Notes_on_Mathematics_for_TMUA_and-ESAT_M2_June2025.pdf

# Algebra and functions This section consolidates foundational algebraic concepts, including the manipulation of indices and surds, and delves into the properties and solutions of quadratic functions [6](#page=6). ### 1.1 Laws of indices Indices, also known as powers or exponents, provide a concise notation for repeated multiplication [6](#page=6). #### 1.1.1 Basic rules of indices The fundamental rules of indices are derived from the definition of exponents: * **Rule 1: Multiplication of powers** When multiplying terms with the same base, add the exponents: $a^m \times a^n \equiv a^{m+n}$ [6](#page=6). This rule holds for all real numbers $m$ and $n$, provided $a > 0$ [8](#page=8). * **Rule 2: Fractional exponents and roots** A fractional exponent indicates a root: $a^{\frac{1}{n}} = \sqrt[n]{a}$ [7](#page=7). For example, $a^{\frac{1}{3}}$ is the cube root of $a$ [7](#page=7). * **Rule 3: Negative exponents** A negative exponent indicates the reciprocal of the base raised to the positive exponent: $a^{-m} = \frac{1}{a^m}$ [7](#page=7). This rule applies when $a > 0$ [7](#page=7). * **Rule 4: Zero exponent** Any positive number raised to the power of zero is one: $a^0 = 1$ [8](#page=8). This rule applies when $a > 0$ [8](#page=8). * **Rule 5: Division of powers** When dividing terms with the same base, subtract the exponents: $a^m \div a^n = \frac{a^m}{a^n} = a^m \times a^{-n} = a^{m-n}$ [8](#page=8). * **Rule 6: Power of a power** When raising a power to another power, multiply the exponents: $(a^m)^n = a^{mn}$ [8](#page=8). It is important to distinguish between $(a^m)^n$ and $a^{m^n}$. For example, $(a^3)^2 = a^6$ while $a^{3^2} = a^9$ [8](#page=8). * **Rule 7: Combined fractional and integer powers** This rule combines fractional and integer exponents: $a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ [8](#page=8). #### 1.1.2 Consistency and extension of rules The index laws are designed to be consistent across all rational and real exponents [7](#page=7) [9](#page=9). * **Irrational exponents:** For a positive base $a$, the index laws still apply when exponents are irrational. The values of $a^m$ where $m$ is irrational are defined using limits, filling the gaps between rational approximations to ensure the continuity of graphs of exponential functions like $y = a^x$ [9](#page=9). * **Zero base:** Issues arise when the base $a=0$. While $0^m = 0$ for $m>0$, the case $0^0$ is indeterminate [10](#page=10). * **Negative base:** The index laws become problematic with negative bases, particularly for fractional exponents, as they can lead to non-real numbers (e.g., the square root of a negative number). For this reason, the laws are typically applied with positive bases [10](#page=10). ### 1.2 Use and manipulation of surds Surds are expressions involving roots, typically square roots, that cannot be simplified to rational numbers. They are used to express irrational numbers exactly [11](#page=11). #### 1.2.1 Convention and simplification By convention, $\sqrt{a}$ always denotes the positive square root [11](#page=11). * **Simplifying roots:** Surds can be simplified by factoring out perfect squares from under the radical sign. Example: $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$ [11](#page=11). * **Multiplying surds:** Treat square roots as variables during multiplication and simplify at the end. Example: $(2 + 3\sqrt{5})^2 = 2^2 + 2 (3\sqrt{5}) + (3\sqrt{5})^2 = 4 + 12\sqrt{5} + 9 = 49 + 12\sqrt{5}$ [11](#page=11) [2](#page=2) [5](#page=5). * **Factorising surds:** This involves recognising patterns, often related to the square of a binomial containing a surd. Example: To factorise $49 + 12\sqrt{5}$, one might suspect it is of the form $(a + b\sqrt{5})^2$. By comparing terms, we find $a=2$ and $b=3$, so $49 + 12\sqrt{5} = (2 + 3\sqrt{5})^2$ [12](#page=12). #### 1.2.2 Rationalising the denominator Rationalising the denominator is the process of removing a surd from the denominator of a fraction by multiplying by a suitable form of 1 [12](#page=12). * **Simple cases:** For a denominator like $\sqrt{a}$, multiply by $\frac{\sqrt{a}}{\sqrt{a}}$: $\frac{1}{\sqrt{a}} = \frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}$ [13](#page=13). * **Complex cases:** For denominators of the form $a + \sqrt{b}$ or $a - \sqrt{b}$, use the difference of two squares formula: $x^2 - y^2 = (x+y)(x-y)$. To rationalise $\frac{3}{2+\sqrt{5}}$, multiply by $\frac{2-\sqrt{5}}{2-\sqrt{5}}$: $\frac{3}{2+\sqrt{5}} \times \frac{2-\sqrt{5}}{2-\sqrt{5}} = \frac{3(2-\sqrt{5})}{2^2 - (\sqrt{5})^2} = \frac{6-3\sqrt{5}}{4-5} = \frac{6-3\sqrt{5}}{-1} = 3\sqrt{5} - 6$ [13](#page=13) [14](#page=14). The key is to multiply by the conjugate of the denominator. ### 1.3 Quadratic functions and their graphs Quadratic functions are polynomial functions of degree two, typically expressed in the form $ax^2 + bx + c$, where $a \neq 0$. They are fundamental in mathematics due to their relative simplicity and rich properties [15](#page=15). #### 1.3.1 Completing the square Completing the square is an algebraic technique used to rewrite a quadratic expression into a form that reveals its vertex and facilitates solving [15](#page=15). For $y = x^2 + bx + c$, completing the square yields: $y = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4} + c$ [16](#page=16). This form shows that the graph of $y = x^2 + bx + c$ is a translation of the graph of $y = x^2$. The vertex (minimum or maximum point) is located at $\left(-\frac{b}{2}, -\frac{b^2}{4} + c\right)$ [16](#page=16). For the more general quadratic $y = ax^2 + bx + c$: $y = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a}$ [17](#page=17). The vertex for this general form is at $\left(-\frac{b}{2a}, -\frac{b^2 - 4ac}{4a}\right)$ [18](#page=18). #### 1.3.2 The discriminant