classes continuous§ 13.2 →
$10$–$19$, $20$–$29$ become $9.5$–$19.5$, $19.5$–$29.5$ (adjust by half the gap)
Median of grouped data§ 13.3 →
$\text{Median}=l+\left(\dfrac{\frac n2-cf}{f}\right)h$ — $l$ = lower limit of the median class, $cf$ = cumulative frequency of the class before it, $f$ = its frequency, $h$ = class size, $n=\sum f_i$ (continuous classes)
Median class§ 13.3 →
First class with cumulative frequency $\ge\dfrac n2$
Median of a list§ 13.3 →
$n$ odd: the $\frac{n+1}{2}$th value; $n$ even: mean of the $\frac n2$th and $(\frac n2+1)$th
Empirical relation§ 13.3 →
$3\,\text{Median}=\text{Mode}+2\,\text{Mean}$ (moderately skewed data, approximate), so $\text{Mode}=3\,\text{Median}-2\,\text{Mean}$
Chapter 14 · Probability
Theoretical probability§ 14.1 →
Equally likely outcomes: $P(E)=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$
Range of a probability§ 14.1 →
$0\le P(E)\le1$; impossible event $P=0$, sure event $P=1$
Complementary events§ 14.1 →
$P(E)+P(\overline{E})=1$, i.e. $P(\overline{E})=1-P(E)$
Counting outcomes§ 14.2 →
$n$ coins: $2^n$ outcomes. Two dice: $6\times6=36$. In general $p\times q$ (Theorem 14.3)
Ways to get each sum with two dice§ 14.2 →
sum $2,3,4,5,6,7,8,9,10,11,12$ in $1,2,3,4,5,6,5,4,3,2,1$ ways out of $36$; doublets: $6$ of $36$
A pack of 52 cards§ 14.2 →
$26$ red, $26$ black; $13$ per suit; $4$ of each rank; $12$ face cards (J, Q, K); $36$ cards numbered $2$–$10$
"At least one"§ 14.2 →
$P(\text{at least one})=1-P(\text{none})$