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NZ Scholarship Calculus formula sheet (formulae and tables booklet)

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Formula sheet

The card below opens today's Scholarship Calculus paper with the sheet in the drawer. The contents follow in our own layout. This is the 2025 booklet, the latest NZQA has published. NZQA's 2026 specification says a Calculus Formulae and Tables booklet will be provided, so expect the same layout.

Formula sheet

NCEA Scholarship Calculus

Open formula sheet

Algebra

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}Quadratic formula, for ax² + bx + c = 0
y=\log_bx\Leftrightarrow x=b^yDefinition of a logarithm
\log_b(xy)=\log_bx+\log_byLog of a product
\log_b\left(\frac{x}{y}\right)=\log_bx-\log_byLog of a quotient
\log_b\left(x^n\right)=n\log_bxLog of a power
\log_bx=\frac{\log_ax}{\log_ab}Change of base

Binomial theorem and first principles

Only in the Scholarship booklet

(a+b)^n=\binom{n}{0}a^n+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^2+\ldots+\binom{n}{r}a^{n-r}b^r+\ldots+\binom{n}{n}b^nBinomial theorem
\binom{n}{r}={}^nC_r=\frac{n!}{(n-r)!r!}Binomial coefficient
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}Derivative from first principles

Complex numbers

z=x+iy=r\ \text{cis}\ \theta=r(\cos\theta+i\sin\theta)Forms of z
\overline{z}=x-iy=r\ \text{cis}(-\theta)=r(\cos\theta-i\sin\theta)Forms of the conjugate
r=|z|=\sqrt{z\overline{z}}=\sqrt{x^2+y^2}Modulus
\theta=\arg z,\ \cos\theta=\frac{x}{r},\ \sin\theta=\frac{y}{r}Argument
(r\ \text{cis}\ \theta)^n=r^n\ \text{cis}(n\theta)De Moivre's theorem, for any integer n

Coordinate geometry

y-y_1=m(x-x_1)Straight line through (x₁, y₁)
(x-a)^2+(y-b)^2=r^2Circle, centre (a, b) and radius r
y^2=4axParabola: focus (a, 0), directrix x = −a
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1Ellipse
e=\frac{c}{a}Ellipse: foci (c, 0) and (−c, 0), with b² = a² − c²; eccentricity
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1Hyperbola
y=\pm\frac{b}{a}x,\ e=\frac{c}{a}Hyperbola: asymptotes; foci (c, 0) and (−c, 0), with b² = c² − a²; eccentricity

Differentiation and integration

Both booklets print tables for the standard derivatives and integrals

\frac{d}{dx}\ln x=\frac{1}{x}Derivative of ln x
\frac{d}{dx}e^{ax}=ae^{ax}Derivative of e^(ax)
\frac{d}{dx}\sin x=\cos xDerivative of sin x
\frac{d}{dx}\cos x=-\sin xDerivative of cos x
\frac{d}{dx}\tan x=\sec^2xDerivative of tan x
\frac{d}{dx}\sec x=\sec x\tan xDerivative of sec x
\frac{d}{dx}\text{cosec}\ x=-\text{cosec}\ x\cot xDerivative of cosec x
\frac{d}{dx}\cot x=-\text{cosec}^2xDerivative of cot x
\int x^ndx=\frac{x^{n+1}}{n+1}+c\ \ (n\neq-1)Integral of a power
\int\frac{1}{x}dx=\ln|x|+cIntegral of 1/x
\int\frac{f'(x)}{f(x)}dx=\ln|f(x)|+cIntegral of f′(x) over f(x)
\frac{dy}{dx}=v\frac{du}{dx}+u\frac{dv}{dx}\ \ (y=uv)Product rule
\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\ \ \left(y=\frac{u}{v}\right)Quotient rule
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\ \ (y=f(u),\ u=g(x))Chain rule
\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}Parametric first derivative
\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot\frac{dt}{dx}Parametric second derivative

