Worked IGCSE Add Math exam questions

Composite Functions and Solving fg(x) = k

Form fg(x) and gf(x), see why order matters, then solve fg(x) = 21, a 0606 functions question that rewards careful substitution.

Domain of a Composite Function

Form a composite involving a square root and find its domain, a 0606 functions question testing which inputs keep the composite defined.

Finding an Inverse Function and Its Domain

Find the inverse of f(x) = 3/(x − 2) and state its domain, a 0606 question testing the swap-and-rearrange method and the domain–range link.

Inverse of a Rational Function

Find the inverse of a rational function like (2x+1)/(x−3), a 0606 question testing the collect-and-factor step that isolates x.

Range and Why a Function Is Not One-One

Find the greatest value and range of 5 − (x − 2)², explain why it is not one-one, and find the domain restriction that makes it invertible, a 0606 functions question.

Range of a Composite Function on an Interval

Form a composite function and find its range over a restricted domain, a 0606 functions question combining composition with interval reasoning.

Showing a Function Is Self-Inverse

Find the inverse of a rational function and show it equals the original, a 0606 functions question on self-inverse functions where f⁻¹(x) = f(x).

Solving a Composite Equation gf(x) = k

Form a composite and solve it equal to a value, a 0606 functions question ending in a square root with two solutions to keep.

Solving a Modulus Function Equation

Solve |2x − 6| = 4 by splitting into two cases, and state the range of the modulus function, a 0606 question on |ax + b| the mark scheme wants seen in full.

Where a Function Meets Its Inverse

Find where a linear function equals its own inverse, and see why for an increasing function this reduces to solving f(x) = x, a 0606 functions question.

A Quadratic in Disguise with a Square Root

Solve x − 5√x + 6 = 0 by substituting u = √x to reveal a quadratic, then squaring back, a 0606 question on quadratic-type equations.

Completing the Square for a Maximum

Write 3 + 4x − 2x² in completed-square form and read off its maximum, a 0606 question on completing the square when the x² coefficient is negative.

Completing the Square to Find the Minimum

Write 2x² − 12x + 5 in the form a(x + b)² + c, then read off the minimum point directly, a 0606 Paper 1 question where the completed square does all the work.

Intersection of a Line and a Quadratic Curve

Find where a line crosses a quadratic curve by solving simultaneously, a 0606 question producing a quadratic with two intersection points.

Maximum Area with Completing the Square

Maximise the area of a rectangular pen against a wall using completing the square, a 0606 quadratic optimisation done without calculus.

No Real Roots: Finding a Range of k

Find the values of k for which a quadratic has no real roots using b² − 4ac < 0, a 0606 discriminant question giving a bounded range.

Range of a Quadratic on a Restricted Domain

Find the range of x² − 4x + 7 over all reals and over 0 ≤ x ≤ 3, a 0606 question where the vertex and the endpoints together decide the answer.

Sketching a Parabola: Intercepts and Vertex

Find the intercepts and vertex of a quadratic to sketch it, a 0606 question pulling together factorising and completing the square for the key features.

Solving a Quadratic Inequality

Solve 2x² − 5x − 3 > 0 by factorising and reading the parabola, the 0606 method that beats sign tables and gets the 'inside or outside' choice right.

Two Distinct Roots: Finding a Range of k

Find the values of k for which a quadratic has two distinct real roots, a 0606 discriminant question that ends in an inequality, not an equation.

Using the Discriminant for Equal Roots

Find the values of k for which a quadratic has two equal roots by setting b² − 4ac = 0, a 0606 discriminant question that reduces to a quadratic in k.

When Is a Line a Tangent to a Curve?

Find k so that a line is a tangent to a quadratic curve, using the discriminant of the combined equation, a 0606 favourite linking lines, curves and roots.

A Cubic with Two Unknown Coefficients

Find two unknown coefficients of a cubic given two factors, a 0606 factor-theorem question that sets up and solves simultaneous equations.

Factorising a Cubic with the Factor Theorem

Use a given factor to factorise x³ − 3x² − 4x + 12 completely, a 0606 question combining the factor theorem with division and a final quadratic factorisation.

