計算機及程式 Calculator and Programs

Quadratic Equations in One Unknown

For the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+c=0$, where $a\ne 0$:

(a) $x=\frac{-b\pm \sqrt{b^2-4\mathrm{ac}}}{2a}$

(b) Discriminant $\Delta =b^2-4\mathrm{ac}$

• $\Delta >0$ for the equation has 2 roots
• $\Delta =0$ for the equation has only 1 root (or 2 equal roots)
• $\Delta <0$ for the equation has no roots
• $\Delta \ge 0$ for the equation has root(s) (can be 1 or 2)

(c)

• (i) Sum of roots = $\alpha +\beta =-\frac{b}{a}$
• (ii) Product of roots = $\mathrm{\alpha \beta }=\frac{c}{a}$

Note: ${\alpha }^2+{\beta }^2=(\alpha +\beta ){}^2-2\mathrm{\alpha \beta }$

New equation: $x^2-(\mathrm{\alpha \text{'}}+\mathrm{\beta \text{'}})x+\mathrm{\alpha \text{'}\beta \text{'}}=0$

Quadratic Graph

• $a>0$: open upwards, minimum
• $a<0$: open downwards, maximum
• $c$ = y-intercept = Sub $(0,y)$, that is $x=0$, find $y$

Vertex = $(-\frac{b}{2a},-\frac{b^2-4\mathrm{ac}}{4a})=(h,k)$

If $y=f\left(x\right)=a(x-h){}^2+k$:

• Axis of symmetry: $x=-\frac{b}{2a}$ [or $x=h$]
• Optimum/Maximum/Minimum = $-\frac{b^2-4\mathrm{ac}}{4a}$ [or $k$]

Other:

1. $x$-intercept = Sub $(x,0)$, that is $y=0$, find $x$
2. Completing square: $y=a{x}^2+\mathrm{bx}+c=a(x+\frac{b}{2a}){}^2+c-\frac{b^2}{4a}$
3. Important Skill: Write down all coordinates. Sub those points lying on the graph into the equations one by one.

Equation of Straight Lines

Find slope first.

1. (a) For the line passing through a point $(x_1,y_1)$ with slope $m$:
   equation of the straight line: $y-y_1=m(x-x_1)$
2. (b) $y=\mathrm{mx}+c$, $m$ is the slope, $c$ is the y-intercept
3. (c) For an equation of the straight line $\mathrm{Ax}+\mathrm{By}+C=0$:
   • slope = $-\frac{A}{B}$
   • y-intercept = $-\frac{C}{B}$
   • x-intercept = $-\frac{C}{A}$

Find intersection point: Solve equations $L_1$ and $L_2$.

More about Polynomials

1. (a) Division algorithm: Dividend = Divisor × Quotient + Remainder
   e.g., $7=2\times 3+1$
   Let remainder be $R$ [constant] or $\mathrm{Ax}+B$ [linear] or $A{x}^2+\mathrm{Bx}+C$ [Quadratic]
2. (b) A polynomial $P\left(x\right)$ is divided by $\mathrm{ax}-b$, the remainder $R$ is $P\left(\frac{b}{a}\right)$
3. (c) Long Division

Exponential and Logarithmic Functions

1. (a) $a^{\frac{m}{n}}=\sqrt[n]{a^m}$, where $m$ and $n$ are integers and $n>0$
2. (b) Laws of indices:
   1. (i) $a^m\times a^n=a^{m+n}$
   2. (ii) $a^m÷a^n=a^{m-n}$
   3. (iii) $\left(a^m\right){}^n=a^{\mathrm{mn}}$
   4. (iv) $\left(\mathrm{ab}\right){}^m=a^m{b}^m$
   5. (v) $\left(\frac{a}{b}\right){}^m=\frac{a^m}{b^m}$
   5 methods to solve:
   1. $logx=logA$ → $x=A$
   2. $A^x=B$ → $xlogA=logB$
   3. ${log}_{A}x=B$ → $x=A^B$
   4. $A^{x+1}+A^x=B$ → Factorize: $A^x(A+1)=B$
   5. $A^{2x+1}+A^x=B$ → Let $y=A^x$: $A{y}^2+y=B$
3. (c) For $M,N>0$, $a>0$ and $a\ne 1$:
   1. (i) ${log}_a\left(\mathrm{MN}\right)={log}_{a}M+{log}_{a}N$
   2. (ii) ${log}_a\left(\frac{M}{N}\right)={log}_{a}M-{log}_{a}N$
   3. (iii) ${log}_a\left(M^n\right)=n{log}_{a}M$
   4. (iv) ${log}_{a}1=0$
   5. (v) ${log}_{a}a=1$
   [Change Base] ${log}_{a}b=\frac{{log}_{c}b}{{log}_{c}a}=\frac{logb}{loga}$
4. (d) $L={log}_{10}\left(\frac{I}{I_0}\right)$, where $I$ is the intensity of sound and $I_0$ is the minimum audible sound intensity
5. (e) $logE=4.8+1.5R$, where $E$ is the energy released (in joules (J)) from an earthquake

