Even a 3-D question,step by step.
A tetrahedron, and a craftsman's claim about the angle between two of its faces. Press Next step. The solid turns, its faces are lifted out flat, and the working is written out line by line.
Question (7 marks)
The figure shows a tetrahedron $ABCD$. Let $P$ be a point lying on $AD$ such that $BP$ is perpendicular to $AD$. A craftsman finds that $ {AC=AD=CD=13\text{ cm}}$, $ {BC=8\text{ cm}}$, $ {BD=12\text{ cm}}$ and $ {\angle ABD=72^\circ}$.
(a) Find
(i) $\angle BAD$,
(ii) $CP$.
(b) The craftsman claims that $\angle BPC$ is the angle between the face $ABD$ and the face $ACD$. Is the claim correct? Explain your answer.
3-D trigonometry
Example questionSolution
- The working appears here, one line per step.
Every DSE paper, 2012–2026
Paper 1 and Paper 2 of all 15 years: 30 papers and 961 questions, each one its own animation, in both English and Chinese. Every question is also filed under its topic, so a topic can be revised across every year at once. The topics below are ranked by the marks they carried; the darker the box, the more marks on that topic that year.
Marks on each topic, year by year (Paper 1 and Paper 2 together, each Paper 2 question 1 mark)
| Topic | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 | 2025 | 2026 | 2012–2026 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Polynomials, Identities and Factorization | 17 | 18 | 15 | 16 | 24 | 18 | 18 | 20 | 16 | 19 | 19 | 16 | 22 | 18 | 20 | 276 |
| Coordinate Geometry | 14.5 | 12.5 | 11.5 | 12.5 | 12.5 | 14.5 | 15.5 | 5 | 13.5 | 14.5 | 24.5 | 18.5 | 21 | 22 | 25 | 238 |
| Statistics | 15 | 19 | 14 | 13 | 18 | 16 | 14.5 | 17 | 15 | 16 | 16 | 16 | 16 | 16 | 15 | 236 |
| Trigonometry | 12 | 12 | 11 | 17 | 10 | 15 | 10 | 9 | 14.5 | 8 | 10 | 8 | 11 | 3.5 | 10 | 161 |
| Deductive Geometry | 3.5 | 5.5 | 7 | 8.5 | 18 | 10.5 | 12.5 | 21.5 | 10 | 10 | 9.5 | 13.5 | 10 | 10 | 10 | 160 |
| Rate, Ratio and Variation | 9 | 8 | 16 | 8 | 8 | 7 | 15 | 11 | 11.5 | 8 | 8 | 9.5 | 12.5 | 11 | 8 | 150 |
| Mensuration | 15 | 10 | 9 | 7 | 8 | 8 | 11 | 8 | 8.5 | 11 | 9 | 9.5 | 9 | 11 | 10 | 144 |
| Arithmetic and Geometric Sequences | 15 | 14 | 6 | 7 | 7 | 6 | 7.5 | 8 | 7 | 9.5 | 5.5 | 10 | 1 | 8 | 9 | 120 |
| Functions and Graphs of Quadratic Functions | 2 | 7 | 2 | 10 | 7 | 10 | 2 | 8 | 9.5 | 8 | 4.5 | 2.5 | 8 | 10 | 8 | 98 |
| Inequalities and Linear Programming | 6 | 6 | 10 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 6 | 6 | 6 | 6 | 6 | 98 |
| Probability | 7 | 5 | 14 | 10 | 6 | 7 | 5 | 3 | 7 | 3.5 | 6 | 6 | 6 | 6 | 6 | 98 |
| Indices, Surds and Logarithms | 5 | 6 | 9 | 5 | 5 | 9 | 6 | 3 | 6 | 6 | 7 | 5 | 5.5 | 10 | 6 | 94 |
| Percentages | 5 | 2 | 6 | 6 | 3 | 1 | 6 | 5 | 5 | 5 | 5 | 5 | 5 | 6 | 9 | 74 |
| Basic Properties of Circles | 4.5 | 2 | 3 | 8 | 2 | 5 | 7 | 10 | 8.5 | 2 | 3 | 6.5 | 2 | 2 | 2 | 68 |
| Locus | 5.5 | 1 | 4.5 | 1 | 2.5 | 1 | 1 | 1 | 1 | 4.5 | 4.5 | 7 | 4 | 4.5 | 1 | 44 |
| Numbers, Estimation and Errors | 2 | 6 | 5 | 3 | 1 | 6 | 4 | 2 | 4 | 1 | 1 | 2 | 1 | 1 | 2 | 41 |
| Quadratic Equations in One Unknown | 1 | 3 | 2 | 3 | 2 | 1 | 2 | 1 | 4 | 2 | 5.5 | 3.5 | 2 | 2 | 2 | 36 |
| More about Equations | 5 | 4 | 1 | 5 | 3 | 4 | 0 | 2.5 | 0 | 4 | 3 | 0 | 1 | 0 | 0 | 32 |
| Permutation and Combination | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 22 |
| More about Graphs of Functions | 1 | 0 | 1 | 0 | 0 | 2 | 1 | 6 | 0 | 0 | 0 | 0.5 | 0 | 0 | 0 | 12 |
Marks in a year: 01–34–67–1011 or more
2026 is the first exam on the revised syllabus.
