Mathematics (9709) AS Standard cheat sheet

    Mathematics (9709) · CAIE · AS

    Standard
    4 pages
    44 formulas, 38 concepts
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    All Mathematics (9709) AS cheat sheets

    P1 - Quadratics

    • The Quadratic Formula
      x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
      Used to find the roots (solutions) of any quadratic equation in the form $ax^2+bx+c=0$. Essential when factorization is not straightforward.
    • The Discriminant
      Δ=b2−4ac\Delta = b^2-4ac
      Used to determine the nature of the roots of a quadratic equation $ax^2+bx+c=0$ without solving it. It tells you if there are two distinct real roots, one repeated real root, or no real roots.
    • Completed Square Form
      a(x+p)2+qa(x+p)^2+q
      A standard form for a quadratic expression. For $ax^2+bx+c$, it is written as $a\left(x+\frac{b}{2a}\right)^2 + \left(c-\frac{b^2}{4a}\right)$.
    • Vertex from Completed Square Form
      (−p,q)(-p, q)
      When a quadratic is written as $a(x+p)^2+q$, the vertex (minimum or maximum point) of the parabola is at the coordinate $(-p, q)$.

    Key concepts: **Using the Discriminant**: For a quadratic equation $ax^2+bx+c=0$: if $b^2-4ac > 0$, there are two distinct real roots; if $b^2-4ac = 0$, there is one repeated real root (or two equal roots); if $b^2-4ac < 0$, there are no real roots., **Completing the Square**: This algebraic process rewrites a quadratic expression $ax^2+bx+c$ into the form $a(x+p)^2+q$, which directly reveals the vertex of the parabola and its line of symmetry ($x=-p$)., **Solving Quadratic Inequalities**: To solve an inequality like $ax^2+bx+c > 0$, first find the roots of the corresponding equation $ax^2+bx+c=0$. Then, sketch the parabola to determine the range(s) of x-values for which the graph is above (for >) or below (for <) the x-axis., **Simultaneous Equations (Linear-Quadratic)**: To solve a system with one linear and one quadratic equation, rearrange the linear equation to make one variable the subject (e.g., $y = mx+c$) and substitute this expression into the quadratic equation. This creates a single quadratic equation in one variable, which can then be solved.

    Exam tips

    • Never try to solve a quadratic inequality without a sketch. Find the roots, sketch the U-shaped or n-shaped parabola, and then read the correct region(s) from your graph. This avoids common errors with signs and ranges.
    • A line is a tangent to a curve if they intersect at exactly one point. After setting the equations equal to each other, the resulting quadratic will have one repeated root. Therefore, set the discriminant $b^2-4ac = 0$ to solve for unknown constants.
    • When solving simultaneous equations, a common mistake is to find the x-values and forget to substitute them back into the linear equation to find the corresponding y-values. The solution is a set of coordinate pairs.

    P1 - Functions

    • Vertical Translation
      y=f(x)+ay = f(x) + a
      Translates the graph of y=f(x) vertically. If a > 0, it moves up by 'a' units. If a < 0, it moves down by 'a' units.
    • Horizontal Translation
      y=f(x+a)y = f(x + a)
      Translates the graph of y=f(x) horizontally. If a > 0, it moves left by 'a' units. If a < 0, it moves right by 'a' units.
    • Vertical Stretch/Compression
      y=af(x)y = a f(x)
      Stretches or compresses the graph of y=f(x) vertically by a factor of 'a'. If a < 0, it is also reflected in the x-axis.
    • Horizontal Stretch/Compression
      y=f(ax)y = f(ax)
      Stretches or compresses the graph of y=f(x) horizontally by a factor of 1/a. If a < 0, it is also reflected in the y-axis.
    • Composite Function
      gf(x)=g(f(x))gf(x) = g(f(x))
      Represents a composite function where the function f(x) is applied first, and then the function g(x) is applied to the result.
    • Inverse Function
      y=f−1(x)y = f^{-1}(x)
      Represents the inverse of a one-to-one function f(x). If f(a) = b, then f^{-1}(b) = a.