Numerical methods

In both rules, h = (b − a) ÷ n and y_r = f(x_r)

\int_a^bf(x)dx\approx\frac{1}{2}h\left[y_0+y_n+2(y_1+y_2+\ldots+y_{n-1})\right]Trapezium rule
\int_a^bf(x)dx\approx\frac{1}{3}h\left[y_0+y_n+4(y_1+y_3+\ldots+y_{n-1})+2(y_2+y_4+\ldots+y_{n-2})\right]Simpson's rule, with n even

Trigonometry

\text{cosec}\ \theta=\frac{1}{\sin\theta}Cosecant
\sec\theta=\frac{1}{\cos\theta}Secant
\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}Cotangent
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}Sine rule
c^2=a^2+b^2-2ab\cos CCosine rule
\cos^2\theta+\sin^2\theta=1Pythagorean identity
\tan^2\theta+1=\sec^2\thetaTangent and secant identity
\cot^2\theta+1=\text{cosec}^2\thetaCotangent and cosecant identity
\sin\theta=\sin\alpha\Rightarrow\theta=n\pi+(-1)^n\alphaGeneral solution: sine
\cos\theta=\cos\alpha\Rightarrow\theta=2n\pi\pm\alphaGeneral solution: cosine
\tan\theta=\tan\alpha\Rightarrow\theta=n\pi+\alphaGeneral solution: tangent
\sin(A\pm B)=\sin A\cos B\pm\cos A\sin BCompound angle: sine
\cos(A\pm B)=\cos A\cos B\mp\sin A\sin BCompound angle: cosine
\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}Compound angle: tangent
\sin2A=2\sin A\cos ADouble angle: sine
\tan2A=\frac{2\tan A}{1-\tan^2A}Double angle: tangent
\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2ADouble angle: cosine
2\sin A\cos B=\sin(A+B)+\sin(A-B)Product to sum: sin cos
2\cos A\sin B=\sin(A+B)-\sin(A-B)Product to sum: cos sin
2\cos A\cos B=\cos(A+B)+\cos(A-B)Product to sum: cos cos
2\sin A\sin B=\cos(A-B)-\cos(A+B)Product to sum: sin sin
\sin C+\sin D=2\sin\frac{C+D}{2}\cos\frac{C-D}{2}Sum to product: sin plus sin
\sin C-\sin D=2\cos\frac{C+D}{2}\sin\frac{C-D}{2}Sum to product: sin minus sin
\cos C+\cos D=2\cos\frac{C+D}{2}\cos\frac{C-D}{2}Sum to product: cos plus cos
\cos C-\cos D=-2\sin\frac{C+D}{2}\sin\frac{C-D}{2}Sum to product: cos minus cos

Measurement

A=\frac{1}{2}ab\sin CArea of a triangle
A=\frac{1}{2}(a+b)hArea of a trapezium
A=\frac{1}{2}r^2\thetaArea of a sector
s=r\thetaArc length
V=\pi r^2hVolume of a cylinder
A=2\pi rhCurved surface area of a cylinder
V=\frac{1}{3}\pi r^2hVolume of a cone
A=\pi rlCurved surface area of a cone, slant height l
V=\frac{4}{3}\pi r^3Volume of a sphere
A=4\pi r^2Surface area of a sphere

Binomial coefficients

The booklet prints a table of nCr, with n from 0 to 12 down the side and r from 0 to 10 across. It is Pascal's triangle, so the values in row 8 are 1, 8, 28, 56, 70, 56, 28, 8 and 1.

Exact value triangles

The trigonometry page prints two labelled triangles: a right triangle with angles π/6 and π/3 and sides 1, √3 and 2, and a right isosceles triangle with angles π/4 and sides 1, 1 and √2.