The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $\Delta = b^2 - 4ac$. It determines the nature of the roots (solutions) [18](#page=18) [19](#page=19): * If $\Delta > 0$, there are two distinct real roots. The graph crosses the x-axis at two distinct points [19](#page=19). * If $\Delta = 0$, there is one repeated real root. The graph touches the x-axis at exactly one point (the vertex) [19](#page=19). * If $\Delta < 0$, there are no real roots. The graph does not intersect the x-axis [19](#page=19). #### 1.3.3 Solving quadratic equations Quadratic equations can be solved using several methods: * **Factorisation:** If the quadratic can be easily factored, this is often the quickest method [15](#page=15). * **Completing the square:** As shown above, this method can be used to derive the quadratic formula [15](#page=15). * **The quadratic formula:** Derived from completing the square, the quadratic formula provides the solutions for $ax^2 + bx + c = 0$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ [18](#page=18). The expression under the square root is the discriminant [18](#page=18). #### 1.3.4 Graphs of quadratic functions The graph of a quadratic function is a parabola [15](#page=15). * **Shape:** If $a > 0$, the parabola opens upwards (U-shaped). If $a < 0$, it opens downwards (∩-shaped) [16](#page=16). * **Vertex:** The minimum point (if $a>0$) or maximum point (if $a<0$) of the parabola. Its x-coordinate is $-\frac{b}{2a}$ [16](#page=16). * **Roots (x-intercepts):** The points where the graph intersects the x-axis, found by setting $y=0$ and solving the quadratic equation [15](#page=15) [16](#page=16). * **Y-intercept:** The point where the graph intersects the y-axis, found by setting $x=0$, which results in $y=c$ [15](#page=15). * **Line of symmetry:** A vertical line passing through the vertex, with the equation $x = -\frac{b}{2a}$ [18](#page=18). > **Tip:** When sketching quadratic graphs, always mark the x-intercepts (roots), the y-intercept, and the coordinates of the vertex [15](#page=15). --- # Simultaneous equations and inequalities Simultaneous equations and inequalities involve finding values that satisfy multiple conditions, either through exact matches (equations) or ranges (inequalities). ## 2. Simultaneous equations and inequalities This section explores the concept of simultaneous equations, focusing on methods for solving them, and then delves into the principles and techniques for solving linear and quadratic inequalities. ### 2.1 Solving simultaneous equations Simultaneous equations require finding values for variables that satisfy all given equations concurrently. Geometrically, the solution to simultaneous equations represents the point(s) where the graphs of the individual equations intersect [21](#page=21) [22](#page=22). #### 2.1.1 Linear simultaneous equations Linear equations, whose graphs are straight lines, can be solved simultaneously using several methods: * **Substitution:** Solve one equation for one variable and substitute that expression into the other equation [22](#page=22). * **Elimination:** Manipulate the equations (e.g., by multiplying by constants) so that the coefficients of one variable are opposites, then add or subtract the equations to eliminate that variable [22](#page=22). * **Graphical:** Plot both lines and identify the coordinates of their intersection point(s) [22](#page=22). The number of solutions for a pair of linear simultaneous equations depends on the relationship between the lines: * **No solutions:** If the lines are parallel and distinct, they never intersect. This occurs when the equations have the same ratio of $x$ to $y$ coefficients but different constant terms, e.g., $y + 2x = 4$ and $y + 2x = 8$ [24](#page=24). * **Infinitely many solutions:** If the lines are identical (parallel and coinciding), every point on the line is a solution. This occurs when one equation is a multiple of the other, e.g., $y + 2x = 4$ and $2y + 4x = 8$ [24](#page=24). * **One solution:** If the lines are not parallel, they will intersect at exactly one point [24](#page=24). #### 2.1.2 Simultaneous equations with one linear and one quadratic equation When solving a system with one linear and one quadratic equation, the general approach involves eliminating one variable to obtain a single quadratic equation. The solutions to this quadratic equation will then determine the nature of the intersection between the line and the quadratic curve [25](#page=25). The graphical interpretation reveals three possibilities: * **Two distinct solutions:** The line intersects the quadratic at two separate points [25](#page=25). * **One repeated solution:** The line is tangent to the quadratic, intersecting at exactly one point [25](#page=25). * **No real solutions:** The line does not intersect the quadratic curve [25](#page=25). The number of real solutions can be determined by examining the discriminant of the resulting quadratic equation after elimination [26](#page=26). **Example:** Solve $y = x^2 + 3x + 2$ and $y = x + 1$ [26](#page=26). Eliminate $y$: $x^2 + 3x + 2 = x + 1$ Rearrange into a quadratic equation: $x^2 + 2x + 1 = 0$ Factorize: $(x + 1)^2 = 0$ This gives a single solution $x = -1$. Substituting back into $y = x + 1$, we get $y = -1 + 1 = 0$. Thus, the solution is $x = -1, y = 0$, indicating the line is tangent to the quadratic [26](#page=26). ### 2.2 Solving linear and quadratic inequalities When working with inequalities, it is crucial to remember that they do not always behave like equations. While addition and subtraction can be performed on both sides without changing the inequality, multiplication, division, raising to an even power, or applying certain functions require careful consideration of the sign of the operation to avoid reversing the inequality or introducing extraneous solutions [27](#page=27). **Key principles