CIE IGCSE Mathematics Comprehensive Study Notes

Number Sequences and Identification of the nth Term

Mathematics at the IGCSE level focuses heavily on the derivation of nth term formulas for various types of sequences, including linear, quadratic, and geometric patterns. For instance, in sequence problems where the nth term is defined as $an^2 + bn - 4$, such as in a problem where the first term is $-3$ and the second term is $2$, equations are formed by substituting $n = 1$ and $n = 2$ to solve for constants $a$ and $b$. Students must differentiate between linear sequences, where the nth term follows the form $dn + c$, and quadratic sequences, which involve an $n^2$ term. Examples from Study Haven past papers include determining the nth term for sequences like $13, 17, 21, 25$, which yields a linear expression, or more complex quadratic patterns such as $8, 17, 32, 53, 80$. Specific sequences analyzed include the powers of three ($3, 9, 27, 81$), which are represented by $3^n$, and sequences where terms decrease, such as $22, 17, 12, 7, 2$, requiring a negative common difference in the formula $-5n + 27$.

Advanced sequence problems involves the identification of patterns within geometric structures, as demonstrated by workers like Meena or Marco. Meena’s pattern uses dots and lines where the number of dots in pattern $n$ can be expressed as a linear function of $n$, such as $2n + 1$, and the number of lines follows a rule like $4n + 1$. If a pattern has $129$ lines, the value of $n$ must be calculated first to determine the corresponding number of dots. Similarly, Marco’s patterns involve grey and white circular mats where the total number of mats follows a triangular number sequence or a cubic summation formula such as $\frac{1}{3}n^3 + an^2 + bn$. Additionally, the Fibonacci sequence is explored, defined by the rule that each term is the sum of the two preceding terms, such as in the sequence $0, 1, 1, 2, 3, 5, 8, 13, 21$, or variations involving variables like $p$ and $q$.

Algebraic Expansion, Simplification, and Factorisation

Algebraic proficiency requires the ability to expand brackets and simplify terms across various complexities. Basic operations include combining like terms, such as simplifying $3x - 4x + 7x$ to $6x$ or $5t + 4t - 2t$ to $7t$. Expansion involves distributing terms, such as $3(2a - b) - b$ resulting in $6a - 4b$. More advanced tasks include the expansion of cubic expressions or the product of three binomials, such as $(x - 2)(2x + 5)(x + 3)$ or $(2x + 3)(x - 2)^2$. Simplification also extends to algebraic fractions, where students must factorise numerators and denominators to cancel common factors, exemplified by expressions like $\frac{2x^2 - 5x - 12}{3x^2 - 12x}$ or $\frac{3xy + 36y - 5x - 60}{2x^2 - 288}$.

Factorisation is categorized into several methods: common factors, grouping, difference of two squares, and quadratic factorisation. Common factor examples include $18px - 27p = 9p(2x - 3)$ and $14xy - 7y^2 = 7y(2x - y)$. Factorisation by grouping is used for four-term expressions like $mt - n - m + nt$, which can be rearranged and factorised as $(m + n)(t - 1)$, or $px + py - x - y$, yielding $(x + y)(p - 1)$. The difference of two squares is frequently applied to expressions like $5x^2 - 20y^2$, which factorises to $5(x - 2y)(x + 2y)$, or $2t^2 - 98m^2$. Completing the square is another vital technique, where quadratic expressions like $x^2 + 8x + 10$ are written in the form $(x + p)^2 + q$, which helps in identifying the vertex of a parabola or solving the equation.

Geometric Principles: Angles, Lines, and Polygons

Geometry topics encompass the relationships within parallel lines, triangles, and complex polygons. When two straight lines intersect two parallel lines, alternate and corresponding angles are created. For example, if interior angles are given as $71^\circ$ and $55^\circ$ in a configuration involving parallel lines, students must use the properties of alternate angles or angles on a straight line to find unknown values like $x$. In triangles, properties such as isosceles triangles (where two sides and two angles are equal) are frequently tested, such as finding the base angles when the vertex angle is $34^\circ$. Specific cases include triangle $ABC$ where $AB = AC$ and angle $BAC = 38^\circ$, leading to the calculation of adjacent exterior angles like angle $ACD$.

Regular polygons are studied through their interior and exterior angle sums. The sum of interior angles of an $n$-sided polygon is $(n - 2) \times 180^\circ$, while the exterior angles always sum to $360^\circ$. For a regular polygon, one exterior angle is $\frac{360^\circ}{n}$. Problems often involve working backwards to find the number of sides, such as when a regular polygon has an interior angle of $178.5^\circ$ or an exterior angle of $36^\circ$. Particular focus is given to regular dodecagons ($12$-sides) with a side length of $6\,cm$, where students must prove the interior angle is $150^\circ$ and calculate the radius of the circumscribed circle as approximately $11.6\,cm$. Other named polygons include pentagons (angles in ratios like $4:5:5:7:9$), octagons, decagons, and regular hexagons (where students might calculate the area given a side length like $7.4\,cm$). Geometrical reasons, such as "alternate angles," "corresponding angles," or "opposite angles in a cyclic quadrilateral are supplementary," must be explicitly stated in proofs.

