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TCE Mathematics Specialised (MTS415118) information sheet: what TASC gives you

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Formula sheet

The card below opens today's paper with the sheet in a drawer. The contents follow in our own layout. TASC publishes the sheet on the TASC Mathematics Specialised page, so check any line against it before you rely on it.

Formula sheet

Maths Extension 2 & Specialist Maths

Open formula sheet

Sequences and series

u_n=a+(n-1)darithmetic sequence, nth term
S_n=\frac{n}{2}\left(2a+(n-1)d\right)=\frac{n}{2}(a+\ell)arithmetic sequence, sum
u_n=ar^{n-1}geometric sequence, nth term
S_n=\frac{a(1-r^n)}{1-r}geometric sequence, sum for r not equal to 1
S_n=nageometric sequence, sum for r = 1
S_\infty=\frac{a}{1-r}sum to infinity, when |r| < 1
\sum_{r=1}^nr=\frac{n(n+1)}{2}sum of the first n integers
\sum_{r=1}^nr^2=\frac{n(n+1)(2n+1)}{6}sum of squares
\sum_{r=1}^nr^3=\frac{n^2(n+1)^2}{4}sum of cubes
f(x)=f(0)+f'(0)\times x+f''(0)\times\frac{x^2}{2!}+f'''(0)\times\frac{x^3}{3!}+\ldots+f^{(n)}(0)\times\frac{x^n}{n!}+\ldotsMaclaurin series

Convergence

A sequence converges to a finite limit L if, for any ε > 0, there is an N(ε) such that |a_n − L| < ε for all n > N. It diverges to positive infinity if, for any κ > 0, there is an N(κ) such that a_n > κ for all n > N, and to negative infinity if a_n < −κ for all n > N.

Matrices and linear transformations

ItemForm
Determinant of a 2 × 2 matrix with entries a, b in the first row and c, d in the seconddet A = |A| = ad − bc
Inverse of that matrix1 ÷ det A times the matrix with first row d, −b and second row −c, a, where ad ≠ bc
Dilation matricesfirst row a, 0 and second row 0, 1; or first row 1, 0 and second row 0, a
Shear matricesfirst row 1, a and second row 0, 1; or first row 1, 0 and second row a, 1
Rotation matrixfirst row cos θ, −sin θ and second row sin θ, cos θ
Reflection matrixfirst row cos 2θ, sin 2θ and second row sin 2θ, −cos 2θ

Conics, planes and lines

(x-h)^2+(y-k)^2=r^2circle, centre (h, k) and radius r
\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1ellipse, centre (h, k), semi-axes a and b
ax+by+cz=dplane
\left(x_0+at,\ y_0+bt,\ z_0+ct\right),\ t\text{ real}line, parametric form
\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}line, symmetric form, with a, b, c not zero

Complex numbers

z=a+ib=r(\cos\theta+i\sin\theta)=r\ \text{cis}\ \theta=re^{i\theta}forms of a complex number
Re(z)=a,\quad Im(z)=b,\quad|z|=\sqrt{a^2+b^2}=rreal and imaginary parts, and modulus
z^n=[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)=r^n\text{cis}\ n\theta=r^ne^{in\theta}de Moivre's theorem

Trigonometry

\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\sec\theta=\frac{1}{\cos\theta},\quad\text{cosec}\ \theta=\frac{1}{\sin\theta},\quad\cot\theta=\frac{1}{\tan\theta}quotient and reciprocal identities
\cos^2\theta+\sin^2\theta=1,\quad1+\tan^2\theta=\sec^2\theta,\quad1+\cot^2\theta=\text{cosec}^2\thetaPythagorean identities
\sin2A=2\sin A\cos Adouble angle, sine
\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2Adouble angle, cosine
\tan2A=\frac{2\tan A}{1-\tan^2A}double angle, 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
2\sin A\cos B=\sin(A+B)+\sin(A-B),\quad2\cos A\sin B=\sin(A+B)-\sin(A-B)product to sum
2\cos A\cos B=\cos(A+B)+\cos(A-B),\quad2\sin A\sin B=\cos(A-B)-\cos(A+B)product to sum, cosines and sines
\sin C+\sin D=2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right),\quad\sin C-\sin D=2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)sum to product, sines
\cos C+\cos D=2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right),\quad\cos C-\cos D=2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{D-C}{2}\right)sum to product, cosines
\text{multiply by }\frac{180}{\pi},\quad\text{multiply by }\frac{\pi}{180}radians to degrees, and back

