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IGCSE Additional Maths Past Papers and Topic Questions

IGCSE  Additional Maths Topic Questions  for / 2024/ 2025/2026/2027 Exams

Following are free downloadable topic Questions on the topic of Functions

Paper 1:

 

 

Paper 2: / Topic Functions

IGCSE Additional Math Keypoints

IGCSE Additional Maths — Functions Keypoints
  • A mapping is a rule for changing one number into another number or set of numbers.
  • A function, f(x), maps one number onto another single number.
  • The graph of a function has only one value of y for each x; two or more x values may give the same y.
  • The domain is the set of input values; the range is the set of output values.
  • One-one: each x gives a unique y; many-one: two or more x values give same y.
  • The inverse reverses the function; only one-one functions have inverses.
  • The modulus function y=|f(x)| reflects negative parts of the graph in the x-axis.

IGCSE Additional Maths — Quadratic Functions Keypoints
  • A quadratic function has the form f(x)=ax²+bx+c, where a≠0. The domain is x-values, range is y-values.
  • The graph is ∪-shaped if a>0, ∩-shaped if a<0; it is symmetrical about a vertical axis.
  • Discriminant Δ=b²−4ac: Δ>0 two real roots, Δ=0 one repeated root, Δ<0 none.
  • Quadratic formula: x=(−b±√(b²−4ac))/(2a)
  • Completing the square: x²+bx+c = (x+b/2)² − (b/2)² + c
  • Factorising: Find p,q with p+q=b, pq=ac ⇒ ax²+bx+c=a(x−p)(x−q)
  • For intersections: substitute one equation into another to obtain a quadratic in x.
  • Solve quadratic inequalities by sketching the related curve.

IGCSE Additional Maths — Factors of Polynomials Keypoints
  • An expression of the form ax³ + bx² + cx + d, where a ≠ 0, is called a cubic expression.
  • The graph of a cubic expression can be plotted by calculating the value of y for each value of x in the given range.
  • The solution to a cubic equation is the set of x-values for which the graph crosses the x-axis (where f(x) = 0).
  • A polynomial is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where coefficients are real numbers and aₙ ≠ 0.
  • The degree of a polynomial is the highest power of x present with a non-zero coefficient.
  • The Factor Theorem states: if (x − a) is a factor of f(x), then f(a) = 0.Conversely, if f(a) = 0, then (x − a) is a factor of f(x).
  • The Remainder Theorem states: for any polynomial f(x), the remainder when f(x) is divided by (x − a) is f(a).
    It can be written as: f(x) = (x − a)g(x) + f(a).
  • If f(x) is divided by (x − a) and the remainder is zero, then (x − a)is a factor of f(x) and x = a is a root of f(x) = 0.
  • To factorise a cubic or higher-degree polynomial, use the Factor Theorem to find one root, then apply polynomial division (long or synthetic) to obtain the remaining factor(s).
  • Solve any quadratic factor using factorisation or the quadratic formula to find all roots of the polynomial.
  • A polynomial of degree n can have at most n real roots.
  • When expressed as f(x) = a(x − α)(x − β)(x − γ)..., each factor represents a root (x-intercept) of the graph.
  • The graph of a cubic function may have up to three real roots. Its shape depends on the sign of the coefficient of x³: ∪∩ (for positive a) or ∩∪ (for negative a).

IGCSE Additional Maths — Equations, Inequalities and Graphs Keypoints
  • For any real number x, the modulus of x is denoted by |x| and is defined as:
    |x| = x if x ≥ 0
    |x| = −x if x < 0.
  • A modulus equation of the form |ax + b| = b can be solved either graphically or algebraically.
  • A modulus equation of the form |ax + b| = |cx + d| can be solved graphically by plotting both graphs on the same axes and then identifying the intersection points algebraically if needed.
  • A modulus inequality of the form |x − a| < b is equivalent to a − b < x < a + b. It can be illustrated on a number line with open circles for endpoints not included, and solid circles for ≤ or ≥ cases.
  • A modulus inequality of the form |x − a| > b or |x − a| ≥ b is represented by the parts of the line outside the above interval.
  • A modulus inequality in two dimensions forms a region on a graph called the feasible region. The unrequired region is usually shaded out to make the feasible region clear.
  • Sometimes, an equation involving both x and √x can be solved by substituting x = u². Always check all solutions in the original equation.
  • The graph of a cubic function has a distinctive shape determined by the sign of the coefficient of x³:
    Positive x³ → rising S-shape.
    Negative x³ → inverted S-shape.

IGCSE Additional Maths — Simultaneous Equations Keypoints
  • Simultaneous equations may be solved using these three methods:
  • Graphically: This method can be used for any two simultaneous equations. The advantage is that it is generally easy to draw graphs, although it can be time-consuming. The disadvantage is that it may not give an answer to the level of accuracy required.
  • Elimination: This is the most useful method when solving two linear simultaneous equations.
  • Substitution: This method is best for one linear and one non-linear equation. You start by isolating one variable in the linear equation and then substituting it into the non-linear equation.