    Key concepts: **Domain**: The set of all possible input values (x-values) for which a function is defined., **Range**: The set of all possible output values (y-values) that a function can produce from its domain., **Function**: A rule that assigns to each input value (from the domain) exactly one output value (in the range). Can be one-to-one or many-to-one., **One-to-One Function**: A function where every distinct input value produces a distinct output value. A function must be one-to-one for an inverse function to exist., **Composite Function**: A function formed by applying one function to the results of another function. The output of the first function becomes the input for the second., **Existence of a Composite Function**: For the composite function gf(x) to exist, the range of the inner function, f(x), must be a subset of (or equal to) the domain of the outer function, g(x)., **Inverse Function**: A function that 'reverses' another function. If f maps x to y, then the inverse function f⁻¹ maps y back to x., **Graph of an Inverse Function**: The graph of y = f⁻¹(x) is a reflection of the graph of y = f(x) in the line y = x., **Restricting the Domain**: The process of limiting the domain of a many-to-one function (like a parabola) to make it one-to-one, which is necessary to find its inverse.

    Exam tips

    • The domain of f⁻¹(x) is the range of f(x), and the range of f⁻¹(x) is the domain of f(x). Always find the range of the original function first!
    • To find the range of a quadratic function, complete the square to find its vertex (turning point). The y-coordinate of the vertex is the minimum or maximum value, which defines the range.
    • A common mistake is mixing up the order. For gf(x), you substitute f(x) into g(x), not the other way around. Work from right to left.
    • Remember that f(x+a) is a shift to the LEFT by 'a' units, and f(x-a) is a shift to the RIGHT. This is often counter-intuitive.

    P1 - Coordinate Geometry

    • Gradient of a Line
      m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
      Calculates the steepness (gradient) of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$.
    • Equation of a Straight Line (Point-Slope Form)
      y−y1=m(x−x1)y - y_1 = m(x - x_1)
      Finds the equation of a straight line given its gradient 'm' and a single point $(x_1, y_1)$ that it passes through.
    • Condition for Perpendicular Lines
      m1×m2=−1m_1 \times m_2 = -1
      The product of the gradients of two perpendicular lines is -1. Use this to find a perpendicular gradient: $m_2 = -\frac{1}{m_1}$.
    • Equation of a Circle (Centre-Radius Form)
      (x−a)2+(y−b)2=r2(x-a)^2 + (y-b)^2 = r^2
      Defines a circle with centre $(a, b)$ and radius 'r'. This is the most common and useful form.
    • The Discriminant
      Δ=b2−4ac\Delta = b^2 - 4ac
      Used on a quadratic equation $ax^2+bx+c=0$ to determine the number of real roots, which corresponds to the number of intersection points.
    • Distance Between Two Points
      d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
      Calculates the length of a line segment between two points $(x_1, y_1)$ and $(x_2, y_2)$. Derived from Pythagoras' theorem.
    • Midpoint of a Line Segment
      M=(x1+x22,y1+y22)M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)
      Finds the coordinates of the midpoint of a line segment connecting points $(x_1, y_1)$ and $(x_2, y_2)$.

    Key concepts: **Parallel and Perpendicular Lines**: Parallel lines have the same gradient ($m_1 = m_2$). Perpendicular lines have gradients whose product is -1 ($m_1 \times m_2 = -1$)., **Intersection of Lines and Curves**: To find intersection points, solve the equations of the line and curve simultaneously. This usually results in a quadratic equation., **Discriminant and Intersection Conditions**: For the resulting quadratic $ax^2+bx+c=0$: $b^2-4ac > 0$ means two distinct intersections; $b^2-4ac = 0$ means one intersection (tangent); $b^2-4ac < 0$ means no intersections., **Completing the Square for Circles**: To find the centre and radius from the expanded form $x^2+y^2+Dx+Ey+F=0$, you must complete the square for the x-terms and y-terms separately to get it into the form $(x-a)^2+(y-b)^2=r^2$.