Polynomial Long Division

Divide a cubic by a linear factor using long division to find the quadratic quotient, then factorise completely, a core 0606 polynomials method.

Remainder Theorem: Finding an Unknown Constant

Use the remainder theorem to find an unknown coefficient given the remainder on division, a 0606 question that turns a division fact into a simple equation.

Solving a Cubic Equation

Solve x³ + 2x² − 5x − 6 = 0 by finding one root with the factor theorem, then factorising, a 0606 question linking division to the roots of a cubic.

A Modulus Equation That Needs Checking

Solve |x − 3| = 2x, where the variable on the right means each answer must be checked, a 0606 question on why one case gets rejected.

Number of Solutions from a Modulus Graph

Use the graph of y = |x² − 4| against a horizontal line to count how many solutions an equation has, a 0606 reasoning question on modulus graphs.

Sketching the Graph of a Modulus Function

Sketch y = |2x − 4|, find its vertex and intercepts, and state its range, the standard 0606 modulus-graph skill that underpins modulus equations.

Solving |2x − 4| = x + 1 Graphically and Algebraically

Solve |2x − 4| = x + 1, interpreting the two solutions as where a V-shaped graph meets a straight line, a 0606 modulus-graph question.

Solving |x + 1| = |2x − 4|

Solve an equation with a modulus on each side by considering both sign cases, a 0606 question where both solutions turn out valid.

Solving a Cubic Inequality Graphically

Solve a factorised cubic inequality by reading the sign of the curve between its roots, a 0606 question using a quick sketch to get the regions right.

Solving a Modulus Inequality

Solve |2x − 1| < 5 by unfolding it into a double inequality, a 0606 question on the 'less than' modulus case that gives a single bounded region.

Simultaneous Equations with a Sum of Squares

Solve x + y = 5 with x² + y² = 13 by substitution, a 0606 simultaneous-equations question where the symmetry of the answers is a built-in check.

Simultaneous Equations with an xy Product

Solve 2x − y = 3 with xy = 20 by substitution, a 0606 non-linear simultaneous question that lands on a quadratic needing careful factorising.

Simultaneous Equations: Line Meets a Curve

Solve one linear and one quadratic equation by substitution to find where a line meets a curve, a 0606 question the mark scheme wants done a specific way.

Sum and Sum-of-Squares of Two Numbers

Find two numbers from their sum and sum of squares using the identity (x+y)²=x²+2xy+y², then a quadratic, an elegant 0606 simultaneous problem.

A Hidden Quadratic in an Exponential Equation

Solve 3^(2x) − 10(3^x) + 9 = 0 with the substitution y = 3^x, a 0606 question that turns a scary exponential into an ordinary quadratic.

A Hidden Quadratic in Logarithms

Solve (log₃x)² − log₃x² = 3 by substituting u = log₃x to reveal a quadratic, then converting back, a common 0606 logarithm question.

An Exponential Growth and Decay Problem

Apply an exponential model to find a value and solve for time using ln, a 0606 question that puts e^x and logarithms in a real context.

Exponential Decay: Finding a Time

Use an exponential decay model to find when a quantity halves, a 0606 question applying ln to a negative exponent in a real context.

Expressing a Logarithm in Terms of Given Logs

Given log_a 2 = p and log_a 3 = q, express log_a 4.5 in terms of p and q, a 0606 question testing fluent use of the three log laws in reverse.

Reducing y = Ab^x to Linear Form

Given a straight-line graph of lg y against x, find A and b in y = Ab^x, a 0606 question on taking logs to linearise a relationship and read off the constants.

Simplifying an Expression with Log Laws

Combine lg 8 + lg 5 − lg 4 into a single logarithm and evaluate it, a 0606 question testing fluent use of the addition and subtraction laws.

Solving a Log Equation with Terms on Both Sides

Solve lg(x + 2) = 1 + lg(x − 1) by collecting logs and handling the lone constant, a 0606 question testing the subtraction law and base-10 conversion.