More about Trigonometry

1. (a) Trigonometric identities:
   1. (i) $90°$, $270°$ changed: sin ↔ cos, cos ↔ sin, tan ↔ 1/tan
   2. (ii) $0°$, $180°$, $360°$ unchanged: sin ↔ sin, cos ↔ cos, tan ↔ tan
2. (b)
   1. (i) area of $△\mathrm{ABC}=\frac{1}{2}\mathrm{ab}sinC$
   2. (ii) Heron's Formula: area of $△\mathrm{ABC}=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{1}{2}(a+b+c)$
   3. (iii) Sine Formula: $\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$
   4. (iv) Cosine Formula: $c^2=a^2+b^2-2\mathrm{ab}cosC$ or $cosC=\frac{a^2+b^2-c^2}{2\mathrm{ab}}$
   $a$, $b$ and $c$ are the opposite sides of $\angle A$, $\angle B$ and $\angle C$ respectively.

Variation

• Directly: $k\times$
• Inversely: $k÷$
• Partly: $k_1$, $k_2$ constant
• square: $\left(\right){}^2$
• square root: $\sqrt{}$
• cube: $\left(\right){}^3$
• cubic root: $\sqrt[3]{}$

Permutation and Combination

1. (a) The number of permutations of $r$ objects from $n$ distinct objects without repetition = ${}_{n}P_r=\frac{n!}{(n-r)!}$ [Consider Order]
2. (b) The number of combinations of $r$ objects from $n$ distinct objects without repetition = ${}_{n}C_r=\frac{{}_{n}P_r}{r!}=\frac{n!}{r!(n-r)!}$ [Not Consider Order]

More about Probability

1. (a) If two events $E$ and $F$ are mutually exclusive, then $P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)$
2. (b) For any two events $E$ and $F$, $P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)-P\left(E\text{and}F\right)$
3. (c) The complementary event of $E$ is denoted by $E\text{'}$. $P(E\text{'})=1-P\left(E\right)$
4. (d) If two events $E$ and $F$ are independent, then $P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F\right)$
5. (e) Conditional probability: For any two events $E$ and $F$, $P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F\text{after}E\text{has occurred}\right)$, where $P\left(E\right)>0$

More about Statistics

1. (a) Range Range = Largest value − Smallest value Range = Highest class boundary − Lowest class boundary
2. (b) Inter-quartile range Inter-quartile range = Upper quartile − Lower quartile
3. (c) Standard deviation $\sigma =\sqrt{\frac{{(x_1-\overline{x})}^2+{(x_2-\overline{x})}^2+\backslash cdots+{(x_n-\overline{x})}^2}{n}}$ where x ¯ is the mean and n is the total number of data.
4. (d) Standard score = $\frac{x-\overline{x}}{\sigma }$ , where x is the value of the datum.
5. Normal distribution: 34%, 13.5%, 2.35%, 0.15%
6. If each datum is added/subtracted by k: mean, mode, median → add/subtract k Range, IQR, S.D., Variance → no change
7. If each datum is multiplied/divided by k: mean, mode, median → multiply/divide by k Range, IQR, S.D. → multiply/divide by k Variance → multiply/divide by k squared