for inequalities:** * Multiplying or dividing by a negative number reverses the inequality sign [27](#page=27). * Squaring both sides can introduce false solutions if one side is negative [27](#page=27). #### 2.2.1 Linear inequalities Linear inequalities can often be solved by simple algebraic rearrangement, similar to solving linear equations, but maintaining the inequality sign [28](#page=28). **Example:** Solve $3x + 2 \leq x + 5$ [28](#page=28). Subtract $x$ from both sides: $2x + 2 \leq 5$ Subtract 2 from both sides: $2x \leq 3$ Divide by 2 (which is positive, so the inequality sign remains): $x \leq \frac{3}{2}$ #### 2.2.2 Quadratic inequalities Quadratic inequalities are typically solved by factorizing the quadratic expression and then considering the sign of the expression for different intervals of $x$ values, often aided by a sketch of the corresponding quadratic graph [28](#page=28). **Example:** Solve $x^2 + 5x + 6 \geq 0$ [28](#page=28). Factorize the quadratic: $(x + 2)(x + 3) \geq 0$ The roots are $x = -2$ and $x = -3$. Sketching the parabola $y = x^2 + 5x + 6$ shows that the graph is above or on the x-axis ($ \geq 0$) when $x \leq -3$ or $x \geq -2$ [28](#page=28). #### 2.2.3 Rational inequalities For rational inequalities (those involving fractions), a common and safe strategy is to multiply both sides by the square of the denominator. This ensures that the multiplier is always positive and does not affect the direction of the inequality [29](#page=29). **Example:** Solve $\frac{2x + 5}{x + 3} > 1$ [29](#page=29). Multiply both sides by $(x + 3)^2$ (which is always positive for real $x$): $(\frac{2x + 5}{x + 3})(x + 3)^2 > 1 \cdot (x + 3)^2$ $(2x + 5)(x + 3) > (x + 3)^2$ Rearrange and simplify: $(2x + 5)(x + 3) - (x + 3)^2 > 0$ $(x + 3)[(2x + 5) - (x + 3)] > 0$ $(x + 3)(x + 2) > 0$ This inequality is solved similarly to quadratic inequalities, yielding $x < -3$ or $x > -2$ [29](#page=29). #### 2.2.4 Inequalities involving modulus The modulus of an expression, $|a|$, represents the positive distance of $a$ from zero. Therefore, $|x - c|$ can be interpreted as the distance between $x$ and $c$ on the number line [30](#page=30) [31](#page=31). Inequalities involving modulus can be solved graphically by sketching the graphs of the modulus functions and comparing their values, or by interpreting them as distance comparisons [31](#page=31). **Example:** Solve $|x - 3| < |x - 5|$ [31](#page=31). This inequality asks for the values of $x$ that are closer to 3 than to 5. On the number line, the midpoint between 3 and 5 is 4. Any value of $x$ less than 4 will be closer to 3 than to 5. Therefore, the solution is $x < 4$ [32](#page=32). **Example:** Solve $|2x - 4| < |x + 2|$ [32](#page=32). This can be rewritten as $2|x - 2| < |x + 2|$. Graphically, this asks for values of $x$ where the graph of $|2x - 4|$ is below the graph of $|x + 2|$. The solutions are found to be $\frac{2}{3} < x < 6$ [32](#page=32). > **Tip:** Understanding the geometric interpretation of inequalities, especially those involving modulus, can often provide a more intuitive approach to solving them. Relying solely on algorithmic manipulation without conceptual understanding can be limiting [33](#page=33). --- # Sequences, series, and binomial expansion This section delves into fundamental mathematical concepts including arithmetic and geometric sequences and series, alongside the principles of binomial expansion. ### 3.1 Sequences and series A **sequence** is an ordered list of numbers, which can be infinite. A **series** is the sum of the terms in a sequence. A **progression** is a general term used for either a sequence or a series [47](#page=47). When working with sequences generated by recurrence relations, such as $x_{n+2} = |x_n - x_{n+1}|$, it is crucial to write out enough terms to identify a stable pattern. The number of terms needed depends on the structure of the recurrence relation, as each term might depend on the preceding few. For instance, a recurrence relation involving two previous terms requires observing a repeat in two adjacent terms before a pattern can be reliably deduced [47](#page=47). **Sigma notation** ($\sum$) is used for sums. When using sigma notation for a sum, it is important to: 1. Pay close attention to the limits of summation; the number of terms is often one more than the difference between the upper and lower limits (the "fence post" issue) [48](#page=48). 2. Write out the first few terms of the sum to understand the pattern [48](#page=48). 3. Consider the behavior at the end of the sum, especially if patterns have been identified [48](#page=48). #### 3.1.1 Arithmetic series An **arithmetic series** (or arithmetic progression, AP) is a sequence where the difference between consecutive terms is constant. Key terminology and formulae include: * First term: $a$ [49](#page=49). * Common difference: $d$ [49](#page=49). * $n$th term: $u_n = a + (n-1)d$ [49](#page=49). * Sum of the first $n$ terms: $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + u_n)$ [49](#page=49). The sum can be interpreted as $S_n = n \times (\text{average of the first and last term})$. The difference between consecutive terms is always the common difference: $u_{n+1} - u_n = d$ [49](#page=49). Linear combinations of arithmetic series are also arithmetic series. For example, the sum of two arithmetic series $S_n$ and $T_n$ with first terms $a$ and $A$ and common differences $d$ and $D$ respectively, results in a new arithmetic series with first term $a+A$ and common difference $d+D$ [49](#page=49). #### 3.1.2 Geometric series A **geometric series** (or geometric progression, GP) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Key terminology and formulae include: * First term: $a$ (can also be written as $ar^0$) [51](#page=51). * Common ratio: $r$ [51](#page=51). * $n$th term: $u_n = ar^{n-1}$ [51](#page=51). * Sum of the first $n$ terms: $S_n = \frac{a(1 - r^n)}{1 - r}$ [51](#page=51). * Sum to infinity (for a convergent series where $|r| < 1$): $S_\infty = \frac{a}{1 - r}$ [51](#page=51). Sigma notation can be used for GPs: $S_n = \sum_{k=1}^{n} ar^{k-1} = \sum_{k=0}^{n-1} ar^k$ [51](#page=51). The ratio of adjacent terms is constant: $\frac{u_{n+1}}{u_n} = r$ [51](#page=51). New GPs can be derived from existing ones. For example, replacing $r$ with $-r$ in a GP results in a GP with the same first term $a$ but a common ratio of $-r$, leading to the sum to infinity $S_\infty = \frac{a}{1 + r}$. Combining GPs can also yield new ones [52](#page=52): * $\frac{1}{2}(S_n + S_n(-r))$ forms a GP with first term $a$ and common ratio $r^2$ [52](#page=52). * Squaring every term of a GP results in a new GP with first term $a^2$ and common ratio $r^2$ [52](#page=52). * Raising every term to the power of $k$ creates a GP with first term $a^k$ and common ratio $r^k$ [53](#page=53). When considering the sum of a part of a GP, such as $ar^m + ar^{m+1} + \dots + ar^n$, it is important to correctly determine the number of terms, which is $n - m + 1$. This sum can be tackled in several ways [53](#page=53): * **Method 1: Difference of two sums**: $S_{n+1} - S_m$. Note the $n+1$ in $S_{n+1}$ [53](#page=53). * **Method 2: Factorization**: Factor out $r^m$ to get $r^m(a + ar + \dots + ar^{n-m})$ or factor out $ar^m$ to get $ar^m(1 + r + \dots + r^{n-m})$. The expression in the bracket is a GP with $u_1=a$ and ratio $r$ and $n-m+1$ terms [54](#page=54). * **Method 3: New GP**: Treat it as a new GP with first term $ar^m$, common ratio $r$, and $n-m+1$ terms [54](#page=54). ### 3.2 Binomial expansion The binomial theorem provides a formula for expanding expressions of the form $(a+f(x))^n$. For examinations, it's essential to know the formulae and their applications. The primary skills required are calculating binomial coefficients and finding specific terms in expansions [55](#page=55). #### 3.2.1 Binomial coefficients and notation The notation $n!$ represents the factorial of $n$, which is the product of all positive integers up to $n$ ($n! = n \times (n-1) \times \dots \times 2 \times 1$). By convention, $0! = 1$ [59](#page=59). The binomial coefficient $\binom{n}{k}$ (read as "n choose k") represents the number of ways to choose $k$ items from a set of $n$ distinct items without regard to the order of selection. It is calculated as [59](#page=59): $$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$ This formula can be understood by considering the number of ordered arrangements ($n!/(n-k)!$) and then dividing by the number of ways to order the chosen $k$ items ($k!$) to account for combinations [60](#page=60). A key property of binomial coefficients is their symmetry: $$ \binom{n}{k} = \binom{n}{n-k} $$ This arises because choosing $k$ items to keep is equivalent to choosing $n-k$ items to discard [60](#page=60). #### 3.2.2 The binomial theorem The binomial theorem states that for any non-negative integer $n$: $$ (a + f(x))^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} [f(x)]^k $$ or equivalently: $$ (a + f(x))^n = \sum_{k=0}^{n} \binom{n}{k} [f(x)]^k a^{n-k} $$ The term $[f(x)]^k a^{n-k}$ arises from choosing $f(x)$ from $k$ of the $n$ brackets and $a$ from the remaining $n-k$ brackets. The number of ways to do this is $\binom{n}{k}$ [61](#page=61) [62](#page=62). **Key points for binomial expansion:** 1. The powers of the terms in the expansion must sum to the overall power $n$ [56](#page=56). 2. The top number in the binomial coefficient $\binom{n}{k}$ is always the power that the bracket is raised to ($n$) [56](#page=56). 3. The bottom number ($k$) in $\binom{n}{k}$ can correspond to the power of either term in the expansion, due to the symmetry $\binom{n}{k} = \binom{n}{n-k}$ [56](#page=56). **Common Pitfalls:** * Forgetting to raise the coefficient of $f(x)$ to the power $k$ [56](#page=56). * Forgetting to include the sign (if negative) when raising a term to a power [57](#page=57). > **Tip:** Using the binomial expansion formula directly is often faster and more efficient than using Pascal's triangle for higher powers [55](#page=55). > **Tip:** When calculating terms, ensure that all parts of the term (including negative signs) are raised to the specified power [57](#page=57). --- # Coordinate geometry and trigonometry This topic covers the fundamental concepts of coordinate geometry, including the equations of straight lines and circles, and extends into trigonometry, encompassing radian measure, arc lengths, sector areas, and trigonometric functions and equations. ### 4.1 Straight lines The equation of a straight line can be represented in several forms, most commonly $y = mx + c$ and $ax + by + c = 0$ [63](#page=63). #### 4.1.1 Gradient ($m$) The gradient ($m$) of a straight line represents its "steepness" [63](#page=63). * A positive gradient indicates a line sloping upwards from bottom left to top right, while a negative gradient indicates a line sloping downwards from top left to bottom right [63](#page=63). * The gradient can be understood as the vertical change for every 1 unit of horizontal change [63](#page=63). * Alternatively, it represents the rate at which $y$ changes relative to $x$; as $x$ increases by 1, $y$ changes by $m$ [63](#page=63). * The gradient is also equal to the tangent of the angle the line makes with the positive $x$-axis, assuming equal scales on both axes. For a line at 45 degrees, the gradient is $\tan 45^\circ = 1$, and for 135 degrees, it is $\tan 135^\circ = -1$ [64](#page=64). #### 4.1.2 Special cases for gradients * **Horizontal lines** have a gradient of zero. Their equation is of the form $y = \text{constant}$. The $x$-axis is a horizontal line with the equation $y = 0$ [64](#page=64). * **Vertical lines** are often