A GCSE × IB MYP Mathematics crossover test. One hundred questions carry you from the shared valley — the foundations both kingdoms teach — up to the GCSE-only summit tower, ending in word problems that call on everything you've learned.
The Shared Kingdom
Chapters 1–9. Core content that both GCSE and IB MYP Mathematics teach: number, algebra, geometry, graphs, statistics and probability.
The GCSE Summit Tower
Chapter 10. Circle theorems, the quadratic formula, and the sine & cosine rules — GCSE Higher-tier content that sits beyond typical MYP scope.
Shared Kingdom Ch.1 Number Foundations
Shared Kingdom Ch.2 Ratio & Proportion
Shared Kingdom Ch.3 Indices, Standard Form & Surds
A delivery fee is modeled as 2x - 5 dollars, where x is the number of items. For which x is the fee greater than 9 dollars? Solve 2x - 5 > 9.
Add 5 to both sides, then divide by 2.
Shared Kingdom
Chapter 5 · Sequences & Patterns
Spotting the rule hidden inside a list of numbers.
Q41Skills Check
Find the next term: 3, 7, 11, 15, ...
Find the common difference between terms.
Q42Skills Check
The nth term of a sequence is 4n - 1. Find the 10th term.
Substitute n=10 into the formula.
Q43Skills Check
Find the nth term rule for the sequence 5, 8, 11, 14, ...
Common difference is 3; adjust the constant so n=1 gives the first term.
Q44Skills Check
Find the next term: 2, 6, 18, 54, ...
Each term is multiplied by the same ratio.
Q45Word Problem
A pattern of dots forms triangular numbers: 1, 3, 6, 10, ... How many dots are in the 5th pattern?
Triangular numbers add one more row each time: +2, +3, +4, +5.
Q46Skills Check
A sequence follows the rule n^2 + 1: 2, 5, 10, 17, 26, ... Find the 8th term.
Substitute n=8 into n^2+1.
Q47Word Problem
Find the sum of the first 20 terms of the sequence 3, 7, 11, 15, ...
Use the sum formula S = n/2 (2a + (n-1)d).
Q48Word Problem
Maya saves $5 in week 1, then $3 more than the previous week, every week. How much has she saved in total after 12 weeks?
This is an arithmetic series with a=5, d=3, n=12.
Q49Skills Check
Find the next term: 1, 1, 2, 3, 5, 8, ...
Each term is the sum of the two terms before it.
Q50Word Problem
A sequence begins 3, 8, 15, 24, 35, following the rule n^2 + 2n. What is the 6th term?
Substitute n=6 into n^2 + 2n.
Shared Kingdom
Chapter 6 · Coordinate Geometry & Graphs
Lines, gradients, and the map of the x-y plane.
Q51Skills Check
What is the gradient (slope) of the line y = 3x + 2?
In y=mx+c, m is the gradient.
Q52Skills Check
What is the y-intercept of the line y = -2x + 7?
The y-intercept is the constant term c in y=mx+c.
Q53Skills Check
Find the gradient of the line joining (2, 3) and (4, 7).
Gradient = (change in y) / (change in x).
Q54Skills Check
Find the midpoint of (2, 5) and (6, 9).
Average the x-coordinates and average the y-coordinates.
Q55Skills Check
Find the distance between (2, 5) and (6, 9).
Use the distance formula: sqrt((x2-x1)^2+(y2-y1)^2).
Q56Skills Check
A line passes through (0, 4) with gradient -1/2. What is its equation?
Use y = mx + c with c as the y-intercept.
Q57Skills Check
A line is parallel to y = 4x - 1. What is its gradient?
Parallel lines always share the same gradient.
Q58Skills Check
A line is perpendicular to y = 2x + 3. What is its gradient?
Perpendicular gradients multiply to give -1.
Q59Word Problem
A phone plan costs y = 0.1x + 20 dollars, where x is the number of texts sent. Find the cost for 150 texts.
Substitute x=150 into the equation.