Equations of Circles

1. For a circle with the centre (h, k) and radius r, the equation of the circle: (x - h)^2 + (y - k)^2 = r^2
2. For an equation of a circle x^2 + y^2 + Dx + Ey + F = 0, centre = ( -D/2, -E/2 ) radius = sqrt{ (D/2)^2 + (E/2)^2 - F }
3. Real Circle: r > 0 Point Circle: r = 0 Imaginary Circle: r < 0
4. *Equation of Tangent: Let y = mx + c. Sub y = mx + c into x^2 + y^2 + Dx + Ey + F = 0, find Δ = b^2 − 4ac.
5. *Geometric Relationships: Collinear: A, B, C forms a straight line. Concyclic: A, B, C, D forms a circle. Perpendicular bisector: straight line passes through mid-point of AB and perpendicular to AB. Angle bisector: two straight lines pass through intersection of AB and CD, and divide angles equally. Parallel: slope of straight line = slope of AB. Perpendicular: (slope of line) × (slope of AB) = -1.

Arithmetic and Geometric Sequences

1. Arithmetic sequence (AS):
2. a = first term, d = common difference, n = number of terms, l = last term
3. General term: T(n) = a + (n - 1)d
4. Sum: S(n) = n/2 [2a + (n - 1)d] or S(n) = n/2 (a + l)
5. Geometric sequence (GS):
6. a = first term, r = common ratio, n = no. of terms
7. General term: T(n) = a r^{n-1}
8. Sum: S(n) = [a(r^n - 1)]/(r - 1) or S(n) = [a(1 - r^n)]/(1 - r)
9. If $-1\mathrm{<\; r\; <}1$, sum to infinity: S(∞)=a/(1 - r)

Compound Inequality

1. “And”: overlap， "Or": All
2. a is positive：
3. ax^2 + bx + c < 0， roots α < x < β
4. ax^2 + bx + c ≤ 0， α ≤ x ≤ β
5. ax^2 + bx + c > 0, x < α or x> β
6. ax^2 + bx + c ≥ 0， x ≤ α or x ≥ β

Percentages

1. Percentage change = (New value − Original value) / Original value × 100%
2. New value = Original value × (1 + Percentage increase)
3. New value = Original value × (1 − Percentage decrease)
4. Profit and loss Percentage = (Selling price − Cost price) / Cost price × 100%
5. If the percentage change > 0, there is a profit; if < 0, there is a loss.
6. Selling price = Cost price × (1 + Profit percentage)
7. Selling price = Cost price × (1 − Loss percentage)
8. Discount percentage = (Marked price − Selling price) / Marked price × 100%
9. Selling price = Marked price × (1 − Discount percentage)

Mensuration

1. Square: Perimeter = 4 × Side length; Area = (Side length)^2
2. Rectangle: Perimeter = 2 × (Length + Width); Area = Length × Width
3. Triangle: Perimeter = sum of three sides; Area = (1/2) × Base × Height
4. Parallelogram: Perimeter = 2 × (sum of 2 sides); Area = Base × Height
5. Trapezium: Perimeter = sum of 4 sides;
6. Area = (1/2) × (sum of two bases) × Height
7. Rhombus: Perimeter = 4 × Side length; Area = Base × Height
8. Cube: Surface area = 6 × (Side length)^2; Volume = (Side length)^3
9. Cuboid: Surface area = 2(Length × Width + Width × Height + Height × Length); Volume = Length × Width × Height
10. Prism: Surface area = All lateral faces + 2 × Base area; Volume = Base area × Height

Coordinate

1. Rotation 90°: (x, y) → (±y, ±x)
2. Rectangular to polar: r = sqrt(x^2 + y^2), θ = arctan(y/x)
3. Polar to rectangular: x = r cos θ, y = r sin θ

Statistics Pie Chart

1. Pie Chart: θ° / 360° = ?% / 100% = Part / Total

Estimation, Approximation and Errors

1. $\mathrm{a)}\text{Absolute error = estimated value - exact value}$
2. $\mathrm{b)}\text{Maximum absolute error = scale ± 2}$
3. $\mathrm{c)}\frac{\text{Relative error}}{\text{Measured absolute error}}\text{or}\frac{\text{Absolute error}}{\text{Exact value}}$
4. $\mathrm{d)}\text{Percentage error = Relative error × 100\%}$

