Cambridge A Level Mathematics 9709 · Pure 1

9709 formula sheet (MF19)

Every 9709 exam comes with MF19, Cambridge’s list of formulae and statistical tables — but it does not contain everything Paper 1 needs. The Pure Mathematics section gives the quadratic formula, the arithmetic and geometric series results, the binomial expansion, arc length and sector area, and the standard trigonometric identities. It does not give the equation of a circle, the chain rule, the discriminant conditions or the volume of revolution — those you must know.

Below: the 51 Pure Mathematics formulae MF19 gives you, the ones Paper 1 actually uses, and the list it leaves out.

Updated 28 September 2026

Not on MF19: the Paper 1 formulas you must memorise

These all come up on Paper 1 and none of them is printed in the booklet. The first three are behind most of the coordinate geometry marks; the last three behind most of the integration marks.

Equation of a circle	$(x - a)^{2} + (y - b)^{2} = r^{2}$	centre (a, b), radius r
Straight line through a point	$y - y_{1} = m (x - x_{1})$	gradient m through (x₁, y₁)
Gradient	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
Perpendicular lines	$m_{1} m_{2} = - 1$	parallel lines have equal gradients
Midpoint	$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
Distance between points	$\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
Discriminant	$b^{2} - 4 a c$	> 0 two roots · = 0 one (tangent) · < 0 none
Radians and degrees	$π rad = 180^{\circ}$
Chain rule	$\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$
Connected rates of change	$\frac{d y}{d t} = \frac{d y}{d x} \times \frac{d x}{d t}$
Stationary point	$\frac{d y}{d x} = 0$	d²y/dx² > 0 minimum, < 0 maximum
Area under a curve	$\int_{a}^{b} y d x$
Volume of revolution about the x-axis	$π \int_{a}^{b} y^{2} d x$
Volume of revolution about the y-axis	$π \int_{c}^{d} x^{2} d y$

Also worth having cold, though they are definitions rather than formulas: the exact values of sin, cos and tan at 30°, 45° and 60°, and the completed-square form a(x + p)² + q, whose vertex is (−p, q).

What MF19 gives you: the Pure Mathematics section

The full Pure Mathematics list as it appears in the booklet. Rows marked P1 are the ones Paper 1 questions lean on most. Most of the rest are for Papers 2 and 3, though the sphere and cone formulas can turn up in a Paper 1 rates-of-change question.

Mensuration

Volume of sphere	$V = \frac{4}{3} π r^{3}$
Surface area of sphere	$A = 4 π r^{2}$
Volume of cone or pyramid	$V = \frac{1}{3} \times base area \times height$
Curved surface of cone	$A = π r \times slant height$
Arc length of circleP1	$s = r θ$	θ in radians
Area of sector of circleP1	$A = \frac{1}{2} r^{2} θ$	θ in radians

Algebra

Quadratic formulaP1	$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$	for ax² + bx + c = 0
Arithmetic series — nth termP1	$u_{n} = a + (n - 1) d$
Arithmetic series — sumP1	$S_{n} = \frac{1}{2} n (a + l) = \frac{1}{2} n {2 a + (n - 1) d}$
Geometric series — nth termP1	$u_{n} = a r^{n - 1}$
Geometric series — sumP1	$S_{n} = \frac{a (1 - r^{n})}{1 - r}$	r ≠ 1
Geometric series — sum to infinityP1	$S_{\infty} = \frac{a}{1 - r}$	\|r\| < 1
Binomial expansionP1	$(a + b)^{n} = a^{n} + (\binom{n}{1}) a^{n - 1} b + (\binom{n}{2}) a^{n - 2} b^{2} + (\binom{n}{3}) a^{n - 3} b^{3} + \dots + b^{n}$	n a positive integer
Binomial coefficientP1	$(\binom{n}{r}) = \frac{n!}{r! (n - r)!}$
Binomial series	$(1 + x)^{n} = 1 + n x + \frac{n (n - 1)}{2!} x^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{3} + \dots$	n rational, \|x\| < 1

Trigonometry

Tangent identityP1	$\tan θ \equiv \frac{\sin θ}{\cos θ}$
Pythagorean identityP1	$\cos^{2} θ + \sin^{2} θ \equiv 1$
Secant identity	$1 + \tan^{2} θ \equiv \sec^{2} θ$
Cosecant identity	$\cot^{2} θ + 1 \equiv {cosec}^{2} θ$
Sine of a sum	$\sin (A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$
Cosine of a sum	$\cos (A \pm B) \equiv \cos A \cos B \mp \sin A \sin B$
Tangent of a sum	$\tan (A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Double angle — sine	$\sin 2 A \equiv 2 \sin A \cos A$
Double angle — cosine	$\cos 2 A \equiv \cos^{2} A - \sin^{2} A \equiv 2 \cos^{2} A - 1 \equiv 1 - 2 \sin^{2} A$
Double angle — tangent	$\tan 2 A \equiv \frac{2 \tan A}{1 - \tan^{2} A}$
Principal values	$- \frac{1}{2} π ⩽ \sin^{- 1} x ⩽ \frac{1}{2} π, 0 ⩽ \cos^{- 1} x ⩽ π, - \frac{1}{2} π < \tan^{- 1} x < \frac{1}{2} π$