What is not on the sheet

  • The derivative of x to the power n. The differentiation table has no power row. Know it.
  • Most integrals. The table has three rows: powers of x, 1/x and f′(x)/f(x). No integration by substitution or by parts, and no integrals of e^(ax) or the trigonometric functions.
  • Differential equations, series and hyperbolic functions. Nothing in the booklet covers them.
  • Anything beyond calculus at levels 7 and 8. NZQA says questions are on patterns and relationships, equations and expressions, and calculus at levels 7 and 8, and could come from any Level 8 mathematics achievement objective.
  • How to explain. NZQA says part-questions may require written responses, such as comparing models or procedures, and that answers will often be needed in an exact form, such as surds.

NZQA also says an approved graphics or CAS calculator is advised, that answers from a CAS calculator need every command shown, and that correct answers only will not be sufficient.

How to use the sheet in the exam

  1. Read the whole question, because Scholarship questions often join areas, such as complex numbers with trigonometry.
  2. Pick the line on the sheet and write it out before you substitute.
  3. Keep exact values, such as surds and π, until you are asked for a decimal.
  4. Say in words what each step does, as NZQA expects the working shown.

Binomial theorem

Find the coefficient of x³ in the expansion of (2 + x)⁸. The general term has the coefficient C(8, r) with the power of 2 falling as r rises. For x³, r = 3, and the table gives C(8, 3) = 56 in row 8. The power of 2 is 8 − 3 = 5:

\binom{8}{3}\times2^5=56\times32=1792

De Moivre's theorem

Find (1 + i)⁸. First write 1 + i in the form r cis θ: the modulus is the square root of 1² + 1², which is √2, and the argument is π/4. De Moivre's line then gives:

(1+i)^8=\left(\sqrt{2}\ \text{cis}\ \frac{\pi}{4}\right)^8=\left(\sqrt{2}\right)^8\text{cis}\left(8\times\frac{\pi}{4}\right)=16\ \text{cis}\ 2\pi=16

First principles

Differentiate f(x) = x² from first principles using the sheet's limit:

f'(x)=\lim_{h\to0}\frac{(x+h)^2-x^2}{h}=\lim_{h\to0}\frac{2xh+h^2}{h}=\lim_{h\to0}(2x+h)=2x

Check yourself with a quick question:

practice
Pick an answer, then check it.
What is (1 + i) to the power 8?

The Scholarship booklet and the others

The Level 3 Calculus booklet is the same without the binomial theorem, first principles and the coefficient table. The Scholarship Physics sheet is separate, so see the Scholarship Physics formula sheet. For how the Scholarship exam works, see the NZ Scholarship Calculus guide.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the booklet open beside you, mark it, and the worked answers show which line each question wanted. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the sheet with today's paper

Uni-Sine publishes a free daily paper on the NZ Scholarship Calculus track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. Open the sheet from the paper itself and practise deciding which line a question wants.

Common mistakes with the Scholarship Calculus sheet

  • Looking for the power rule. The table does not have it.
  • Using C(n, r) with the wrong r. The table gives C(8, 3) = 56, and C(8, 5) = 56 too, by symmetry. The power of the first term falls as r rises.
  • Quoting a CAS answer. NZQA says every command must be shown.
  • Rounding exact answers. NZQA says answers will often be needed in an exact form.

Common questions

Do you get a formula sheet in NZ Scholarship Calculus?
Yes. NZQA provides a Calculus Formulae and Tables booklet with the exam for 93202.
When is the 2026 Scholarship Calculus exam?
Wednesday 25 November 2026, in the morning session, which starts at 9.30 am.
Is the binomial theorem on the Scholarship Calculus booklet?
Yes, with a table of binomial coefficients up to n = 12. The NCEA Level 3 booklet does not have it.
Is differentiation from first principles on the sheet?
Yes, as a limit. It is in the Scholarship booklet but not the Level 3 one.

Revision cards

Say the answer first, then flip the card to check it.

Where to go next

  • Solve it

    Binomial Expansion Calculator

    Expand brackets to any power with Pascal's triangle

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  • Learn it

    Calculus

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  • Test it

    NCEA Scholarship Calculus

    A new exam-style paper every day, with instant marking and a mark scheme

    Start today's paper
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