Q60Word Problem
A line passes through (1, 2) and (4, 8). At what x-value does this line cross the x-axis?
Find the gradient and equation first, then set y=0.
Shared Kingdom
Chapter 7 · Geometry: Angles, Area & Volume
Shapes, space, and the rules that hold them together.
Q61Skills Check
Two angles of a triangle are 65 degrees and 72 degrees. Find the third angle.
Angles in a triangle sum to 180 degrees.
Q62Skills Check
Three angles of a quadrilateral are 90, 115, and 80 degrees. Find the fourth angle.
Angles in a quadrilateral sum to 360 degrees.
Q63Skills Check
Find the perimeter of a rectangle with length 8 cm and width 5 cm.
Perimeter = 2 x (length + width).
Q64Skills Check
Find the area of a triangle with base 10 cm and height 6 cm.
Area of a triangle = 1/2 x base x height.
Q65Skills Check
Find the area of a circle with radius 7 cm, to 1 decimal place. (Use pi = 3.14159)
Area of a circle = pi x r^2.
Q66Skills Check
Find the volume of a cuboid with dimensions 4 cm x 5 cm x 6 cm.
Volume of a cuboid = length x width x height.
Q67Word Problem
A cylindrical tank has radius 3 m and height 10 m. Find its volume, to 1 decimal place.
Volume of a cylinder = pi x r^2 x h.
Q68Word Problem
An isosceles triangle has a top angle of 40 degrees. Find each base angle.
The two base angles are equal; all three angles sum to 180.
Q69Word Problem
A window is a 10 cm x 6 cm rectangle topped with a semicircle of radius 3 cm. Find its total area, to 1 decimal place.
Add the rectangle's area to the semicircle's area (half a circle).
Q70Word Problem
Two similar cones have a linear scale factor of 2 (large:small). The small cone has volume 20 cm^3. Find the volume of the large cone.
Volume scale factor = (linear scale factor)^3.
Shared Kingdom
Chapter 8 · Pythagoras & Trigonometry
Right triangles and the ratios that unlock their angles.
Q71Skills Check
A right triangle has legs of 3 cm and 4 cm. Find the hypotenuse.
Use Pythagoras: a^2+b^2=c^2.
Q72Skills Check
A right triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the other leg.
Rearrange Pythagoras: b^2=c^2-a^2.
Q73Skills Check
What is sin(30 degrees)?
This is one of the standard triangle angle values.
Q74Word Problem
In a right triangle, the hypotenuse is 10 cm and one angle is 40 degrees. Find the length of the side opposite that angle, to 1 decimal place.
Use sin(angle) = opposite / hypotenuse.
Q75Word Problem
In a right triangle, the hypotenuse is 15 cm and one angle is 55 degrees. Find the length of the side adjacent to that angle, to 1 decimal place.
Use cos(angle) = adjacent / hypotenuse.
Q76Word Problem
A right triangle has an opposite side of 8 cm and adjacent side of 6 cm. Find the angle, to 1 decimal place.
Use tan(angle) = opposite / adjacent, then inverse tan.
Q77Word Problem
A 6 m ladder leans against a wall, with its foot 2.5 m from the wall base. How high up the wall does it reach, to 1 decimal place?
The ladder is the hypotenuse; use Pythagoras.
Q78Word Problem
From a point 50 m from a tower's base, the angle of elevation to the top is 25 degrees. Find the tower's height, to 1 decimal place.
Use tan(angle) = height / distance.
Q79Word Problem
A box has a rectangular base 6 cm by 8 cm and height 24 cm. Find the length of the longest diagonal inside the box.
First find the base diagonal, then use it with the height in Pythagoras again.
Q80Skills Check
What is tan(45 degrees)?
This is a standard triangle angle value.
Shared Kingdom
Chapter 9 · Statistics & Probability
Making sense of data and predicting chance.
Q81Skills Check
Find the mean of 4, 8, 6, 5, 7.
Add all the values, then divide by how many there are.
