9709 Pure Mathematics 1 Past Papers 2020 to 2025: Papers, Mark Schemes, Examiner Insights

An answer with no method attached scores nothing

This is the single most repeated line across every report we read. The front cover of the paper says unsupported answers from a calculator earn no marks, and examiners enforce it. Your calculator can solve the quadratic. It cannot earn you the method mark.

• Solving a quadratic: show factorisation, a completed square, or the actual numbers substituted into the formula. Quoting the formula and then writing the roots is not enough.
• If you factorise, check your factors expand back to the coefficients you started with. Examiners spot factors reverse-engineered from calculator roots.
• Definite integrals: write the integrated expression, then show both limits substituted. A calculator's numerical value alone is not creditable.
• Use the calculator to check the answer you got on paper, not to replace the working.

The form of the answer is part of the answer

A correct value in the wrong form loses the accuracy mark. Reports flag this in every session, and it is the cheapest set of marks on the paper to get back.

• Default accuracy is three significant figures, and one decimal place for angles in degrees, unless the question says otherwise.
• Carry at least four significant figures through your intermediate working, or your three-figure answer drifts.
• If the question says exact, a decimal is wrong. Surds stay as surds, fractions stay as fractions.
• The reverse also bites. If the answer comes out exact, do not round it to three significant figures out of habit.
• In circular measure, converting radians to degrees midway is where accuracy quietly disappears.

Half the solutions lost, the impossible one kept

Two opposite errors, both reported repeatedly. Students find one root when there are four, and students keep a root the question has already ruled out. Both come from not asking what the equation and the context allow.

• Hidden quadratics: a quartic in k, or a quadratic in sin squared theta or in cos theta, has more solutions than the quadratic you solved.
• Taking a square root gives a positive and a negative value, and reports show candidates finding only one of them. Where a root really is invalid, say why you are rejecting it.
• Trigonometric equations: find every angle inside the stated interval, not just the one your calculator returns. Check the quadrants.
• Then reject properly, with a reason. A length cannot be negative. A convergent geometric progression needs a ratio between minus one and one.
• If a diagram shows one intersection and your algebra gives two, one of them is not the answer.

Circular measure runs on radians, and segments are a subtraction

MF19 gives arc length and sector area in their radian forms only. You can work in degrees by taking a fraction of the whole circle, and reports show candidates doing so correctly, but it is the slower route and it is where accuracy quietly disappears. Reports also single out candidates who read an angle off the diagram instead of calculating it.

• Stay in radians for the whole question. Converting to degrees to find a fraction of the circumference costs marks on accuracy, not on method.
• Segment area is sector area minus triangle area. Learn it as one move.
• For the triangle, half a b sin C is usually cleaner and more accurate than half base times height, because you are not stacking two rounded lengths.
• Never assume a triangle is equilateral or right-angled because the diagram looks that way. Calculate the angle.
• Annotate the diagram with what you have worked out. Examiners repeatedly note that candidates who did this had a strategy and the others did not.

Transformations need a name, a scale factor, and a direction

Describing a graph transformation is a mark-scheme exercise in precision, and it is one of the weakest areas in the reports. “Moved”, “shifted” and “squeezed” earn nothing.

• A translation is described with a column vector, or with an explicit direction and distance.
• A stretch needs both the direction and the scale factor, and the scale factor is often the reciprocal of the number in the bracket.
• If the question asks for a single transformation, give one. Describing a stretch and a translation when one transformation was asked for loses the mark.
• Order matters. A translation applied after a horizontal stretch needs a different vector from the same translation applied before it.
• Apply transformations one at a time and write down the intermediate result. Reports note that candidates who worked in two stages were consistently more successful.
• Do not describe the transformation in reverse. Read carefully which graph maps to which.

Rates of change mean differentiate, and integration always brings a constant

Two calculus habits that reports flag every year. Both are about recognising what kind of question you are in before you start writing.

• A question about how fast something is changing needs differentiation and the chain rule. Substituting a value into the volume formula scores nothing.
• Going back from the second derivative to the curve means integrating twice, which means two constants of integration, each found from its own condition.
• The condition on the first integration is usually a gradient value, often zero at a stationary point, not a y-value. Substituting the y-value there is a common and fatal slip.
• Reverse chain rule: divide by the new power and by the coefficient of x. Dropping the second division is the most reported integration error.
• If the question asks for the equation of a curve, the answer starts with “y =”. An expression alone does not get the final mark.

All 9709 thresholds and how they are set