Functional Notation and Composite Functions

The study of functions involves evaluating expressions for specific inputs, finding inverse functions, and working with composite functions. For a given set of functions such as $f(x) = 5x - 3$ and $g(x) = \frac{10}{x - 2}$, the composite function $gf(x)$ is found by substituting $f(x)$ into $g(x)$, resulting in $g(5x - 3) = \frac{10}{(5x - 3) - 2}$. Inverse functions, denoted as $f^{-1}(x)$, are found by setting $y = f(x)$ and rearranging to make $x$ the subject. For instance, if $j(x) = 5 - 2x$, its inverse involves a multi-step rearrangement. Composite operations can become iterative, as in $ff(x - 1)$ or $jj(2)$.

Variables and coefficients within functions are often determined through specific solve-for-variable tasks. If $f(x) = kx^2$ and $f(-5k) = 675$, the value of $k$ is found by solving $k(-5k)^2 = 675$. Exponential functions, such as $h(x) = 2^x$ or $h(x) = 3^x$, introduce logarithmic concepts when solving for $x$ in equations like $h^{-1}(x) = 10$ or $h(x) = 256$. Students are also required to perform algebraic operations on functions, such as writing $f(x) - h(x)$ as a single fraction over a common denominator or finding constants $a, b, c$ in equations like $(f(x))^2 - ff(x) = ax^2 + bx + c$.

Rearranging Formulas and solving Equations

Rearranging formulas, or "changing the subject," is a fundamental algebraic skill. Students must isolate a specific variable in formulas like $P = 2r + \pi r$ (rearranging for $r$) or $A = \pi rl + \pi r^2$ (rearranging for $l$). More complex forms include variables in denominators or under square roots, such as $m = 2p + \sqrt{\frac{x + 1}{x}}$ or $y = \frac{3x - 2}{1 - x}$, requiring cross-multiplication and factoring to isolate the target variable.

Equation solving spans linear, quadratic, and simultaneous systems. Simultaneous equations are solved using elimination or substitution methods. Typical systems include $4x - 3y = 26$ and $5x + 6y = 13$, or application-based problems such as buying items: Natalie buys $4$ tomato plants and $3$ pepper plants for $\$9.35$, while Samir buys $2$ tomato plants and $11$ pepper plants for $\$16.55$. Quadratic equations like $3x^2 + 6x - 22 = 0$ are solved using the quadratic formula, giving solutions correct to a specified number of significant figures, such as $4\,sig\,figs$ or $2\,decimal\,places$. Non-linear simultaneous equations, such as the intersection of a line $y = 2x - 3$ and a curve $y = x^2 - 9x + 21$, provide the coordinates of intersection points $A$ and $B$.

Proportionality and Variation

Relationships between variables are categorized as direct or inverse proportionality. In direct variation, $y$ is proportional to a power of $x$, expressed as $y = kx^n$, such as $y$ being proportional to $(x - 1)^2$. If $x = 3$ and $y = 24$, the constant $k$ is calculated before finding $y$ for other values of $x$. Inverse variation follows the form $y = \frac{k}{x^n}$, often involving the square root of a variable, as in $y$ being inversely proportional to $\sqrt{x - 2}$.

Real-world applications of variation include physics problems, such as the force of attraction $F$ in Newtons between two magnets being inversely proportional to the square of the distance $d\,cm$ between them. In another scenario, a ball falls a distance $d\,metres$ in $t\,seconds$ where $d$ is directly proportional to $t^2$; given that a ball falls $44.1\,m$ in $3\,seconds$, a formula is derived to predict the distance for other time intervals, such as $t = 2$. Variation can also be linked, where $y$ is inversely proportional to $\sqrt{x}$ and $x$ is directly proportional to $w^2$, requiring an expression for $y$ in terms of $w$.

Inequalities and Linear Programming

Linear programming involves defining a region $R$ on a coordinate grid using a set of inequalities. Students must identify the boundaries, such as $y \leq 2$, $x < 3$, and $y \leq x + 4$, and shade the unwanted regions to leave the feasible region $R$ clear. Inequalities can be derived from practical constraints, such as Raheem making baskets ($x$) and mats ($y$) where he works a maximum of $22.5\,hours$ and takes specific times for each ($2\,hours$ per basket, $1.5\,hours$ per mat), leading to the inequality $3x + 2y \leq 30$.

Once the region $R$ is defined, students often perform optimization, such as finding the largest value of a profit function like $40x + 28y$ for integer coordinates within the region. Simple inequalities include finding all positive integers $n$ satisfying $3n - 8 < 5n - 15$ or solving $15 < 4n < 28$ for integer values. Another example includes a car hire company with $x$ small cars and $y$ large cars where constraints like "the number of large cars is less than or equal to the number of small cars" are translated into $y \leq x$.

Indices and Radical Expressions

The laws of indices govern the multiplication, division, and exponentiation of powers. Operations include multiplying terms like $2x^2 \times 5x^5 = 10x^7$ and dividing terms like $y^3 \div y^3 = 1$ or $32g^{32} \div 4g^4 = 8g^{28}$. Zero and negative indices are also explored, such as $7x^0 = 7$ (provided $x \neq 0$) and $4^{-2} = \frac{1}{16}$. Fractional indices represent roots, with expressions like $(81x^{16})^{\frac{3}{4}}$ requiring the application of the power to both the coefficient and the variable. More complex power equations involve solving for unknowns in the exponent, such as finding $x$ when $32 \times 3^x = 81$ or finding $k$ when $16 = 64^k$. Prime factorisation is also utilized to simplify powers, such as writing $243 \times 27^{2n}$ as a single power of $3$.