said to have an infinite or negative infinite gradient, but it is best to treat them as a special case. Their equation is always of the form $x = \text{constant}$. The $y$-axis is a vertical line with the equation $x = 0$ [64](#page=64). #### 4.1.3 Parallel and perpendicular lines * **Parallel lines** (excluding horizontal and vertical lines) have the same gradient. Two lines $y = m_1x + c_1$ and $y = m_2x + c_2$ are parallel if and only if $m_1 = m_2$ [65](#page=65). * **Perpendicular lines** have a relationship between their gradients such that $m_1m_2 = -1$ [65](#page=65). #### 4.1.4 The $y$-intercept ($c$) In the equation $y = mx + c$, the value of $c$ represents the $y$-intercept, which is the $y$-coordinate where the line crosses the $y$-axis [66](#page=66). #### 4.1.5 Finding the equation of a line The equation of a line can be uniquely specified by two pieces of information: * **Case 1: A point and the gradient** Given a point $(x_1, y_1)$ and gradient $m$. The gradient of the line joining any point $(x, y)$ on the line to $(x_1, y_1)$ must be constant: $$ \frac{y - y_1}{x - x_1} = m $$ Rearranging this gives the point-slope form: $y - y_1 = m(x - x_1)$ [67](#page=67). Alternatively, substitute the point $(x_1, y_1)$ into $y = mx + c$ to find $c$ [67](#page=67). * **Case 2: Two distinct points** Given two points $(x_1, y_1)$ and $(x_2, y_2)$. Let $(x, y)$ be a general point on the line. The gradient can be calculated in two ways and equated: $$ \frac{y - y_1}{x - x_1} = \frac{y_1 - y_2}{x_1 - x_2} $$ Rearrange this equation to find the line's equation [67](#page=67). Alternatively, calculate the gradient $m = \frac{y_1 - y_2}{x_1 - x_2}$ first, and then use $y = mx + c$, substituting one of the points to find $c$ [68](#page=68). The form $ax + by + c = 0$ is also common, and one should be able to convert between $y = mx + c$ and $ax + by + c = 0$ [68](#page=68). ### 4.2 Coordinate geometry of the circle The equation of a circle can be expressed in two standard forms: * **Standard form:** $(x - a)^2 + (y - b)^2 = r^2$, where $(a, b)$ is the centre and $r$ is the radius [70](#page=70). This form is derived from the distance formula (Pythagoras' theorem), stating that the distance from any point $(x, y)$ on the circle to the centre $(a, b)$ is equal to the radius $r$ [70](#page=70). For a circle with its centre at the origin $(0, 0)$, the equation simplifies to $x^2 + y^2 = r^2$ [69](#page=69). * **General form:** $x^2 + y^2 + ax + by + c = 0$ [71](#page=71). To find the centre and radius from this form, "completing the square" is used: 1. Group the $x$ terms and $y$ terms: $(x^2 + ax) + (y^2 + by) + c = 0$. 2. Complete the square for each group: $(x + \frac{a}{2})^2 - (\frac{a}{2})^2 + (y + \frac{b}{2})^2 - (\frac{b}{2})^2 + c = 0$. 3. Rearrange to match the standard form: $(x + \frac{a}{2})^2 + (y + \frac{b}{2})^2 = (\frac{a}{2})^2 + (\frac{b}{2})^2 - c$. From this, the centre is $(-\frac{a}{2}, -\frac{b}{2})$ and the radius is $r = \sqrt{(\frac{a}{2})^2 + (\frac{b}{2})^2 - c}$ [71](#page=71). #### 4.2.1 Conditions for the general form to represent a circle Not all equations of the form $x^2 + y^2 + ax + by + c = 0$ represent a circle. For it to be a circle, the term under the square root for the radius must be positive: $(\frac{a}{2})^2 + (\frac{b}{2})^2 - c > 0$. If this value is zero, it represents a single point (the centre); if it's negative, there are no real solutions, and it's not a circle [72](#page=72). For an equation $px^2 + qy^2 + ax + by + c = 0$ to represent a circle, $p$ must equal $q$, and both must be non-zero. Additionally, the conditions for completing the square must result in a positive radius squared [73](#page=73). #### 4.2.2 Circle properties and intersections * **Tangency:** A line is tangent to a circle if it intersects the circle at exactly one point. This can be found by substituting the line's equation into the circle's equation and solving the resulting quadratic. For tangency, the discriminant of the quadratic must be zero [74](#page=74). * **Distance from a line to a circle:** The closest distance between a line and a circle is found by determining the distance from the centre of the circle to the line and then subtracting the radius [75](#page=75). ### 4.3 Circle properties (Theorems) The following circle properties are essential: * The perpendicular from the centre to a chord bisects the chord [77](#page=77). * The tangent at any point on a circle is perpendicular to the radius at that point [77](#page=77). * The angle subtended by an arc at the centre of a circle is twice the angle subtended by the arc at any point on the circumference [77](#page=77). * The angle in a semicircle is a right angle (90°) [77](#page=77). * Angles in the same segment are equal [77](#page=77). * Opposite angles in a cyclic quadrilateral add up to 180° [77](#page=77). * The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment [77](#page=77). Techniques for solving problems involving circle theorems include angle chasing, rotating diagrams, adding lines, and using dynamic geometry methods [78](#page=78). ### 4.4 Trigonometry #### 4.4.1 Area of a triangle The area of a triangle can be calculated using the formula: $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$ where the height is perpendicular to the base [79](#page=79). An alternative formula using trigonometry is: $$ \text{Area} = \frac{1}{2}ab\sin C $$ where $a$ and $b$ are the lengths of two sides, and $C$ is the angle between them [80](#page=80). #### 4.4.2 The sine rule The sine rule can be derived from the area formula. For a triangle with sides $a, b, c$ and opposite angles $A, B, C$: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$ This rule is useful for finding unknown sides or angles when: * Two angles and one side are known [81](#page=81). * Two sides and one angle (not between