Solving an Equation with Different Bases

Solve 2ˣ = 3ˣ⁻¹ by taking logs of both sides and collecting the x terms, a 0606 exponential equation where the bases don't match.

Solving an Equation with the Laws of Logarithms

Solve log₃(x + 1) + log₃(x − 1) = 1 by combining logs, then reject the invalid root, a 0606 question where checking the domain earns the final mark.

Solving an Exponential Equation with ln

Solve 5e^(2x) = 40 exactly and as a decimal, a 0606 question on taking natural logs to bring the power down, with the exact-form mark explained.

Solving Simultaneous Logarithmic Equations

Solve a pair of simultaneous equations in log₂x and log₂y by treating the logs as the unknowns, then converting back, a neat 0606 logs question.

Using Logs to Solve an Exponential Equation

Solve 2^x = 7 by taking logs and using the power law, a 0606 question on bringing the exponent down when the base isn't e.

Area of a Quadrilateral from Its Vertices

Find the area of a quadrilateral from its four vertices using the array (shoelace) method, extending the triangle-area formula to four points, a 0606 question.

Area of a Triangle from Its Vertices

Find the area of a triangle from three coordinate vertices using the shoelace formula, a 0606 straight-line-graphs question with a reliable, error-resistant layout.

Equation of a Line Through Two Points

Find the equation of the straight line through two given points, a foundational 0606 coordinate-geometry skill: gradient first, then a point.

Equation of a Perpendicular Bisector

Find the perpendicular bisector of two points by combining the midpoint and the perpendicular gradient, a 0606 straight-line question in three clean steps.

Finding an Endpoint from the Midpoint

Find the other endpoint of a line segment given its midpoint and one end, a short 0606 coordinate-geometry question running the midpoint formula in reverse.

Line Perpendicular to a Given Line Through a Point

Find the line through (2, −1) perpendicular to 3x − 2y = 6, a 0606 straight-line question that starts by extracting a gradient from implicit form.

Point of Intersection of Two Lines

Find where two straight lines meet by solving them simultaneously, a foundational 0606 coordinate-geometry step that underpins many longer questions.

Showing Three Points Are Collinear

Show that three points lie on the same straight line by comparing gradients, a short 0606 coordinate-geometry proof with a clean logical finish.

Using the Distance Formula to Find an Unknown

Given the length between two points, find an unknown coordinate, a 0606 straight-line question that squares the distance formula into a quadratic with two answers.

Circle Equation from Centre and a Point

Find a circle's equation given its centre and a point on it, a 0606 circle question using the distance formula to get the radius, then the standard form.

Does a Line Intersect a Circle?

Decide whether a line cuts, touches, or misses a circle using the discriminant of the combined equation, a 0606 circle question linking to nature of roots.

Equation of a Circle from a Diameter

Given the two endpoints of a diameter, find the equation of the circle, a 0606 circle question combining the midpoint and distance formulae.

Equation of a Circle Touching the x-axis

Find the equation of a circle given its centre and the fact that it touches the x-axis, using the tangent-radius condition, a 0606 circle-geometry question.

Finding a Circle's Centre and Radius

Find the centre and radius of x² + y² − 6x + 4y − 12 = 0 by completing the square in x and y, a 0606 question on the newly added circle topic.

Length of a Tangent from an External Point

Find the length of the tangent from an external point to a circle using the right-angle between tangent and radius, a 0606 circle-geometry question.

Tangent to a Circle at a Given Point

Find the tangent to a circle at a point on it, using the fact that the tangent is perpendicular to the radius, a 0606 circle-geometry question.

Tangent to a Circle with an Off-Origin Centre

Find the tangent to a circle centred away from the origin at a point on it, a 0606 circle question using the perpendicular-to-radius property.

Where a Line Meets a Circle

Find the two points where y = x + 1 crosses x² + y² = 25, a 0606 circle question solved by substitution into a single quadratic.

Arc Length and the Perimeter of a Sector

Find the arc length and full perimeter of a sector with radius 8 cm and angle 1.2 radians, a 0606 circular-measure question with the classic perimeter trap.

Area and Perimeter of an Annular Sector

Find the area and perimeter of the region between two concentric arcs, combining sector formulae with the two straight edges, a 0606 circular-measure question.