    Exam tips

    • A common mistake is forgetting to both flip the fraction and change the sign for a perpendicular gradient. If $m = \frac{2}{3}$, the perpendicular gradient is $m_{\perp} = -\frac{3}{2}$, NOT $-\frac{2}{3}$ or $\frac{3}{2}$.
    • After finding the x-coordinates of intersection, substitute them back into the LINEAR equation to find the y-coordinates. It's simpler and less prone to errors than using the circle equation.
    • If a question states a line is a tangent to a curve (circle, parabola, etc.), it's a strong hint to set up a quadratic equation and use the discriminant condition $b^2 - 4ac = 0$.

    P1 - Circular Measure

    • Degrees to Radians Conversion
      radians=degrees×π180\text{radians} = \text{degrees} \times \frac{\pi}{180}
      Used to convert an angle from degrees to radians. This is essential as all circular measure formulas require the angle to be in radians.
    • Radians to Degrees Conversion
      degrees=radians×180π\text{degrees} = \text{radians} \times \frac{180}{\pi}
      Used to convert an angle from radians to degrees. Useful for sense-checking an answer or when the final answer is required in degrees.
    • Arc Length
      s=rθs = r\theta
      Calculates the length of an arc of a circle, where 'r' is the radius and 'θ' is the angle at the center in RADIANS.
    • Area of a Sector
      A=12r2θA = \frac{1}{2}r^2\theta
      Calculates the area of a sector (a 'slice') of a circle, where 'r' is the radius and 'θ' is the angle at the center in RADIANS.
    • Area of a Triangle within a Sector
      A=12r2sin⁡θA = \frac{1}{2}r^2\sin\theta
      A specific application of the A=½ab sin(C) formula to find the area of the triangle formed by two radii and a chord. 'θ' must be in RADIANS if used alongside sector formulas.
    • Area of a Segment
      A=12r2(θ−sin⁡θ)A = \frac{1}{2}r^2(\theta - \sin\theta)
      Calculates the area of a segment (region between a chord and an arc). This is derived from: Area of Sector - Area of Triangle.

    Key concepts: **Radian Measure**: A radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. All key formulas for arc length and sector area require the angle to be in radians., **Sector**: A sector is the region of a circle enclosed by two radii and the arc between them, resembling a slice of a pizza., **Segment**: A segment is the region of a circle enclosed by a chord and the arc it cuts off. Its area is found by subtracting the area of a triangle from the area of a sector.

    Exam tips

    • The most common error is using the wrong calculator mode. ALWAYS check if your calculator is in Radians (RAD) mode before using s=rθ or A=½r²θ. Switch to Degrees (DEG) for standard trigonometry if angles are given in degrees.
    • Don't confuse 'arc length' with 'perimeter of a sector'. The perimeter includes the two straight sides (radii). Perimeter = Arc Length + 2r = rθ + 2r.
    • For complex shaded region problems, express the required area as a sum or difference of simpler shapes. The most common strategy is: Shaded Area = Area of Sector - Area of Triangle.