the given sides) are known [81](#page=81). **Ambiguous case:** When using the sine rule with two sides and an angle (SSA), there can be one, two, or no possible triangles. This arises because for any angle $\theta$ (other than 90°), there are two angles between 0° and 180° that have the same sine value: $\theta$ and $180^\circ - \theta$ (#page=81, 82) [81](#page=81) [82](#page=82). #### 4.4.3 The cosine rule The cosine rule is a generalization of Pythagoras' theorem for non-right-angled triangles: $$ a^2 = b^2 + c^2 - 2bc\cos A $$ This rule can be rearranged to find an angle: $$ \cos A = \frac{b^2 + c^2 - a^2}{2bc} $$ The cosine rule is used when: * Three sides are known, to find an angle [86](#page=86). * Two sides and the angle between them are known, to find the third side [86](#page=86). #### 4.4.4 Radian measure Angles can be measured in radians as well as degrees. One radian is the angle subtended by an arc of length equal to the radius of the circle [87](#page=87). * One revolution is $360^\circ$ or $2\pi$ radians [87](#page=87). * Conversion formulas: * Degrees to radians: $\theta \text{ degrees} = \frac{\theta}{360} \times 2\pi \text{ radians}$ [88](#page=88). * Radians to degrees: $\alpha \text{ radians} = \frac{\alpha}{2\pi} \times 360 \text{ degrees}$ [88](#page=88). Standard angle conversions: * $30^\circ = \frac{\pi}{6}$ rad * $45^\circ = \frac{\pi}{4}$ rad * $60^\circ = \frac{\pi}{3}$ rad * $90^\circ = \frac{\pi}{2}$ rad * $180^\circ = \pi$ rad * $360^\circ = 2\pi$ rad [88](#page=88). #### 4.4.5 Arc length and area of a sector For a sector of a circle with radius $r$ and angle $\alpha$ (in radians): * Arc length: $L = r\alpha$ [89](#page=89). * Area of sector: $A = \frac{1}{2}r^2\alpha$ [89](#page=89). #### 4.4.6 Trigonometric values for standard angles The values of sine, cosine, and tangent for $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ should be known or quickly derivable using standard triangles [90](#page=90). * **45°:** Isosceles right-angled triangle with sides 1, 1, $\sqrt{2}$. * **30° and 60°:** Half of an equilateral triangle with sides 2, 1, $\sqrt{3}$. #### 4.4.7 Trigonometric functions: graphs, symmetries, and periodicity The graphs of $y = \sin x$, $y = \cos x$, and $y = \tan x$ have distinct shapes, symmetries, and periods [91](#page=91). * **Sine and Cosine:** Period $2\pi$ (or 360°). Symmetries about the $y$-axis (cosine) and origin (sine). * **Tangent:** Period $\pi$ (or 180°). Has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$ (or $90^\circ + n \times 180^\circ$). Modifications to the basic functions, such as $y = A \sin(Bx + C) + D$, affect amplitude ($A$), period ($B$), phase shift ($C$), and vertical shift ($D$) [91](#page=91). Sine and cosine can be visualized as "projection operators" onto the $x$ and $y$ axes, respectively. Tangent relates $x$-axis projections to $y$-axis projections (#page=92, 93) [92](#page=92) [93](#page=93). #### 4.4.8 Trigonometric identities Two fundamental identities are: * $\tan \theta = \frac{\sin \theta}{\cos \theta}$ [94](#page=94). * $\sin^2 \theta + \cos^2 \theta = 1$ [94](#page=94). These identities hold for any angle $\theta$. The second identity is a direct consequence of Pythagoras' theorem when considering trigonometric functions in a right-angled triangle [94](#page=94). #### 4.4.9 Solution of trigonometric equations Solving simple trigonometric equations involves finding values of the variable within a given interval [95](#page=95). * **Method:** Typically involves reducing the equation to a standard form (e.g., $\sin? = k$, $\cos? = k$, $\tan? = k$) using identities if necessary (#page=95, 98) [95](#page=95) [98](#page=98). * **Finding solutions:** Solutions can be found using a calculator for the principal value, then considering the periodicity and symmetries of the trigonometric function, often aided by graphs or CAST diagrams (#page=95, 96) [95](#page=95) [96](#page=96). * **Important Note:** When solving equations like $\sin^2(2x + 60^\circ) = \frac{1}{4}$, it is crucial to take both positive and negative square roots, leading to $\sin(2x + 60^\circ) = \frac{1}{2}$ and $\sin(2x + 60^\circ) = -\frac{1}{2}$ [95](#page=95). * **Order of operations:** It is best to find all possible values for the argument of the trigonometric function (e.g., $2x + 60^\circ$) before rearranging to solve for the variable (e.g., $x$), especially when the argument is multiplied by a constant, to avoid losing solutions (#page=96, 97) [96](#page=96) [97](#page=97). * **Mixed equations:** Trigonometric equations can be combined with other algebraic forms, such as quadratics. This often requires using identities (like $\cos^2 x = 1 - \sin^2 x$) to express the equation in terms of a single trigonometric function [98](#page=98). --- # Calculus: Differentiation and Integration This topic explores the fundamental concepts of differentiation as a measure of rates of change and the geometric interpretation of derivatives, alongside the inverse operation of integration, focusing on its applications in finding areas and solving differential equations . ### 5.1 Differentiation The derivative of a function $f(x)$ represents the gradient of the tangent to the graph $y = f(x)$ at a specific point. It also quantifies the rate of change of one measure with respect to another. For instance, speed is the rate of change of distance with respect to time, and acceleration is the rate of change of speed with respect to time. Similarly, the gradient of a line is the rate of change of $y$ with respect to $x$. For a curve, the rate of change at a point is defined as the gradient of the tangent to the curve at that point . #### 5.1.1 Notation for derivatives Various notations are used to represent derivatives: * $\frac{dy}{dx}$ . * $\frac{d^2y}{dx^2}$ (second-order derivative) . * $f'(x)$ . * $f''(x)$ (second-order derivative) . The "dot" notation, such as $\dot{t}$ for $\frac{dt}{dt}$ and $\ddot{t}$ for $\frac{d^2t}{dt^2}$, is commonly used in physics when differentiating with respect to time . #### 5.1.2 Differentiation of powers of x The fundamental rule for differentiating powers of $x$ is: $$ \frac{d}{dx} x^n = nx^{n-1} $$ . This rule applies for rational values of $n$, where $n \neq -1$ for integration . When differentiating an expression, it is often beneficial to simplify it into a sum of powers of $x$ first, then differentiate term by term. Differentiation can be performed term by term : $$ \frac{d}{dx} (x^3 + 7x^2 - 3x + 11) = \frac{d}{dx} x^3 + \frac{d}{dx} 7x^2 - \frac{d}{dx} 3x + \frac{d}{dx} 11 = 3x^2 + 14x - 3 + 0 $$ . > **Tip:** The TMUA/ESAT specification focuses on differentiating simple expressions involving sums of powers of $x$ and those that can be simplified to this form. Advanced rules like the chain rule or product rule are not required . #### 5.1.3 Applications of differentiation Differentiation has several applications, including: * **Gradients of tangents:** Finding the gradient of a curve at a given point . * **Tangents and normals:** Determining the equations of tangent and normal lines to a curve at a point . * **Stationary points:** Identifying points where the tangent to the curve is horizontal, meaning the gradient is zero ($\frac{dy}{dx} = 0$). These can be local maxima or minima . * **Classification of stationary points:** * **Using the second derivative:** If $\frac{d^2y}{dx^2} > 0$ at a stationary point, it is a minimum. If $\frac{d^2y}{dx^2} < 0$, it is a maximum . * **Caution:** The conditions $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} > 0$ (or $<0$) are sufficient but not always necessary. If $\frac{d^2y}{dx^2} = 0$, the point could be a minimum (e.g., $y = x^4$) or a maximum (e.g., $y = -x^4$) . * **Strictly increasing and decreasing functions:** * A function is strictly increasing if its derivative is positive: if $f'(x) > 0$, then the function is strictly increasing . * A function is strictly decreasing if its derivative is negative: if $f'(x) < 0$, then the function is strictly decreasing . > **Tip:** The specification uses "strictly increasing" and "strictly decreasing," which are slightly narrower definitions than "increasing" and "decreasing" used in some contexts. For the TMUA/ESAT, focus on continuous functions without sharp corners, such as polynomials . * **Points of inflexion:** A qualitative understanding of points of inflexion is expected, particularly in simple polynomial functions (where the concavity of the curve changes), but detailed identification using differentiation is not examined . ### 5.2 Integration Integration can be understood in two ways: as the reverse of differentiation, or as finding the area between a curve and the x-axis . #### 5.2.1 Indefinite and definite integrals * **Indefinite integral:** This is the reverse of differentiation. For example, integrating $x^2$ asks "what must be differentiated to get $x^2$?" The answer is $\frac{x^3}{3} + c$, where $c$ is the constant of integration. The constant $c$ is added because the derivative of any constant is zero . * **Definite integral:** This involves finding the "area" between a curve and the x-axis between specified limits (upper and lower bounds). Notation: $\int_{a}^{b} f(x) dx$ . > **Tip:** The symbol $\int$ is an elongated "s," signifying "sum." . #### 5.2.2 Definite integration and area A definite integral calculates the sum of areas above the x-axis minus the sum of areas below the x-axis. To find the actual total area between a curve and the x-axis, areas below the x-axis must be added as positive values. This often requires calculating integrals for segments above and below the x-axis separately and then summing their absolute values . > **Example:** To find the area between $y = x^2 - 1$ and the x-axis from $x=0$ to $x=2$: > Calculate $\int_{0}^{1} (x^2 - 1) dx = [ \frac{x^3}{3} - x ]_{0}^{1} = (\frac{1}{3} - 1) - = -\frac{2}{3}$. This represents an area of $\frac{2}{3}$ below the x-axis . > Calculate $\int_{1}^{2} (x^2 - 1) dx = [ \frac{x^3}{3} - x ]_{1}^{2} = (\frac{8}{3} - 2) - (\frac{1}{3} - 1) = \frac{4}{3}$. This represents an area of $\frac{4}{3}$ above the x-axis. > The total area is $\frac{2}{3} + \frac{4}{3} = \frac{6}{3} = 2$ . Integrals can also be with respect to $y$ (e.g., $\int y^3 dy$), requiring the integrand to be expressed in terms of $y$ . #### 5.2.3 Integration "tricks" using symmetry Symmetry can simplify definite integrals: * If a graph $y = f(x)$ is symmetric about the y-axis (even function), then $\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$ . * If a graph $y = f(x)$ is antisymmetric (odd function), then $\int_{-a}^{a} f(x) dx = 0$ . #### 5.2.4 Finding integrals of powers of x The rule for integrating powers of $x$ (for $n \neq -1$) is: $$ \int kx^n dx = \frac{k x^{n+1}}{n+1} + c $$ . where $k$ and $c$ are real constants and $n$ is any real number except $-1$ . Integration can be performed term by term for sums and differences. Expressions may need simplification before integration . > **Tip:** Advanced integration methods like substitution or integration by parts are not required. Questions can be efficiently solved using the basic rules . #### 5.2.5 The Trapezium Rule The Trapezium Rule approximates the area under a curve using a series of equal-width trapezoids. The formula is : $$ \text{Approximate Area} = \frac{h}{2} (y_0 + 2y_1 + 2y_2 + \dots + 2y_{n-1} + y_n) $$ . where $h$ is the width of each trapezoid and $y_0, y_1, \dots, y_n$ are the function values at the boundaries of the trapezoids . It's important to determine whether this approximation constitutes an overestimate or underestimate based on the curve's shape . ### 5.3 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration . #### 5.3.1 Significance of the theorem The theorem has two key forms: 1. **Relating definite integrals and antiderivatives:** $$ \int_{a}^{b} f(x) dx = F(b) - F(a) $$ where $F'(x) = f(x)$. This form shows how to calculate definite integrals using an antiderivative $F(x)$ . A consequence is that swapping the limits of integration negates the integral: $\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx$ . Also, integrals can be split across contiguous ranges: $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$ . 