Area of a Segment of a Circle

Find the area of a minor segment using sector area minus triangle area, a 0606 circular-measure question and the topic's most examined 'combined' shape.

Finding the Angle from the Arc Length

Find a sector's angle from its arc length and radius, then its area, a 0606 circular-measure question running the arc formula in reverse.

Perimeter of a Segment

Find the perimeter of a segment using the arc length and the chord (via the cosine rule), a 0606 circular-measure question combining two skills.

Sector Area, and Finding the Angle from the Perimeter

Given a sector's perimeter and radius, find its angle in radians and its area, a 0606 circular-measure question that runs the formulae in reverse.

Working in Radians with Exact Values

Convert an angle to radians, then find an exact arc length and sector area in terms of π, a 0606 Paper 1 circular-measure question with no decimals allowed.

Amplitude, Period and Counting Solutions

Read the amplitude, period, maximum and minimum of y = a + b sin(cx), then count how many solutions an equation has in a given range, a 0606 graph question.

Exact Trig Values from a Given Ratio

Given sinθ and that θ is obtuse, find the exact values of cosθ and tanθ, a 0606 Paper 1 question that tests the quadrant sign rule and the Pythagorean identity.

Exact Values of Special Angles

Evaluate an expression exactly using the special-angle values, a 0606 Paper 1 non-calculator question testing recall of sin, cos and tan at 30, 45 and 60 degrees.

Exact Values Using Double-Angle Formulae

Given sin θ, find sin 2θ and cos 2θ exactly using the double-angle formulae and a right triangle, a standard 0606 trigonometry question.

Proving a Reciprocal Trig Identity

Prove sec²x + cosec²x = sec²x·cosec²x by writing everything over a common denominator, a 0606 identity proof using the reciprocal functions.

Proving a tan² Identity

Prove that (1 − cos²x)/cos²x = tan²x using the Pythagorean identity, a short 0606 identity proof built on two familiar relationships.

Proving a Trigonometric Identity

A 0606 identity proof worked from one side only, the disciplined layout examiners reward, with the Pythagorean identity doing the heavy lifting.

Proving an Identity with tan and cot

Prove that tan x + cot x = 1/(sin x cos x) by converting to sines and cosines, a 0606 identity proof that shows the go-to first move.

Solving a Cosine Equation in Radians

Solve cos x = −0.5 over 0 to 2π, a 0606 trig question testing radian solutions and the quadrants where cosine is negative.

Solving a Quadratic Trigonometric Equation

Solve a 0606 trig equation that hides a quadratic: swap sin² for 1 − cos², factorise, reject the impossible root, then find every angle in range.

Solving a Tangent Equation in a Range

Solve tan(2x) = 1 over a given interval, a 0606 trig question where tan's 180-degree period and the doubled argument both matter.

Solving a Trig Equation Using an Identity

Turn a mixed tan-and-cos equation into a quadratic in sin using Pythagorean identities, then solve over 0 to 360 degrees, a classic 0606 question.

Solving a Trig Equation with a Multiple Angle

Solve sin(2x) = 0.5 in a given range, a 0606 trig question where the doubled argument means widening the search interval to catch every solution.

The R-Formula: Solving and Finding a Maximum

Express a sinθ + b cosθ as R sin(θ + α), then use that form to solve an equation and read off the maximum. Extension practice, not in the 0606 syllabus.

Transformations of a Trig Graph

Sketch y = 2 sin x + 1, reading the amplitude, period and range from the transformation, a standard 0606 trig-graph question.

A Committee with At Least Two Men

Count committees with at least two men by adding the qualifying cases, a 0606 combinations question on handling 'at least' by cases.

Arrangements Around a Circular Table (Beyond 0606)

Count seatings around a round table with (n−1)! and the block method. Extension practice: circular arrangements are not examined in Cambridge 0606.

Arrangements of a Word with Repeated Letters (Beyond 0606)

Count distinct arrangements of the letters of BANANA by dividing n! by the factorials of the repeats. Extension practice: repeated-object cases are not in 0606.