    P1 - Trigonometry

    • Pythagorean Identity
      sin⁡2θ+cos⁡2θ≡1\sin^2\theta + \cos^2\theta \equiv 1
      Used to relate sin²θ and cos²θ. Essential for simplifying expressions and solving equations where both sin and cos appear.
    • Tangent Identity
      tan⁡θ≡sin⁡θcos⁡θ\tan\theta \equiv \frac{\sin\theta}{\cos\theta}
      Used to relate tanθ, sinθ, and cosθ. Key for solving equations like a sinθ = b cosθ.
    • Exact Values for Special Angles
      sin⁡(30∘)=12,cos⁡(30∘)=32,tan⁡(30∘)=13\sin(30^{\circ})=\frac{1}{2}, \cos(30^{\circ})=\frac{\sqrt{3}}{2}, \tan(30^{\circ})=\frac{1}{\sqrt{3}} \\ sin⁡(45∘)=12,cos⁡(45∘)=12,tan⁡(45∘)=1\sin(45^{\circ})=\frac{1}{\sqrt{2}}, \cos(45^{\circ})=\frac{1}{\sqrt{2}}, \tan(45^{\circ})=1 \\ sin⁡(60∘)=32,cos⁡(60∘)=12,tan⁡(60∘)=3\sin(60^{\circ})=\frac{\sqrt{3}}{2}, \cos(60^{\circ})=\frac{1}{2}, \tan(60^{\circ})=\sqrt{3}
      Memorize these values (or the two special triangles) for non-calculator questions. Angles are often given in degrees (30, 45, 60) or radians (π/6, π/4, π/3).
    • Symmetry and Periodicity Rules
      sin⁡θ=sin⁡(180∘−θ)\sin\theta = \sin(180^{\circ}-\theta) \\ cos⁡θ=cos⁡(360∘−θ)\cos\theta = \cos(360^{\circ}-\theta) or cos⁡θ=cos⁡(−θ)\cos\theta = \cos(-\theta) \\ tan⁡θ=tan⁡(180∘+θ)\tan\theta = \tan(180^{\circ}+\theta)
      These rules are used to find all solutions to a trigonometric equation within a given interval, after finding the principal value. These are derived from the CAST diagram or graph symmetries.

    Key concepts: **The CAST Diagram**: A diagram of the four quadrants that indicates which trigonometric functions are positive in each: All (1st), Sin (2nd), Tan (3rd), Cos (4th). It is essential for finding all possible solutions to trigonometric equations in a given range., **Trigonometric Graphs**: The graphs of y=sin(x), y=cos(x), and y=tan(x) show their periodic nature. Sin(x) and cos(x) have a period of 360° (2π) and an amplitude of 1, while tan(x) has a period of 180° (π) and asymptotes., **Principal Value**: When you use an inverse trigonometric function on a calculator (e.g., sin⁻¹(0.5)), it gives you the principal value, which is only one of many possible solutions. You must use the CAST diagram or graph symmetry to find all other solutions in the required interval., **Solving Quadratic Trigonometric Equations**: Equations of the form a sin²θ + b sinθ + c = 0 can be solved by treating them as a standard quadratic. Let x = sinθ, solve for x, then substitute back to solve for θ.

    Exam tips

    • Always check the required interval for your solutions (e.g., 0° ≤ θ ≤ 360°) and ensure you find ALL solutions within it. Also, double-check if the interval is in degrees or radians and set your calculator to the correct mode (DEG or RAD).
    • When solving an equation like sinθ cosθ = sinθ, do not divide by sinθ. Instead, rearrange to sinθ cosθ - sinθ = 0 and factorise to sinθ(cosθ - 1) = 0. Dividing would lose the solutions where sinθ = 0.
    • A quick sketch of the trigonometric graph and a horizontal line (e.g., y=0.5) can help you visualize how many solutions to expect in a given interval and where they should be located. This is a great way to check your answers.

    P1 - Series

    • Binomial Expansion
      (a+b)n=an+(n1)an−1b+(n2)an−2b2+...+(nr)an−rbr+...+bn(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ... + \binom{n}{r}a^{n-r}b^r + ... + b^n
      To expand expressions of the form (a+b)ⁿ for a positive integer n. The term \binom{n}{r} is the binomial coefficient.
    • Arithmetic Progression (AP): nth Term
      un=a+(n−1)du_n = a + (n-1)d
      To find the value of the nth term of an AP, where 'a' is the first term and 'd' is the common difference.
    • Arithmetic Progression (AP): Sum of n Terms
      Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d)
      To find the sum of the first n terms of an AP when the first term 'a' and common difference 'd' are known.
    • Geometric Progression (GP): nth Term
      un=arn−1u_n = ar^{n-1}
      To find the value of the nth term of a GP, where 'a' is the first term and 'r' is the common ratio.
    • Geometric Progression (GP): Sum of n Terms
      Sn=a(1−rn)1−rS_n = \frac{a(1-r^n)}{1-r} or Sn=a(rn−1)r−1S_n = \frac{a(r^n-1)}{r-1}
      To find the sum of the first n terms of a GP. Use the form that keeps the denominator positive (e.g., first form for |r|<1).
    • Geometric Progression (GP): Sum to Infinity
      S∞=a1−rS_\infty = \frac{a}{1-r}
      To find the sum of an infinite GP. This formula is only valid if the series converges, which occurs when |r| < 1.
    • Binomial Coefficient
      (nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}
      Calculates the coefficient of the term with b^r in the expansion of (a+b)ⁿ. Most calculators have a dedicated nCr button.