Differentiation

Power	$\frac{d}{d x} x^{n} = n x^{n - 1}$
Natural log	$\frac{d}{d x} \ln x = \frac{1}{x}$
Exponential	$\frac{d}{d x} e^{x} = e^{x}$
Sine	$\frac{d}{d x} \sin x = \cos x$
Cosine	$\frac{d}{d x} \cos x = - \sin x$
Tangent	$\frac{d}{d x} \tan x = \sec^{2} x$
Secant	$\frac{d}{d x} \sec x = \sec x \tan x$
Cosecant	$\frac{d}{d x} cosec x = - cosec x \cot x$
Cotangent	$\frac{d}{d x} \cot x = - {cosec}^{2} x$
Inverse tangent	$\frac{d}{d x} \tan^{- 1} x = \frac{1}{1 + x^{2}}$
Product rule	$\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$
Quotient rule	$\frac{d}{d x} ⁣ (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$
Parametric	$\frac{d y}{d x} = \frac{d y}{d t} \div \frac{d x}{d t}$	for x = f(t), y = g(t)

Integration

Power	$\int x^{n} d x = \frac{x^{n + 1}}{n + 1}$	n ≠ −1; constants omitted throughout
Reciprocal	$\int \frac{1}{x} d x = \ln ∣ x ∣$
Exponential	$\int e^{x} d x = e^{x}$
Sine	$\int \sin x d x = - \cos x$
Cosine	$\int \cos x d x = \sin x$
Secant squared	$\int \sec^{2} x d x = \tan x$
Inverse tangent form	$\int \frac{1}{x^{2} + a^{2}} d x = \frac{1}{a} \tan^{- 1} ⁣ (\frac{x}{a})$
Partial fraction form	$\int \frac{1}{x^{2} - a^{2}} d x = \frac{1}{2 a} \ln ∣ \frac{x - a}{x + a} ∣$	x > a
Partial fraction form	$\int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} \ln ∣ \frac{a + x}{a - x} ∣$	x < a
Integration by parts	$\int u \frac{d v}{d x} d x = u v - \int v \frac{d u}{d x} d x$
Log form	$\int \frac{f^{'} (x)}{f (x)} d x = \ln ∣ f (x) ∣$

Vectors

Scalar (dot) product

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = ∣ a ∣ ∣ b ∣ \cos θ

for a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k

MF19 is shared with Further Mathematics (9231), so the booklet also carries Further Pure, Mechanics and Statistics sections, plus the normal distribution and other tables. Ignore them for Paper 1.

Using the booklet in the exam

Series: MF19 gives the geometric sum to infinity but not its condition in words — it only converges when |r| < 1, and a question that asks you to justify using it wants that stated.
Circular measure: the arc length and sector formulas are only valid with θ in radians. Using degrees in s = rθ is the most common way to lose every mark on a circular measure question.
Binomial: Paper 1 uses the expansion for a positive integer n only. The version with rational n and |x| < 1 is Paper 3. Binomial expansion guide →
Differentiation and integration tables: Paper 1 needs only the power rule from each. The rest (ln x, eˣ, trig, product and quotient rules) are Paper 3.

Common questions

Do you get a formula sheet in 9709 Paper 1?

Yes — MF19, Cambridge’s list of formulae and statistical tables, is supplied in every 9709 exam. It covers the quadratic formula, series, binomial expansion, arc length, sector area and trigonometric identities, but not the circle equation, chain rule or volume of revolution.

Is the chain rule on the 9709 formula sheet?

No. MF19 lists the derivatives of standard functions and the product, quotient and parametric rules, but not the chain rule. Paper 1 uses it constantly, so it has to be memorised.

Is the volume of revolution formula given?

No. V = π∫y² dx (about the x-axis) is not in MF19, yet area and volume of revolution came up on 33 of the 42 Paper 1s Quanta holds.

What is MF19?

The code for Cambridge’s combined formula list for AS & A Level Mathematics (9709) and Further Mathematics (9231). The same booklet is used for every 9709 paper, and a copy is printed in the syllabus.

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