SUEC 高中 Adv Maths (Linear Equation) (Part 3)

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Sources: Visual - various maths sites (credits to original creator) Questions - Dong Zong's Textbook suitable for SUEC (Maths), SPM (Maths and Add Maths) too

Education

𝒑𝒈 𝟒𝟐
"the two lines are the same"
they "intersect" everywhere

The document discusses random phenomena and probability. It defines a random phenomenon as one where individual outcomes are uncertain. It provides examples of sample spaces and sample points for events like goals in a game or coin flips. It also includes examples of calculating probabilities of certain outcomes occurring based on the sample space and equally likely outcomes, such as the probability of getting 3 heads in a row or having at least 1 head.

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1. There are 6 math books and 5 language books on different shelves. The number of ways to choose 1 of each is 6 × 5 = 30. 2. There are 5 colors of tops and 4 colors of skirts. The total number of dress combinations is 5 × 4 = 20. There are 3 styles of shoes, so the total number of styles is 3. 3. The number of 3-digit numbers that can be formed without repeating digits is 100 × 99 × 98 = 9,702. The number of ways for 2 boys to sit in 5 chairs is 5 × 4 = 20.

(1) The document discusses finding equations of tangent lines to circles and the intersections of those tangent lines. It provides examples of finding the slopes and equations of tangent lines given the circle's center and a point on the circle. (2) Methods are described for finding the angles between two tangent lines to a circle based on their slopes. Examples are given of solving systems of equations to find the points where tangent lines intersect. (3) One example determines the equation of a circle given that it passes through two known points and is tangent to another circle at a third point.

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The document contains calculations of distances between various geographic points using latitude and longitude coordinates. It includes the distances between points Q and A, which is 319.2 km, and the distance from a point at 42°N 33°27'E or 42°N 6°33'W to 40°N 33°47'E, which is calculated as 8,895.35 km or 4,800 nautical miles. It also contains a calculation using trigonometric functions that finds the distance between two points is 6,560 km or 3,540 nautical miles.

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This document contains calculations and solutions to trigonometry problems involving angles, sides of triangles, and distances. Various trigonometric functions are used to calculate unknown angles and distances. Measurements include distances between points, lengths of sides of triangles, angles of triangles, and distances between locations. The document demonstrates applying trigonometric concepts and relationships to solve for unknown values in different geometric scenarios and problems.

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The document contains examples of solving various logarithmic and algebraic equations. It begins by solving equations involving logarithms such as logabc = loga bc - logb a ∙ logc a. It then solves equations involving logarithms of both sides being equal, leading to the determination that x = abc. Further examples include solving quadratic equations that arise from rewriting the original equations in terms of new variables, and determining the solutions for x in each case.

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The document discusses different types of progressions. It covers arithmetico-geometric progressions on pages 138-141, discussing the definitions and formulas. Harmonic progressions are then introduced on pages 144-146, defining them as sequences where the reciprocals of the terms form an arithmetic progression. Both geometric and harmonic progressions are compared to arithmetic progressions, as the document seeks to outline key properties and relationships between these different progression types.

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This document discusses using linear programming to solve optimization problems by finding the maximum or minimum value of an objective function subject to certain constraints. It provides two examples: 1) minimizing the cost of feeding buckwheat with two types of feed and nutrient constraints, and 2) maximizing the revenue from transporting goods with two factories subject to weight and volume constraints. Both examples set up the objective function and constraints, then solve to find the optimal value.

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