    Key concepts: **Arithmetic Progression (AP)**: A sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference (d)., **Geometric Progression (GP)**: A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number. This fixed number is called the common ratio (r)., **Convergence of a GP**: An infinite geometric series converges (its sum approaches a finite value) if and only if the absolute value of the common ratio is less than 1, i.e., |r| < 1 or -1 < r < 1., **Sum to Infinity (S∞)**: The finite value that the sum of an infinite converging geometric series approaches. If a GP does not converge (|r| ≥ 1), it does not have a finite sum to infinity., **Binomial Expansion**: A method for expanding a binomial expression raised to a positive integer power, such as (a+b)ⁿ, into a sum of terms involving powers of a and b.

    Exam tips

    • Before applying any formula, carefully read the question to determine if it describes an Arithmetic Progression (common difference) or a Geometric Progression (common ratio). Misidentifying the series is a fatal error.
    • Be extremely careful with negative signs and coefficients within the binomial. For $(2-3x)^5$, the 'b' term is $(-3x)$. Every term in the expansion must have this full term raised to the appropriate power, e.g., $\binom{5}{2}(2)^3(-3x)^2$.
    • Always state and check the condition |r| < 1 before using the sum to infinity formula. Questions may ask for the range of values of a variable for which a series converges.
    • Many harder problems give you information about two different terms or sums (e.g., the 3rd term is 10 and the sum of the first 5 terms is 50). This requires you to set up and solve a pair of simultaneous equations using the standard formulas.

    P1 - Differentiation

    • Power Rule for Differentiation
      ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}
      Used to differentiate any term that can be expressed as a constant multiplied by a power of x. This is the most fundamental rule in differentiation.
    • Chain Rule
      dydx=dydu×dudx\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}
      Used to differentiate composite functions (a function of a function), such as $y = (ax+b)^n$. Also the core principle for solving connected rates of change problems.
    • Equation of a Straight Line
      y−y1=m(x−x1)y - y_1 = m(x - x_1)
      Used to find the equation of a tangent or a normal once you have a point $(x_1, y_1)$ on the line and its gradient, $m$.
    • Gradient of the Normal
      mnormal=−1mtangentm_{\text{normal}} = -\frac{1}{m_{\text{tangent}}}
      Used to find the gradient of the normal to a curve at a point. It is the negative reciprocal of the tangent's gradient at that same point.
    • Second Derivative Test Conditions
      IfIf \frac{d^2y}{dx^2} > 0,localminimum.If, local minimum. If \frac{d^2y}{dx^2} < 0,localmaximum., local maximum.
      Used to classify the nature of a stationary point $(x, y)$ after finding it by solving $\frac{dy}{dx} = 0$.

    Key concepts: **The Derivative (Gradient Function)**: The derivative of a function, denoted $\frac{dy}{dx}$ or $f'(x)$, gives the gradient of the tangent to the curve at any point $x$. It also represents the instantaneous rate of change of $y$ with respect to $x$., **Stationary Points**: A stationary point on a curve is a point where the gradient is zero, i.e., $\frac{dy}{dx} = 0$. These points can be local maxima, local minima, or stationary points of inflexion., **Tangent and Normal**: A tangent is a straight line that touches a curve at a single point and has the same gradient as the curve at that point. A normal is a straight line that is perpendicular to the tangent at that same point., **Connected Rates of Change**: This technique uses the chain rule to link the rates of change of two or more variables that are connected by a known formula (e.g., volume and radius of a sphere).