Q82Skills Check
Find the median of 3, 9, 5, 7, 1.
Put the numbers in order, then find the middle one.
Q83Skills Check
Find the mode of 2, 2, 3, 5, 5, 5, 7.
The mode is the value that appears most often.
Q84Skills Check
Find the range of 2, 2, 3, 5, 5, 5, 7.
Range = highest value - lowest value.
Q85Skills Check
A fair six-sided die is rolled. What is the probability of rolling a 4?
There is one favorable outcome out of six equally likely outcomes.
Q86Skills Check
A fair six-sided die is rolled. What is the probability of NOT rolling a 4?
P(not A) = 1 - P(A).
Q87Word Problem
A fair coin is flipped twice. What is the probability of getting heads both times?
Multiply the probability of each independent event.
Q88Word Problem
A bag has 4 red and 6 blue balls. Two balls are drawn without replacement. Find the probability both are red.
Multiply the probabilities, reducing the total each time (no replacement).
Q89Word Problem
A survey of pets per household: 2 pets (3 households), 4 pets (5 households), 6 pets (2 households). Find the mean number of pets.
Multiply each value by its frequency, sum, then divide by the total number of households.
Q90Word Problem
P(rain) = 3/10. If it rains, P(late) = 1/2. If it doesn't rain, P(late) = 1/5. Find the overall probability of being late.
Add the probabilities of each path to being late (rain-and-late, no-rain-and-late).
GCSE Summit Tower
Chapter 10 · GCSE-Only Extension & Word Problems
Circle theorems, the quadratic formula, sine & cosine rule — content that goes beyond typical MYP scope, plus a final climb of mixed word problems.
Q91Skills Check
The angle at the circumference is 70 degrees. Find the angle at the center subtended by the same arc. (Circle theorem — GCSE Higher only, beyond typical MYP scope.)
The angle at the center is twice the angle at the circumference.
Q92Skills Check
A triangle is drawn inside a circle with one side as the diameter. What is the angle opposite the diameter? (Circle theorem — GCSE Higher only.)
The angle in a semicircle is always a right angle.
Q93Skills Check
Solve x^2 + 2x - 8 = 0 using the quadratic formula. (GCSE Higher only — beyond typical MYP scope.)
Use x = (-b +/- sqrt(b^2-4ac)) / 2a with a=1, b=2, c=-8.
Q94Skills Check
Write x^2 + 6x + 4 in the form (x + a)^2 + b. (GCSE Higher only.)
Halve the x-coefficient to get a, then adjust the constant.
Q95Skills Check
Solve simultaneously: 2x + y = 11 and x - y = 1.
Add the two equations to eliminate y.
Q96Skills Check
Solve simultaneously: y = x + 1 and y = x^2 - 5. (One linear, one quadratic — GCSE Higher only.)
Substitute the linear equation into the quadratic, then solve for x.
Q97Word Problem
In a triangle, side a = 8 cm is opposite angle A = 40 degrees, and angle B = 65 degrees. Find side b using the sine rule, to 1 decimal place. (Sine rule — GCSE Higher only.)
Sine rule: a/sin(A) = b/sin(B).
Q98Word Problem
A triangle has sides 7 cm and 9 cm with an included angle of 55 degrees. Find the third side using the cosine rule, to 1 decimal place. (Cosine rule — GCSE Higher only.)
Cosine rule: c^2 = a^2 + b^2 - 2ab cos(C).
Q99Skills Check
Which expression proves that the sum of two consecutive integers is always odd? (Algebraic proof — GCSE Higher only.)
Let the integers be n and n+1, then simplify their sum.
Q100Word Problem
A ramp's cross-section is a 3:4:5 right triangle with the base (3 side) measuring 24 m. A handrail runs along the hypotenuse (5 side) and costs $12.50 per meter, plus 10% extra for material waste. Find the total handrail cost.
Scale the 3:4:5 ratio to find the hypotenuse, then apply cost and waste.
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Questions to revisit
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