GCE 'O' and 'N' EM Notes

By u/One_Wishbone_4439 Page 1 of 51 GCE 'O' and 'N' EM Notes Number and algebra Positive numbers: greater than 0. eg. 1, 2, 3, 4, 5 Negative numbers: less than 0. eg. −1, −2, −3, −4, −5 Prime numbers: have exactly two factors: 1 and itself. eg. 2, 3, 5, 7, 11 Composite numbers: have more than two factors. Eg. 4, 6, 8, 9, 12 ➢ 1 and 0 are NOT prime numbers. ➢ 2 is the ONLY even prime number. Rational numbers: Can be represented in the form of P Q where Q ≠ 0. Irrational numbers: Cannot be represented in the form of P Q . HCF – Highest Common Factor ➢ Look at the lowest power. LCM – Lowest Common Multiple ➢ Look at the highest power. Prime factorisation ➢ Only can divide by prime numbers. Significant figures (5 rules) ➢ All non-zero digits are significant. (eg. 211.8 has 4sf) ➢ All zeros that are found between nonzero digits are significant. (eg. 20007 has 5sf) ➢ Leading zeros (to the left of the first nonzero digit) are not significant. (eg. 0.0085 has 2sf) ➢ Trailing zeros for a whole number that ends with a decimal point are significant. (eg. 320 can be 2sf or 3sf) ➢ Trailing zeros to the right of the decimal place are significant. (eg. 12.000 has 5sf)

By u/One_Wishbone_4439 Page 2 of 51 Standard form: expressed as A × 10n where A must be 1 ≤ A < 10 and n is an integer. Common prefixes Power of 10 English word SI prefix Symbol 1012 trillion tera T 109 billion giga G 106 million mega M 103 thousand kilo K 10-3 thousandth milli m 10-6 millionth micro µ 10-9 billionth nano n 10-12 trillionth pico P Law of indices ➢ Zero indices: a0 = 1, a ≠ 0 and a < 0 ➢ Negative indices: a-n = 1 an, a ≠ 0 and a < 0 ➢ Rational indices: a m n = √amn = (√a n ) m , a ≤ 0 ➢ Law 1: am × an = am+n, if a > 0 ➢ Law 2: am ÷ an = am-n, if a > 0 ➢ Law 3: (am)n = amn, if a > 0 ➢ Law 4: an × bn = (a × b)n, if a, b > 0 ➢ Law 5: an ÷ bn = (a b) n , if a, b > 0 Inequality Signs Definitions How is it represented on a number line? < more than > less than ≤ more than or equal to ≥ less than or equal to Basic Four Operations of Algebraic Fractions ➢ Addition: a b + c d = ad + bc bd ➢ Subtraction: a b − c d = ad − bc bd ➢ Multiplication: a b × c d = ac bd ➢ Division: a b ÷ c d = a b × d c = ad bc

By u/One_Wishbone_4439 Page 3 of 51 ➢ Conditions* 1. Addition: If A < B, then A + c < B + c 2. Subtraction: If A < B, then A − c < B − c 3. Multiplication: If A < B, then cA < cB If A < B, then − cA > − cB 4. Division: If A < B, then A c < B c If A < B, then A − c > B − c *Note: For multiplication and division of negative numbers, the inequality sign must flip (shown in red). Algebraic expression ➢ The square of sum: (a + b)2 = (a2 + 2ab + b2) ➢ The square of difference: (a - b)2 = (a2 - 2ab + b2) ➢ The difference of two squares: a2 - b2 = (a + b)(a - b) Factorisation methods: ➢ Divide by HCF ➢ Algebraic expression ➢ Grouping ➢ Completing the square in the form of y = (x – h)2 + k where (h, k) is the turning point. ➢ Quadratic equation: 𝑥 = −b ± √b2 −4ac 2a ➢ Cross product Ratio ➢ a : b = a b

By u/One_Wishbone_4439 Page 4 of 51 Proportion Direct proportion y = kx Inverse proportion y = k x Map scale ➢ Length → Map : Actual = 1 : n best to change them to cm. ➢ Area → 12 : n2 Percentage ➢ Percentage increase/decrease = Increase/Decrease Original Value × 100% ➢ Tax Relief ➢ Commission ➢ Profit/Discount ➢ Income Tax/GST (9%) * *Note: GST is not always 9%, it changes over the years. y x y x

