Physics Senior Secondary Cheatsheet

Cheatsheet Content

1. Whole Numbers, Decimals, Fractions & Operations Number Types: Whole Numbers: $0, 1, 2, 3, \dots$ Integers: $\dots, -2, -1, 0, 1, 2, \dots$ Rational Numbers: Can be expressed as $\frac{p}{q}$ ($q \ne 0$). Irrational Numbers: Cannot be expressed as $\frac{p}{q}$ (e.g., $\sqrt{2}, \pi$). Fundamental Operations: Addition, Subtraction, Multiplication, Division. Order of Operations (BODMAS/PEMDAS): Brackets/Parentheses, Orders/Exponents, Division/Multiplication (left to right), Addition/Subtraction (left to right). Fractions: $\frac{\text{Numerator}}{\text{Denominator}}$. Addition/Subtraction: Find common denominator. $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$. Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$. Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. Decimals: Positional number system with base 10. Unit Digit: Cyclicity of powers (e.g., $2^1=2, 2^2=4, 2^3=8, 2^4=6, 2^5=2 \dots$ cycle of 4 for 2). LCM & HCF: Product of two numbers = HCF $\times$ LCM. HCF of fractions = $\frac{\text{HCF of numerators}}{\text{LCM of denominators}}$. LCM of fractions = $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$. 2. Percentages, Ratio, Proportion, Square Roots & Averages Percentages: $x\% = \frac{x}{100}$. Percentage Change: $\frac{\text{Change}}{\text{Original Value}} \times 100$. Successive Percentage Change: If an amount changes by $x\%$ then by $y\%$, effective change is $(x+y+\frac{xy}{100})\%$. Ratio: Comparison of two quantities $a:b$ or $\frac{a}{b}$. Proportion: Equality of two ratios. If $a:b::c:d$, then $ad=bc$. Mean Proportional of $a,b$: $\sqrt{ab}$. Third Proportional of $a,b$: $b^2/a$. Fourth Proportional of $a,b,c$: $bc/a$. Square Roots: $\sqrt{x}$. Property: $\sqrt{xy} = \sqrt{x}\sqrt{y}$. Averages: $\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}$. Profit & Loss: Profit ($P$) = SP - CP. Loss ($L$) = CP - SP. Profit % = $\frac{P}{CP} \times 100$. Loss % = $\frac{L}{CP} \times 100$. SP = $CP \times (1 \pm \frac{\text{Profit/Loss}\%}{100})$. Discount = MP - SP. Discount % = $\frac{\text{Discount}}{MP} \times 100$. 3. Partnership, Mixture & Alligation, Time & Work, Time & Distance Partnership: Profit sharing ratio is proportional to (Investment $\times$ Time). Mixture and Alligation: Rule of Alligation: $\frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \frac{\text{Price of Dearer} - \text{Mean Price}}{\text{Mean Price} - \text{Price of Cheaper}}$. Time & Work: If A does a work in $x$ days, A's 1-day work = $\frac{1}{x}$. Work = Efficiency $\times$ Time. $M_1 D_1 H_1 / W_1 = M_2 D_2 H_2 / W_2$. (M=men, D=days, H=hours, W=work) Time & Distance: Speed = $\frac{\text{Distance}}{\text{Time}}$. $1 \text{ km/hr} = \frac{5}{18} \text{ m/s}$. Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$. Relative Speed: Same direction: $S_1 - S_2$. Opposite direction: $S_1 + S_2$. Boats & Streams: $S_D = S_B + S_S$, $S_U = S_B - S_S$. ($S_D$=downstream, $S_U$=upstream, $S_B$=boat, $S_S$=stream). 4. Algebra: Basic Algebraic Identities and Linear Graphs Identities: $(a+b)^2 = a^2 + b^2 + 2ab$ $(a-b)^2 = a^2 + b^2 - 2ab$ $a^2 - b^2 = (a-b)(a+b)$ $(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)$ $a^3+b^3 = (a+b)(a^2-ab+b^2)$ $a^3-b^3 = (a-b)(a^2+ab+b^2)$ $(a+b)^3 = a^3+b^3+3ab(a+b)$ $(a-b)^3 = a^3-b^3-3ab(a-b)$ $a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ If $a+b+c=0$, then $a^3+b^3+c^3 = 3abc$. Linear Equations: $ax+b=0$. Solution $x = -b/a$. Linear Graphs: Represent equations of the form $y = mx+c$. $m$: slope. $c$: y-intercept. 