IGCSE Additional Maths — Logarithmic and Exponential Functions Keypoints
  • A logarithm represents a power or index. It tells us what power a base must be raised to in order to get a particular number.
  • The laws of logarithms apply for any base greater than 0 (and not equal to 1). These laws correspond directly to the laws of indices.
  • Summary of laws:
    Operation Law of Indices Law of Logarithms
    Multiplicationaˣ × aʸ = aˣ⁺ʸlogₐ(xy) = logₐx + logₐy
    Divisionaˣ ÷ aʸ = aˣ⁻ʸlogₐ(x/y) = logₐx − logₐy
    Powers(aˣ)ⁿ = aⁿˣlogₐ(xⁿ) = n·logₐx
    Roots(aˣ)¹⁄ⁿ = aˣ⁄ⁿlogₐ(ⁿ√x) = (1/n)·logₐx
    Log of 1a⁰ = 1logₐ1 = 0
    Reciprocals1/aˣ = a⁻ˣlogₐ(1/x) = −logₐx
    Log to its basea¹ = alogₐa = 1
  • For the graph of y = logₐx:
    • It exists only for x greater than 0.
    • The y-axis is a vertical asymptote.
    • It rises slowly with a positive slope.
    • It passes through the point (1, 0) for every base a.
  • Notation reminders:
    • logₐx means “logarithm of x to base a”.
    • log₁₀x or lgx refers to base 10.
    • lnx represents a logarithm to base e.
  • An exponential function has the general form y = aˣ, where a is a positive constant greater than 1.
  • The exponential and logarithmic functions are inverses of each other:
    y = logₐx ⇔ aʸ = x
  • When a > 1, the graph of y = aˣ:
    • Has the x-axis as a horizontal asymptote.
    • Rises steadily with a positive gradient.
    • Passes through (0, 1).
  • When a > 1, the graph of y = a⁻ˣ:
    • Also has the x-axis as a horizontal asymptote.
    • Falls with a negative gradient.
    • Passes through (0, 1).

IGCSE Additional Maths — Straight-Line Graphs Keypoints
  • An equation of the form y = mx + c represents a straight line with gradient m and y-intercept c. The line crosses the y-axis at (0, c).
  • The midpoint of the line joining the points (x1, y1) and (x2, y2) is given by midpoint = ((x1 + x2)/2 , (y1 + y2)/2).
  • The distance between the points (x1, y1) and (x2, y2) is given by length = √((x2 − x1)² + (y2 − y1)²).
  • Two lines are parallel if they have equal gradients.
  • Two lines are perpendicular if they meet at 90 degrees. The product of their gradients is −1.
  • Logarithmic equations can also represent straight lines:
    1. For y = a xⁿ, taking logarithms gives log y = log a + n log x. A graph of log y against log x produces a straight line with gradient n and y-intercept log a.
    2. For y = A bˣ, taking logarithms gives log y = log A + x log b. A graph of log y against x produces a straight line with gradient log b and y-intercept log A.

IGCSE Additional Maths — Coordinate Geometry of the Circle Keypoints
  • Circle with centre (a,b) and radius r- (standard form):
    (x−a) 2 + (y−b) 2 = r2
  • General form and parameters:
    x2+ y2+ 2gx+ 2fy+ c=0 Centre:(−g,−f) Radius: r= g2+ f2−c
  • To recover centre and radius from any quadratic in x and y- complete the square in both variables to reduce to standard form.
  • Intersection with a line (e.g.y=mx+c:)-Substitute into the circle equation and solve the resulting quadratic. Use the discriminant Δ to classify:
    • Δ>0 → two points (a chord)
    • Δ=0 → one point (tangent)
    • Δ<0→ no="no" intersection
  • A radius is perpendicular to the tangent at the point of contact. Tangent to (x−a)^2+(y−b)^2=r^2 at x1,y1:
    (x1−a) (x−a) + (y1−b) (y−b) = r2
  • Two circles with centres C1, C2, radii r1, r2. Distance between centres:
    d= (x2−x1) 2 + (y2−y1) 2
    Classification:
    • |r1−r2|<d<r1+r2 → two intersections
    • d=r1+r2 or d=|r1−r2| → circles touch (external/internal)
    • d>r1+r2 or d<|r1−r2| → no="no" intersection
  • Common chord of two circles: subtract their equations to get the chord’s straight-line equation; solve simultaneously with either circle to find intersection points.
  • Shortest distance from centre (x0,y0) to line ax+by+c=0:
    d= | ax0+ by0+c | a2+b2
    Compare d with r to decide tangent (d=r), chord (d<r), or no="no" contact (d>r).
  • If the endpoints of a diameter are (x1,y1) and (x2,y2), then the centre is their midpoint and the radius is half their distance.

IGCSE Additional Maths — Circular Measure Keypoints
  • Angles are measured in degrees or radians. 180° = π radians.
  • One radian is the angle at the centre of a circle subtended by an arc whose length equals the radius.
  • Circle formulas work in either unit: area A = π r^2, circumference C = 2 π r.
  • Conversions: θ (radians) = θ° × π / 180, and θ° = θ (radians) × 180 / π.
  • When θ is in radians:
    • Arc length s = r θ
    • Sector area A_sector = 1/2 r^2 θ
  • Reference angles: full circle 2π, semicircle π, quadrant π/2.
  • Always state the angle unit and set the calculator to the correct mode.