Identities

1. $\mathrm{a)}{\mathrm{(a\; +\; b)}}^2=a^2+\mathrm{2ab}+b^2$
2. $\mathrm{b)}{\mathrm{(a\; -\; b)}}^2=a^2-\mathrm{2ab}+b^2$
3. $\mathrm{c)}a-b=(a+b)(a-b)$

Laws of Indices

1. $\mathrm{i)}{a}^m\times a^n=a^{m+n}$
2. $\mathrm{ii)}{a}^m÷a^n=a^{m-n}$
3. $\mathrm{iii)}{\left(a^m\right)}^n=a^{m\times n}$
4. $\mathrm{iv)}{\left(\mathrm{ab}\right)}^n=a^n\times b^n$
5. $\mathrm{v)}\frac{a^m}{b^m}={(}^{\frac{a}{b}}m$
6. $\mathrm{vi)}{a}^{-n}=\frac{1}{a^n}=\frac{1}{a^{-n}}$

Surds

1. $\mathrm{a)}\sqrt{\mathrm{ab}}=\sqrt{a}\sqrt{b}$
2. $\mathrm{b)}\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$
3. $\mathrm{c)}{a}^2+b^2=c^2\text{(Pyth. Thm)}$

More about Trigonometry

1. $\text{sin θ =}\frac{\text{opposite}}{\text{H}},\text{cos θ =}\frac{\text{adjacent}}{\text{H}},\text{tan θ =}\frac{\text{opposite}}{\text{adjacent}}$
2. $\mathrm{i)}{\text{sin}}^2\theta +{\text{cos}}^2\theta =1$
3. $\mathrm{ii)}\text{tan θ}=\frac{\text{sin θ}}{\text{cos θ}}$
4. $\mathrm{iii)}\text{sin(90° - θ) = cos θ}$
5. $\mathrm{iv)}\text{cos(90° - θ) = sin θ}$
6. $\mathrm{v)}\text{tan(90° - θ) =}\frac{1}{\text{tan θ}}$
7. $\text{Others:}\mathrm{1\; -\; sin²\; \theta \; =\; cos²\; \theta }$
8. $\mathrm{1\; -\; cos²\; \theta \; =\; sin²\; \theta }$
9. $\frac{1}{\text{sin θ}}=\text{cosec θ}$
10. $\frac{1}{\text{tan(90° - θ)}}=\text{tan θ}$

Mensuration

1. $\mathrm{a)}\text{Circle}\mathrm{i)}\text{Perimeter = 2 × π × Radius}$
2. $\mathrm{ii)}\text{Area = π × (Radius)²}$
3. $\mathrm{b)}\text{Sector}$
4. $\mathrm{i)}\text{Arc length = 2 × π × Radius ×}\frac{\text{Angle at centre}}{\mathrm{360°}}$
5. $\mathrm{ii)}\text{Area = π × (Radius)² ×}\frac{\text{Angle at centre}}{\mathrm{360°}}$
6. $\mathrm{c)}\text{Cylinder}\text{Let r and h be base radius and height of the cylinder respectively.}$
7. $\mathrm{i)}\text{Curved surface area = 2πrh}$
8. $\mathrm{ii)}\text{Total surface area = 2πrh + 2πr²}$
9. $\mathrm{iii)}\text{Volume = πr²h}$

1. (a)(i) Total surface area of a pyramid: $\text{Total surface area}=\text{All lateral faces}+\text{Base area}$
2. (a)(ii) Volume of a pyramid: $\text{Volume}=\frac{1}{3}\times \text{Base area}\times \text{Height}$
3. (b)(i) Curved surface area of a right circular cone: $\text{Curved surface area}=\pi rl$
4. (b)(ii) Total surface area of a right circular cone: $\text{Total surface area}=\pi rl+\pi r^2$
5. (b)(iii) Volume of a right circular cone: $\text{Volume}=\frac{1}{3}\pi r^{2}h$
6. (c)(i) Total surface area of a sphere: $\text{Total surface area}=4\times \pi \times {\text{Radius}}^2$
7. (c)(ii) Volume of a sphere: $\text{Volume}=\frac{4}{3}\times \pi \times {\text{Radius}}^3$
8. Similar figures: area and volume ratios: $\frac{A_1}{A_2}={\frac{l_1}{l_2}}^2\text{and}\frac{V_1}{V_2}={\frac{l_1}{l_2}}^3$