Practise by skill
Choose something you want to strengthen, try the linked question, then compare your reasoning with its working. Review the idea if you get stuck.
These 30 pathways connect selected questions to useful ideas. They are a starting point; use chapter practice for the rest of the course.
Pure mathematics
Use a perpendicular foot to check a reflection
Projection connects the equation of a line to distance and reflection. A geometric check can catch a sign or parameter error in the coordinates.
After your attempt: check your approach
Keep y = 4 as a separate Cartesian equation. Check that N lies on AB and CN is perpendicular to AB, then use N as the midpoint of CD.
Check the whole range before composing
A valid formula is not enough: every output of the first function must be accepted by the second. This range check also supports inverse functions and graph reasoning.
After your attempt: check your approach
Compare the entire inner range with the outer domain in each order, then explain why one composite exists and the other does not.
Carry a sign diagram into a new input
Zeros, excluded values and input restrictions connect algebraic inequalities to logarithmic and exponential functions.
After your attempt: check your approach
Distinguish zeros from poles in part (i). In part (ii), keep the positive-input condition and translate every interval through the logarithm.
Model repeated change in the right order
A recurrence turns a repeated process into a model. Geometric sums then connect the step-by-step rule to a formula and its long-run behaviour.
After your attempt: check your approach
Apply the daily loss before the top-up. Derive the formula using a geometric sum and explain the limiting volume in context.
Use the normal to connect distance and reflection
The same plane normal locates a perpendicular foot, measures distance and identifies the midpoint needed for a reflection.
After your attempt: check your approach
Check that the foot lies on the plane and that its displacement from the original point is parallel to the normal. Use the foot as the midpoint of the reflected pair.
Use a known complex root to unlock a polynomial
Real coefficients connect non-real roots in conjugate pairs. Factoring once can then solve a related equation through a change of variable.
After your attempt: check your approach
Explain why the conjugate must also be a root. Before using reciprocal roots in part (ii), check that zero is excluded.
Calculus
Check where an approximation can be used
Combining expansions combines their conditions too. A function can have a real value at an input where its series does not converge.
After your attempt: check your approach
Intersect the validity intervals before substituting. Explain why x = -1 fails, then isolate the required radical after using x = 1/9.
Check series coefficients by a second method
A derivative relation and a product of standard series describe the same function. Comparing them connects differentiation, approximation and algebra.
After your attempt: check your approach
Evaluate the derivatives at zero and divide by the correct factorials. Verify the coefficients through the cubic term by multiplying the standard series.
Recognise a composite before differentiating
The chain rule connects differentiation to changing quantities and substitution in integration.
After your attempt: check your approach
Identify the inner function and explain where its derivative enters your answer.
Connect quantities before calculating a rate
An equation between the quantities turns a word problem into a rate calculation.
After your attempt: check your approach
Name the changing quantities, write their relation, and check the units of your final rate.
Build the model before finding an optimum
Choosing the quantity and its feasible domain matters as much as differentiating it.
After your attempt: check your approach
Explain your constraint and why the chosen value gives the required optimum.
Recognise what a substitution changes
A change of variable connects integration with the chain rule you already know.
After your attempt: check your approach
Change both the expression and the differential, then differentiate your result to check it.
Choose the region before choosing the integral
A signed integral, a total area and a rotation volume can use the same curve while describing different regions and quantities.
After your attempt: check your approach
Mark the sign changes before splitting the absolute integral. For the volume, identify which lobe the stated boundary selects and square the radius.
Build a rate equation from inflow and outflow
A rate model connects the physical balance, initial condition and long-run behaviour. Solving the equation is only one part of explaining the process.
After your attempt: check your approach
Use inflow minus outflow, calibrate the constant from the given net rate, and check both the initial volume and the limiting balance.
Statistics
Choose a distribution for the sample mean
Distinguishing an observation from a sample mean connects probability models to inference. Sample size alone does not justify every approximation.
After your attempt: check your approach
State the sampling mean and variance. Explain whether normality is exact or approximate and which assumptions justify it.