Form 1 to Form 6, chapter by chapter
Each chapter is one lesson in five parts, every question animated. After it comes a Chapter Test with different questions each time it is sat, marked at once.
- Summary
- Level 1
- Level 2
- Level 3
- Test and exam questions
- Chapter Test
Junior secondary Form 1 to Form 3, 37 chapters Show the chapters
- Revision on Fundamental Arithmetic
- Basic Mathematics
- Directed Numbers and the Number Line
- Introduction to Algebra
- Linear Equations in One Unknown
- Introduction to Geometry
- Introduction to Statistics and Statistical Charts
- Percentages (I)
- Approximate Values and Numerical Estimation
- Areas and Volumes (I)
- Manipulation of Simple Polynomials
- Angles related to Lines
- Introduction to Coordinates
- Errors in Measurement
- Identities and Factorization
- Algebraic Fractions and Formulas
- Angles related to Rectilinear Figures
- Congruence
- More about Statistical Charts
- Rate, Ratio and Proportion
- Similar Triangles
- Linear Equations in Two Unknowns
- Pythagoras' Theorem and Irrational Numbers
- Areas and Volumes (II)
- Trigonometric Ratios
- More about Factorization of Polynomials
- Laws of Integral Indices
- Linear Inequalities in One Unknown
- Percentages (II)
- Quadrilaterals
- Special Lines and Centres in a Triangle
- Areas and Volumes (III)
- Coordinate Geometry of Straight Lines
- Trigonometric Relations
- Applications of Trigonometry
- Measures of Central Tendency
- Introduction to Probability
Senior secondary Form 4 to Form 6, 25 chapters Show the chapters
- Quadratic Equations in One Unknown (I)
- Quadratic Equations in One Unknown (II)
- Functions and Graphs
- Equations of Straight Lines
- More about Polynomials
- Exponential Functions
- Logarithmic Functions
- More about Equations
- Variations
- More about Trigonometry
- Basic Properties of Circles
- Tangents to Circles
- Inequalities
- Linear Programming
- Applications of Trigonometry in 2-dimensional Problems
- Applications of Trigonometry in 3-dimensional Problems
- Equations of Circles
- Locus
- Measures of Dispersion
- Permutation and Combination
- More about Probability
- Arithmetic and Geometric Sequences
- Summation of Arithmetic and Geometric Sequences
- More about Graphs of Functions
- Uses and Abuses of Statistics
AI keeps track of every student.Parents never have to guess.
Every step watched and every test question answered goes into the record. From it, AI works out each student's level chapter by chapter, forecasts the next test or exam, and works out what to revise first and for how long. Parents see it all on their own account.
Records
Every step watched and every test question answered, automatically.
Assesses
The level of every chapter and every DSE topic, updated after each test.
Forecasts
The next Chapter Test and the next school test or exam: a score and its range. Every forecast is checked against the real result.
Advises
Which chapters to revise first, and what 5, 10 or 20 more hours would bring, at the student's own pace.
- Questions watched
- 412
- Chapter Tests sat
- 18
- Studied
- 5 of the last 7 days
- Target
- 75%
If the Uniform Test were tomorrow
66%
An estimated range from your child's learning data, not a promise: eight times in ten the result should land inside it.
Level by chapter
- Quadratic Equations in One Unknown (I)Secure
- Equations of Straight LinesMostly there
- More about PolynomialsNeeds work
- Functions and GraphsNeeds work
Study time
3 h 30 min
this week
18 h 15 min in the last 8 weeks
More revision
| 5 more hours | about 71% |
| 10 more hours | about 74% |
| 20 more hours | about 77% |
To reach 75%: about 12.5 hours of revision.
Revise first
How the next 10 hours are best spent
- 1Functions and Graphsabout 2 h 10 min
- 2More about Polynomialsabout 1 h 40 min
- 3Equations of Straight Linesabout 1 h 15 min
If the Uniform Test were tomorrow, the learning data point to about 66% overall. Functions and Graphs needs the most work: about 2 h 10 min on it first, then More about Polynomials. About 12.5 hours of revision should reach the 75% target.
Every answer leaves a trace.Answers from AI do too.
A chatbot answers any homework question in seconds. The homework looks perfect, and what was never learnt shows only in the exam. The platform's AI looks at how every Chapter Test and class quiz is answered, not only at the answers, and picks out work that does not look like the student's own.
Analyses
How every Chapter Test and class quiz is answered, from the first question to the hand-in, not just the answers.
Flags
When the signs add up, the case goes to the teacher with the evidence as soon as the paper is handed in.
Teacher confirms
The teacher weighs the evidence. Nothing changes until it is confirmed.
Shows parents
Once confirmed, the result counts as zero, and the parent's page shows it in red, at the top.
At the top of the parent's page, in red
⚠ Answering honestly: please read
- Chapter Test: Logarithmic Functions, Paper 1 · 1 Oct, 21:42
Counts as zero.
The teacher has confirmed signs that this was answered with the help of AI or another tool, so it counts as zero. Please talk it over with your child: a test is there to find what is not yet understood, and answers that are not their own hide exactly that from the teacher and from you.