IGCSE Additional Maths — Trigonometry Keypoints
  • The six trigonometric functions are sine, cosine, tangent, cosecant, secant and cotangent. They can be used for angles of any size, in degrees or radians.
  • The amplitude of a trigonometric graph is the maximum vertical distance from the x-axis. The period is the horizontal distance after which the graph repeats its shape.
  • Changing the constants a, b and c in y = a sin(bx) + c, y = a cos(bx) + c and y = a tan(bx) + c affects the height, frequency and position of the graph. The graphs may be drawn in degrees or radians. Any asymptotes for tangent graphs must be clearly shown.
  • Useful trigonometric identities:
    • sin² A + cos² A = 1
    • sec² A = 1 + tan² A
    • cosec² A = 1 + cot² A
  • Trigonometric equations can be solved using these identities. Examples:
    • 4 cot θ = tan θ
    • 2 sec² θ + tan θ − 3 = 0
    • 5 sin(θ / 3) + 2 cos(θ / 3) = 0
    • 3 cosec(2θ − π / 12) = 4
  • Trigonometric proofs and simplifications often use the same standard relationships. Examples:
    • sin x tan x + cos x = sec x
    • (sin θ) / (1 + cos θ) + (1 + cos θ) / (sin θ) = 2 cosec θ
  • Always state the domain and units (degrees or radians) when working with trigonometric graphs or equations.

IGCSE Additional Maths — Permutations and Combinations Keypoints
  • The number of ways of arranging n different objects in a line is written as n! (n factorial).
  • n! = n × (n − 1) × (n − 2) × … × 3 × 2 × 1, where n is a positive integer.
  • By convention, 0! = 1.
  • The number of permutations of r objects taken from n objects is given by nPr = n! / (n − r)!
  • The number of combinations of r objects taken from n objects is given by nCr = n! / [(n − r)! r!]
  • In permutations, the order of arrangement is important. In combinations, the order does not matter.
  • Problems may involve arrangements or selections using these rules. Cases with repetition, circular arrangements, or combined permutation-combination problems are not included.

IGCSE Additional Maths — Series Keypoints
  • A binomial expression has the form (a x + b)^n where n is a positive integer.
  • Binomial coefficients (n choose r) can be obtained from Pascal’s triangle, from tables, or by the formula C(n, r) = n! / [r! (n − r)!], for 0 ≤ r ≤ n.
  • Binomial expansion:
    (1 + x)^n = 1 + n x + n(n − 1)/2! x^2 + n(n − 1)(n − 2)/3! x^3 + … + x^n.
  • A sequence is an ordered list of numbers u1, u2, …, uk, …, un with a general term uk.
  • Arithmetic progression (first term a, common difference d):
    • Recurrence: u(k+1) = u(k) + d
    • kth term: u(k) = a + (k − 1) d
    • Last term: l = a + (n − 1) d
    • Sum of first n terms: S(n) = n(a + l)/2 = n/2 [2a + (n − 1) d]
  • Geometric progression (first term a, common ratio r):
    • Recurrence: u(k+1) = r · u(k)
    • kth term: a r^(k − 1)
    • Last term: a r^(n − 1)
    • Sum of first n terms:
      S(n) = a (r^n − 1)/(r − 1) for r > 1
      S(n) = a (1 − r^n)/(1 − r) for r < 1 and r ≠ 1
  • Infinite geometric series converges when −1 < r < 1. Sum to infinity: S(∞) = a / (1 − r).

IGCSE Additional Maths — Calculus Keypoints
  • Differentiation is the process of finding the rate of change of a function. If y=fx, then the derivative is written as dydx or f'(x).
  • Basic rule: ddx (xn)= nxn−1.
  • Standard derivatives:
    • ddxsinx=cosx
    • ddxcosx=−sinx
    • ddxtanx=sec2x
    • ddxex=ex
    • ddxlnx=1x
  • Rules of differentiation:
    • Sum rule:
    • Product rule:
    • Quotient rule:
    • Chain rule:
  • Applications: The derivative gives the gradient of a curve, rate of change, stationary points and tangents.
  • Integration is the reverse of differentiation and represents the area under a curve. The general integral is written as ∫f(x)dx.
  • Basic rule: ∫xndx= xn+1 n+1+C, for n≠−1.
  • Standard integrals:
    • ∫sinxdx=−cosx+C
    • ∫cosxdx=sinx+C
    • ∫exdx=ex+C
    • ∫1xdx=ln|x|+C
  • Definite integral:gives the exact area between a curve and the x-axis between two limits. ∫ab f(x)dx= F(b)−F(a).
  • Integration by substitution: substitute u=g(x), then du=g'(x)dx.
  • Integration by parts: ∫udv=uv− ∫vdu.
  • Applications of integration: find area under curves, area between two curves, and displacement from velocity–time graphs.