    Exam tips

    • Always rewrite expressions into the form $ax^n$ before applying the power rule. For example, change $\frac{3}{x^2}$ to $3x^{-2}$, $\sqrt{x}$ to $x^{1/2}$, and expand any brackets like $(x+2)^2$ into $x^2+4x+4$ if possible.
    • A frequent error is using the tangent's gradient ($m_t$) for the normal's equation. Always remember to calculate the normal's gradient using $m_n = -1/m_t$ before finding its equation.
    • If you find that $\frac{d^2y}{dx^2} = 0$ at a stationary point, the test is inconclusive. You must then check the sign of the first derivative ($\frac{dy}{dx}$) just before and just after the point to determine its nature (e.g., positive -> zero -> positive is a stationary point of inflexion).

    P1 - Integration

    • Power Rule for Integration
      ∫xn dx=xn+1n+1+C(for n≠−1)\int x^n \,dx = \frac{x^{n+1}}{n+1} + C \quad (\text{for } n \neq -1)
      The fundamental rule for integrating polynomial terms. Increase the power by 1 and divide by the new power.
    • Integration of (ax+b)ⁿ
      ∫(ax+b)n dx=(ax+b)n+1a(n+1)+C(for n≠−1)\int (ax+b)^n \,dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C \quad (\text{for } n \neq -1)
      Used for integrating linear expressions raised to a power. It is a reverse of the chain rule.
    • Definite Integral Evaluation
      ∫abf(x) dx=[F(x)]ab=F(b)−F(a)\int_a^b f(x) \,dx = [F(x)]_a^b = F(b) - F(a)
      The Fundamental Theorem of Calculus. Used to evaluate a definite integral, where F(x) is the antiderivative of f(x).
    • Area Under a Curve (x-axis)
      A=∫aby dxA = \int_a^b y \,dx
      Calculates the area between the curve y=f(x), the x-axis, and the vertical lines x=a and x=b.
    • Volume of Revolution (about x-axis)
      V=π∫aby2 dxV = \pi \int_a^b y^2 \,dx
      Calculates the volume of the solid formed when the area under the curve y=f(x) is rotated 360° about the x-axis.

    Key concepts: **Indefinite Integration**: The process of finding the antiderivative of a function. It is the reverse of differentiation and yields a family of functions, not a single value., **Constant of Integration (+C)**: Since the derivative of any constant is zero, we must add an arbitrary constant 'C' to any indefinite integral to represent all possible antiderivative functions., **Definite Integration**: The process of finding the integral between two limits, 'a' and 'b'. It represents the net signed area between the function's graph and the x-axis., **Area as a Definite Integral**: The definite integral $\int_a^b f(x) \,dx$ gives the area bounded by the curve $y=f(x)$, the x-axis, and lines $x=a, x=b$. If the area is below the x-axis, the integral is negative., **Volume of Revolution**: A 3D solid created by rotating a 2D area (defined by a function and boundaries) around an axis (usually x or y). Its volume is found by integrating the area of an infinitesimally thin disc.

    Exam tips

    • For any indefinite integral, you MUST add the constant of integration, '+C'. Forgetting it is one of the most common ways to lose a mark.
    • You cannot integrate products or quotients directly (e.g., $(x+1)(x-2)$ or $\frac{x^3+x}{x}$). You must first expand brackets or rewrite expressions as a sum of terms in the form $kx^n$.
    • If a region is below the x-axis, its definite integral will be negative. The physical area is the absolute value of this result. For regions both above and below, you must split the integral at the x-intercepts.
    • A very common mistake is to calculate $\pi \int y \,dx$ instead of $\pi \int y^2 \,dx$ for volume of revolution about the x-axis. Always square the function before integrating.