Choose hypotheses that match the claim
The claim determines the direction of the test before you calculate a probability.
After your attempt: check your approach
Define the population mean, state the test statistic and its null distribution, and justify the alternative and assumptions.
Turn a p-value into a contextual conclusion
A probability becomes useful evidence when you compare it with the level and answer the original claim.
After your attempt: check your approach
Compare the unrounded p-value with the level and explain what the evidence supports.
Decide which variable you are predicting
Choosing the regression direction and checking the observed range makes a prediction meaningful.
After your attempt: check your approach
Name the response variable and justify whether your prediction is reliable.
Connect a distribution to its mean and spread
Expectation describes a long-run average; variance describes spread. These ideas carry into binomial models, normal models and sampling.
After your attempt: check your approach
Check that the probabilities sum to one. Calculate the mean and variance, and explain why variances add for the independent difference in part (i).
Decide whether a binomial model fits
Fixed trials and two outcomes are only part of the decision. Independence and a constant success probability determine whether the model fits the situation.
After your attempt: check your approach
Explain why sampling without replacement fails the model in part (i), then check what replacement changes in part (iii).
Find the most likely count by comparing probabilities
The expected count need not be the most likely count. Comparing probability masses also helps you detect a tie between two modes.
After your attempt: check your approach
Compare candidate counts and their neighbours using unrounded probabilities; distinguish a modal count from an expected number of samples.
Distinguish repeated observations from a multiple
A sum of independent measurements has a different spread from a multiple of one measurement. This distinction carries into sample means and comparisons.
After your attempt: check your approach
In part (iii), distinguish four independent times from four times one time. Square coefficients in the variance and state the independence assumption.
Translate an interval into the correct normal tails
A sketch connects the wording, the probability and the calculator input. It also helps recover an unknown standard deviation from a central interval.
After your attempt: check your approach
For part (iii), divide the remaining probability between the two tails before choosing a quantile. Distinguish standard deviation from variance.
Check who the sample represents
A larger sample can reduce sampling variability while leaving selection bias in place. Conclusions depend on how the observations were obtained.
After your attempt: check your approach
Describe a random selection from the day's production in part (i), rather than simply choosing readily available bars.
Calculate estimates before carrying out inference
Reliable inference starts with correct estimates. Frequency tables and shifted summaries describe the same underlying quantities in different forms.
After your attempt: check your approach
Count the observations using the frequencies and divide the corrected sum of squares by n minus one when estimating population variance.
Explain what a regression line minimises
Least squares measures discrepancies in the response variable. Understanding that direction explains why swapping the variables generally produces a different line.
After your attempt: check your approach
In part (b)(iv), identify the response variable and explain which squared residuals the fitted line minimises; no derivation of the regression formula is needed.
Compare models against the shape and context
A strong correlation does not settle whether a model is suitable. Scatter, physical limits and the prediction range all contribute to the decision.
After your attempt: check your approach
Explain why unbounded linear growth is unsuitable here, compare the proposed transformations, and interpret the limiting area in context.
Identify the population you are conditioning on
Reversing a conditional probability changes the denominator. Keeping the shared intersection connects probability trees, set diagrams and statistical interpretation.
After your attempt: check your approach
Write both conditional probabilities using their intersection numerator. Recover the marginal probabilities before interpreting the union with a complement.
Separate finite sampling from independent trials
Counting a sample without replacement and conditioning two independent people require different models, even when both questions ask for an exact count.
After your attempt: check your approach
Explain why the cupcake counts change without replacement. In part (b), condition on flu before finding the vaccinated proportion and state the independent-person assumption.
Decide exactly what the arrangement restriction means
Separating all vowels differs from separating two repeated letters. Defining the restriction first helps you choose gaps, complements or disjoint cases without overcounting.
After your attempt: check your approach
Compare the restrictions in parts (i) and (ii), account for identical letters, and explain why each chosen gap or case is counted once.
After checking your solution, try again later without the working. Note which step you could explain unaided and which idea needs another attempt.