2. **Differentiating an integral:** $$ \frac{d}{dx} \int_{a}^{x} f(t) dt = f(x) $$ This form indicates that differentiating an integral (with a variable upper limit) returns the original function . #### 5.3.2 Solving differential equations The theorem is fundamental to solving differential equations of the form $\frac{dy}{dx} = f(x)$. To solve for $y$, one integrates both sides with respect to $x$ : $$ y = \int f(x) dx $$ . This process yields a general solution that includes a constant of integration, $c$ . > **Example:** To solve $\frac{dy}{dx} = 3x^2 + 4x - 3$: > Integrate both sides: $y = \int (3x^2 + 4x - 3) dx = x^3 + 2x^2 - 3x + c$ . * **With initial/boundary conditions:** If a specific value of $y$ is known for a given $x$ (e.g., $y=5$ when $x=1$), this condition can be used to find the specific value of $c$ . * **Method 1:** Substitute the known values into the general solution to solve for $c$ . * **Method 2:** Use the given values as limits in a definite integral formulation . $$ \int_{1}^{x} \frac{dy}{dx} dx = \int_{1}^{x} (3x^2 + 4x - 3) dx $$ $$ y - 5 = [x^3 + 2x^2 - 3x]_{1}^{x} $$ $$ y - 5 = (x^3 + 2x^2 - 3x) - (1^3 + 2 ^2 - 3 ) $$ [1](#page=1). $$ y = x^3 + 2x^2 - 3x + 5 $$ . --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Indices | Symbols placed above and to the right of a number or variable to indicate how many times it is to be multiplied by itself. | | Rational exponents | Exponents that are rational numbers, meaning they can be expressed as a fraction p/q where p and q are integers and q is not zero. | | Surds | Expressions involving roots, typically square roots, that cannot be simplified to a rational number, used to express irrational numbers exactly. | | Discriminant | A part of a quadratic equation, specifically $b^2 - 4ac$, used to determine the nature of the roots of the equation. | | Quadratic functions | Functions of the form $ax^2 + bx + c$, where $a \neq 0$, which produce a parabolic graph. | | Completing the square | An algebraic technique used to rewrite a quadratic expression into the form $(x+a)^2 + b$, which helps in solving equations and identifying graph properties. | | Simultaneous equations | A set of equations with multiple variables where the goal is to find the values of the variables that satisfy all equations concurrently. | | Inequalities | Mathematical statements that compare two values using symbols like <, >, ≤, or ≥, indicating that one value is less than, greater than, less than or equal to, or greater than or equal to another. | | Recurrence relation | An equation that defines a sequence where each term is defined as a function of preceding terms, often used to generate sequences. | | Arithmetic series | A sequence of numbers such that the difference between consecutive terms is constant, known as the common difference. | | Geometric series | A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. | | Binomial expansion | A formula for expanding expressions of the form $(x+y)^n$, which results in a sum of terms involving binomial coefficients and powers of x and y. | | Factor Theorem | A theorem in algebra stating that a polynomial $f(x)$ has a factor $(x-a)$ if and only if $f(a) = 0$. | | Remainder Theorem | A theorem in algebra stating that when a polynomial $f(x)$ is divided by $(x-a)$, the remainder is $f(a)$. | | Radian measure | A unit of angle measurement where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. | | Trigonometric functions | Functions of an angle that relate the angles of a triangle to the lengths of its sides, such as sine, cosine, and tangent. | | Exponentials | A mathematical expression that involves a base number raised to a power, represented as $a^x$. | | Logarithms | The inverse operation to exponentiation, answering the question of how many times a base number must be multiplied by itself to obtain a certain number. | | Derivative | The instantaneous rate of change of a function with respect to its variable, often representing the gradient of the tangent to the function's graph. | | Differentiation | The process of finding the derivative of a function. | | Stationary points | Points on a curve where the derivative is zero, indicating a horizontal tangent; these can be local maxima, minima, or points of inflexion. | | Integration | The process of finding the antiderivative of a function or calculating the area under a curve. | | Definite integral | An integral that evaluates to a numerical value, representing the net area between a function's curve and the x-axis over a specified interval. | | Fundamental Theorem of Calculus | A theorem that links differentiation and integration, stating that the definite integral of a function can be found by evaluating its antiderivative at the limits of integration. | | Trapezium rule | A numerical method used to approximate the definite integral of a function by dividing the area under the curve into trapezoids. | | Differential equations | Equations that relate a function with its derivatives, used to model phenomena where rates of change are involved. | | Transformations | Operations applied to a graph to alter its position, size, or orientation, such as translations, stretches, and reflections. | | Function composition | Applying one function to the result of another, denoted as $f(g(x))$, where the output of $g(x)$ becomes the input for $f(x)$. |