This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here is a syllabus for Cambridge O Level Mathematics (4024), often referred to as "Syllabus A": 1. Number Number and place value: Integers, fractions, decimals, percentages. Powers and roots: Squares, cubes, square roots, cube roots, indices. Ratio, proportion, and rate: Direct and inverse proportion, speed, density. Estimation and approximation: Rounding, significant figures, decimal places, upper and lower bounds. Standard form: Scientific notation. Financial mathematics: Simple and compound interest, profit and loss, taxation, currency exchange. Set language and notation: Union, intersection, complement, subsets, Venn diagrams. 2. Algebra Algebraic expressions: Simplification, expansion, factorization (common factors, difference of two squares, quadratic trinomials). Formulae: Construction, substitution, rearrangement. Equations and inequalities: Linear, quadratic (factorization, formula, completing the square), simultaneous (linear), linear inequalities. Sequences: Arithmetic and geometric progressions, finding the nth term. Functions and graphs: Linear, quadratic, reciprocal, exponential graphs; gradient, intercepts, turning points. Variation: Direct and inverse variation. Matrices: Addition, subtraction, scalar multiplication, multiplication of 2x2 matrices, inverse of a 2x2 matrix. 3. Geometry Angles: Properties of parallel lines, polygons, circles (tangents, chords, angles in a circle). Symmetry: Line and rotational symmetry. Congruence and similarity: Conditions for congruence, properties of similar figures, scale factors. Mensuration: Perimeter, area (triangles, quadrilateres, circles, sectors), volume and surface area (prisms, pyramids, cylinders, cones, spheres). Pythagoras' theorem: In 2D and 3D. Trigonometry: Sine, cosine, tangent of angles of any magnitude; sine rule, cosine rule, area of a triangle (1/2 ab sin C). Vectors: Magnitude, direction, addition, subtraction, scalar multiplication, position vectors. Coordinate geometry: Gradient, distance between two points, midpoint, equation of a straight line. 4. Statistics and Probability Data handling: Collection, organization, and representation of data (bar charts, pie charts, histograms, frequency polygons, cumulative frequency graphs, box-and-whisker plots). Measures of central tendency: Mean, median, mode. Measures of spread: Range, interquartile range. Probability: Single events, combined events, tree diagrams, mutually exclusive and independent events.