Arrangements with a Restriction: Letters Together

Count arrangements of a word where two letters must stay together using the block method, a 0606 permutations question on the 'glue them together' technique.

Arrangements with Two People Not Together

Count arrangements where two people must not sit together using the total-minus-together method, a 0606 permutations question on the complement approach.

Arrangements: Forming Numbers Without Repetition

How many 4-digit numbers can be formed from seven digits with no repetition, a 0606 permutations question, and why order makes this nPr not nCr.

Combinations: Selecting a Committee

Count committees with a fixed number of men and women using the multiplication principle with nCr, a 0606 combinations question on independent selections.

Forming Numbers with a Restriction

Count how many 4-digit even numbers can be formed with no repetition, a 0606 permutations question using the 'fill the restricted position first' rule.

Selections with an 'At Least One' Condition

Count teams containing at least one woman by subtracting the all-men case from the total, a 0606 combinations question on the complement method.

A Geometric Series Problem: The Bouncing Ball

Find the total distance a bouncing ball travels using a geometric series and sum to infinity, a classic 0606 applied GP question.

An Arithmetic Progression Savings Problem

Model a weekly savings plan as an arithmetic progression and find the total saved, a 0606 applied AP question testing the translation to a and d.

Arithmetic Progression from Two Given Terms

Given the 5th and 12th terms of an AP, find the first term, common difference and the sum of 20 terms, a 0606 question solved by simultaneous equations.

Binomial Expansion: Finding an Unknown Constant

Given the coefficient of x² in (2 + kx)⁶, find k, a 0606 binomial question that isolates one term with the general term rather than expanding everything.

Binomial: The Term Independent of x

Find the constant term in a binomial expansion by setting the power of x to zero, a 0606 question using the general term to target one specific term.

Coefficient in a Product with a Binomial Expansion

Find the coefficient of x² in (1+2x)(1−x)⁶ by expanding only the terms that can produce x², a common 0606 binomial question.

Finding n and a from Binomial Coefficients

Given two coefficients in the expansion of (1+ax)ⁿ, set up simultaneous equations to find n and a, a frequently examined 0606 binomial question.

Finding the First Term of an AP from a Sum

Given the sum of the first n terms and the common difference, find the first term of an arithmetic progression, a 0606 question running the sum formula in reverse.

Geometric Progression from Two Given Terms

Given the 2nd and 5th terms of a GP, find the common ratio, first term and the sum of 8 terms, a 0606 question solved by dividing the term equations.

Geometric Series: Salary with Annual Rises

A salary rising by a fixed percentage each year forms a geometric progression; find a later year's pay and the total over ten years, a 0606 series application.

How Many AP Terms Are Needed to Exceed a Value

Find how many terms of an arithmetic progression are needed for the sum to first exceed 200, a 0606 question that ends in a quadratic inequality in n.

Recurring Decimal as a Geometric Series

Convert 0.474747… to a fraction using the sum to infinity of a geometric series, a neat 0606 application of S∞ = a/(1−r).

Sum of Multiples in a Range

Add up all the multiples of 3 below 100 by treating them as an arithmetic progression, finding n first, then applying the sum formula, a 0606 series question.

Sum to Infinity of a Geometric Progression

Given the first term and sum to infinity of a GP, find the common ratio and a later term, a 0606 question testing the |r| < 1 convergence condition.

The First Negative Term of an AP

Find the first negative term of a decreasing arithmetic progression by solving an inequality for n, a 0606 question ending with a whole-number check.

Collinear Points Using Vectors

Show three points are collinear using position vectors and the idea of a scalar multiple, a 0606 vectors question on parallel vectors sharing a point.

Expressing a Vector as a Combination of Two Others

Find scalars λ and μ so that c = λa + μb by comparing components, a 0606 vectors question that becomes simultaneous equations.

Finding λ So That Two Vectors Are Parallel

Find the value of λ making two vectors parallel using the scalar-multiple condition, a short 0606 vectors question on the meaning of parallel.

Magnitude and the Unit Vector

Find the magnitude of a vector and the unit vector in its direction, a 0606 vectors question on the Pythagoras-style modulus and scaling to length one.