5. Geometry Triangles: Sum of angles = $180^\circ$. Exterior angle = Sum of two opposite interior angles. Pythagoras Theorem: $a^2+b^2=c^2$ (for right-angled triangle). Area = $\frac{1}{2} \times \text{base} \times \text{height}$. Equilateral Triangle: Area = $\frac{\sqrt{3}}{4}a^2$. Height = $\frac{\sqrt{3}}{2}a$. Similarity: Ratio of areas = square of ratio of corresponding sides. Congruence: SSS, SAS, ASA, AAS, RHS. Centroid: Intersection of medians. Divides median in $2:1$. Incenter: Intersection of angle bisectors. Center of inscribed circle. Circumcenter: Intersection of perpendicular bisectors. Center of circumscribed circle. Orthocenter: Intersection of altitudes. Quadrilaterals: Sum of interior angles = $360^\circ$. Parallelogram: Opposite sides parallel and equal. Area = base $\times$ height. Rectangle: Parallelogram with all angles $90^\circ$. Area = $l \times b$. Square: Rectangle with all sides equal. Area = $a^2$. Rhombus: Parallelogram with all sides equal. Area = $\frac{1}{2}d_1 d_2$. Trapezium: One pair of parallel sides. Area = $\frac{1}{2}(a+b)h$. Regular Polygons (n sides): Sum of interior angles = $(n-2) \times 180^\circ$. Each interior angle = $\frac{(n-2) \times 180^\circ}{n}$. Sum of exterior angles = $360^\circ$. Each exterior angle = $\frac{360^\circ}{n}$. Number of diagonals = $\frac{n(n-3)}{2}$. Circle: Circumference = $2\pi r$. Area = $\pi r^2$. Angle at center = $2 \times$ Angle at circumference (subtended by same arc). Angles in same segment are equal. Angle in semicircle is $90^\circ$. Chords: Perpendicular from center bisects chord. Tangents: Perpendicular to radius at point of contact. Lengths of tangents from external point are equal. Tangent-Secant Theorem: $PT^2 = PA \times PB$. Intersecting Chords Theorem: $PA \times PB = PC \times PD$. Cyclic Quadrilateral: All vertices on circle. Sum of opposite angles = $180^\circ$. 6. Mensuration (3D Shapes) Cube (side $a$): Volume = $a^3$. Total Surface Area (TSA) = $6a^2$. Lateral Surface Area (LSA) = $4a^2$. Diagonal = $a\sqrt{3}$. Cuboid (l, b, h): Volume = $l \times b \times h$. TSA = $2(lb+bh+hl)$. LSA = $2h(l+b)$. Diagonal = $\sqrt{l^2+b^2+h^2}$. Cylinder (radius $r$, height $h$): Volume = $\pi r^2 h$. Curved Surface Area (CSA) = $2\pi r h$. TSA = $2\pi r(h+r)$. Cone (radius $r$, height $h$, slant height $l$): $l = \sqrt{r^2+h^2}$. Volume = $\frac{1}{3}\pi r^2 h$. CSA = $\pi r l$. TSA = $\pi r(l+r)$. Sphere (radius $r$): Volume = $\frac{4}{3}\pi r^3$. Surface Area = $4\pi r^2$. Hemisphere (radius $r$): Volume = $\frac{2}{3}\pi r^3$. CSA = $2\pi r^2$. TSA = $3\pi r^2$. Prism: Volume = Base Area $\times$ height. LSA = Perimeter of Base $\times$ height. Pyramid: Volume = $\frac{1}{3} \times$ Base Area $\times$ height. LSA = $\frac{1}{2} \times$ Perimeter of Base $\times$ slant height. 7. Trigonometry Trigonometric Ratios (Right-angled triangle): $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$. $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$. $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$. $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\cot\theta = \frac{1}{\tan\theta}$. $\tan\theta = \frac{\sin\theta}{\cos\theta}$. Identities: $\sin^2\theta + \cos^2\theta = 1$. $1 + \tan^2\theta = \sec^2\theta$. $1 + \cot^2\theta = \csc^2\theta$. Standard Angle Values: Angle $0^\circ$ $30^\circ$ $45^\circ$ $60^\circ$ $90^\circ$ $\sin\theta$ 0 $1/2$ $1/\sqrt{2}$ $\sqrt{3}/2$ 1 $\cos\theta$ 1 $\sqrt{3}/2$ $1/\sqrt{2}$ $1/2$ 0 $\tan\theta$ 0 $1/\sqrt{3}$ 1 $\sqrt{3}$ Undefined Complementary Angles: $\sin(90^\circ-\theta) = \cos\theta$. $\tan(90^\circ-\theta) = \cot\theta$. $\sec(90^\circ-\theta) = \csc\theta$. Height and Distance: Use trigonometric ratios to solve problems involving angles of elevation/depression. 8. Statistics & Probability Statistics: Mean: Sum of values / Number of values. Median: Middle value of ordered data. (Average of two middle for even count). Mode: Most frequent value. Range: Max value - Min value. Probability: $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$. $0 \le P(\text{Event}) \le 1$. $P(\text{A or B}) = P(A) + P(B) - P(\text{A and B})$. $P(\text{A and B}) = P(A) \times P(B)$ (for independent events).