By u/One_Wishbone_4439 Page 5 of 51 Number pattern General term, Tn = an + b where n is the term number, a is the common difference between two consecutive terms, and b is the starting term, which is the value of the sequence when n = 0. For example, the first five terms form a number pattern: 5, 8, 11, 14, 17 Common difference: 8 – 5 = 3 Now, the equation is 3n + b. If n = 1, 3(1) + b = 5 → b = 5 – 3 = 2 Hence, Tn = 3n + 2. Rate ➢ Rate is always over time (s). Speed ➢ Speed (m/s) = Distance (m) Time (s) ➢ Average speed = Total Distance (m) Time (s) ➢ Acceleration: It is an increase in speed over time. ➢ Deceleration: It is a decrease in speed over time. D S T × ÷

By u/One_Wishbone_4439 Page 6 of 51 Speed-time graphs From the speed-time graph above, ➢ from 0 to t1, the car is increasing speed over time, meaning the car is moving quickly ➢ from t1 to t2, the car is at constant speed, meaning the speed does not change ➢ from t2 to t3, the car is decreasing speed over time, meaning the car is slowing down ➢ the distance travelled by the car can be determined by finding the area under the graph ➢ to find acceleration and deceleration (negative acceleration) of the car, find the gradient of the line. ➢ gradient for constant speed is always zero. Displacement-time graphs From the displacement-time graph above, ➢ from 0 to t1, the car does not change in speed. ➢ from t1 to t2, the car does not move. ➢ from 0 to t1, the car does not change in speed. ➢ to find the speed of the car, find the gradient of the line. ➢ gradient for stationary object is always zero. Speed (m/s) Time (s) Constant speed Area under graph = distance travelled t1 t2 0 t3 Displacement (m) Stationary object t1 t2 0 t3 Time (s)

By u/One_Wishbone_4439 Page 7 of 51 Simple interest: It is an interest charge that borrowers pay lenders for a loan. I = PRT 100 where P is the principal amount R is the rate of interest T is the number of years Compound interest: It is the interest calculated on both the initial principal and all of the previously accumulated interest. A = P (1 + r 100) n where A is the total amount P is the principal amount r is the rate of interest n is the number of years Exchange rate: a relative price of one currency expressed in terms of another currency. For example, the exchange rate between Singapore Dollars (SGD) and US Dollars (USD) is now S$1 = $0.75 USD. If I want to purchase a bag that costs $300, how much will I need to pay in USD? S$1 = $0.75 USD S$300 = $0.75 × $300 = $225 USD Hence, I need to pay $225 USD for the same bag in Singapore. Hire purchase: It is an arrangement made while buying expensive goods. Hire purchase = Deposit + Monthly payment When you cannot afford to pay the item in full amount, you pay a deposit to the seller when you first agree to buy the item. Deposit is usually a small percentage of the cash price. Then, you pay the remaining amount in small chucks monthly. After you have fully pay including your monthly payment, then you will get the item you want. Taxation: It is a term for when a taxing authority, usually a government, levies or imposes a financial obligation on its citizens or residents.

By u/One_Wishbone_4439 Page 8 of 51 Set notation Set language Definition A ∪ B union of A and B A ∩ B intersection of A and B n(A) number of elements in set A ∈ an element of ∉ not an element of A′ complement of set A ∅ empty set  universal set A ⊆ B A is a subset of B A ⊈ B A is a not a subset of B A ⊂ B A is a (proper) subset of B A ⊄ B A is a not a (proper) subset of B Set: A list of elements. In simple words, sets are collection of objects such as pile of books and bunch of keys. The collective nouns “pile” and “bunch” are sets. The words “books” and “keys” are elements. For example, let A be the set of the first five prime numbers. It will be written like this: A = {2, 3, 5, 7, 11}. The curly brackets “{…}” are used to show a set. To find the number of elements in a set, we use this notation: n(A). For set A, the number of elements will be 5. Now let’s look at this Venn diagram below. We can observe from the above Venn diagram that the set of elements belonging to  but not to A is called the complementary of the set A, denoted as A′. A  2 3 5 7 11 1 4 6 8 9 10 12

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