Position of a Particle Moving with Constant Velocity

Find a particle's position after a given time from its start point and constant velocity vector, a 0606 vectors kinematics question using r = r₀ + tv.

Position Vector of a Point Dividing a Line

Find the position vector of a point that divides AB in a given ratio using OP = OA + (fraction)·AB, a standard 0606 vectors question.

Position Vectors and the Displacement Between Points

Find the displacement vector AB from two position vectors and its magnitude, a 0606 vectors question on the 'destination minus origin' rule.

Resultant of Two Vectors and Its Magnitude

Add two vectors and find the magnitude of the resultant, a foundational 0606 vector question combining component addition with Pythagoras.

Area Between a Curve and a Line

Find the area enclosed between a curve and a line by integrating the difference, a 0606 calculus question that starts with finding the intersection points.

Area Between a Curve and the x-axis

A 0606 definite-integral question where the region sits below the x-axis, worked to the mark, including why the integral comes out negative and how to report area.

Connected Rates of Change: Expanding Circle

A circle's radius grows at a known rate; find how fast the area grows using the chain rule dA/dt = dA/dr · dr/dt, a 0606 connected-rates question.

Connected Rates of Change: Expanding Sphere

A 0606 Paper 2 connected-rates question worked step by step using the chain rule, the topic students most often freeze on, made mechanical.

Connected Rates: Filling a Cylindrical Tank

Find how fast the water level rises in a cylindrical tank from the rate of volume increase, a 0606 connected-rates question with a clean chain rule.

Differentiation: Product, Quotient and Chain Rule

Three short 0606 differentiation parts, one each for the product, quotient and chain rule, worked with the factorising step examiners expect in the answer.

Finding the Equation of a Curve from Its Gradient

Given dy/dx and a point on the curve, integrate and find the constant to recover y, a 0606 question that tests integration plus the +C step students forget.

Finding Where a Tangent Is Parallel to a Line

Find the point where a curve's tangent is parallel to a given line by matching gradients, a 0606 calculus question linking differentiation to straight lines.

Greatest and Least Values on a Closed Interval

Find the greatest and least values of a cubic on a closed interval by comparing stationary points with the endpoints, a 0606 calculus question.

Integrating a Bracket Raised to a Power

Integrate a linear bracket raised to a power using the reverse chain rule, then evaluate a definite integral, a core 0606 integration technique.

Kinematics: Finding Velocity from Acceleration

Integrate acceleration to get velocity and displacement, using initial conditions to fix the constants, a core 0606 kinematics question.

Kinematics: Total Distance from a Velocity Function

A 0606 kinematics question: find when a particle is at rest, then the total distance travelled, the part where distance and displacement differ.

Maximum Volume of an Open Box

Maximise the volume of an open box made by cutting squares from a sheet, a classic 0606 optimisation question with a rejected root to watch for.

Optimisation: Minimum Surface Area of a Cylinder

A 0606 applied maximum/minimum question: form the surface-area function from a volume constraint, then minimise it and prove it is a minimum.

Small Changes and Approximation

Use the derivative as a rate to approximate a small change in y, the standard 0606 'small increments' technique with δy ≈ (dy/dx)·δx.

Stationary Points and the Second Derivative Test

Find both stationary points of a cubic and classify each as a maximum or minimum using the second derivative, a standard 0606 calculus question.

Tangent and Normal to a Cubic Curve

A full 0606 Paper 1 question worked to the mark: differentiate a cubic, then find the tangent and normal at a point, in the exact form examiners want.

Tangent to an Exponential Curve

Find the equation of the tangent to y = e^(2x) at a point, differentiating the exponential and using the chain rule, a 0606 calculus question.

Velocity and Acceleration from Displacement

Differentiate a displacement function to find velocity and acceleration, then find the velocity when acceleration is zero, a 0606 kinematics question.

Where a Function Is Increasing or Decreasing

Find the intervals where a cubic is increasing or decreasing using the sign of dy/dx, a 0606 calculus question that ends in a quadratic inequality.