Trigonometry: exact values

θsin θcos θtan θ
0010
π/61/2√3/2√3/3
π/4√2/2√2/21
π/3√3/21/2√3
π/210undefined
π0−10
3π/2−10undefined
2π010

Calculus: definition of the derivative

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}derivative from first principles

Calculus: differentiation formulae

FunctionDerivative
xⁿnxⁿ⁻¹
sin xcos x
cos x−sin x
tan xsec² x or 1/cos² x
eˣeˣ
logₑ x or ln x1/x
arcsin x1/√(1 − x²)
arccos x−1/√(1 − x²)
arctan x1/(1 + x²)
aˣaˣ ln a
logₐ x1/(x ln a)

Calculus: integration formulae

FunctionIntegral
aax + c
xⁿxⁿ⁺¹/(n + 1) + c
(ax + b)ⁿ(ax + b)ⁿ⁺¹/(a(n + 1)) + c
eˣeˣ + c
1/xln|x| + c
sin x−cos x + c
cos xsin x + c
1/√(a² − x²)arcsin(x/a) + c, or −arccos(x/a) + c
1/(a² + x²)(1/a) arctan(x/a) + c
aˣaˣ/ln a + c
logₐ x(x ln|x| − x)/ln a + c

Calculus: differentiation rules, integration by parts, areas and volumes

\frac{d}{dx}\left(f(x)g(x)\right)=f'(x)g(x)+f(x)g'(x)\quad\text{or}\quad v\frac{du}{dx}+u\frac{dv}{dx}product rule
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f'(x)g(x)-f(x)g'(x)}{\left(g(x)\right)^2}\quad\text{or}\quad\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}quotient rule
\frac{d}{dx}g(f(x))=g'(f(x))f'(x)\quad\text{or}\quad\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}chain rule
\int f(x)g'(x)dx=f(x)g(x)-\int f'(x)g(x)dx+cintegration by parts
A=\int_a^by\ dx\quad\text{or}\quad A=\int_a^bx\ dyarea under a curve
V=\pi\int_a^by^2\ dx\quad\text{or}\quad V=\pi\int_a^bx^2\ dyvolume of revolution

Algebraic formulae

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}quadratic formula
a^x\times a^y=a^{x+y},\quad a^x\div a^y=a^{x-y},\quad\left(a^x\right)^y=a^{x\times y}index laws
(a)^{\frac{1}{y}}=\sqrt[y]{a},\quad(a)^{\frac{x}{y}}=\sqrt[y]{a^x}fractional indices
y=a^x\Leftrightarrow x=\log_ay,\ a>0,\ y>0logarithm definition
\log_ax+\log_ay=\log_axy,\quad\log_ax-\log_ay=\log_a\frac{x}{y},\quad\log_ax^n=n\log_axlog laws
\log_ax=\frac{\log_bx}{\log_ba},\quad\log_a1=0,\quad\log_aa=1change of base and special values
n!=n(n-1)(n-2)\times\ldots\times2\times1,\quad\binom{n}{r}=\frac{n!}{r!(n-r)!}factorial and binomial coefficient
(x+y)^n=\binom{n}{0}x^n+\binom{n}{1}x^{n-1}y+\binom{n}{2}x^{n-2}y^2+\ldots+\binom{n}{n-1}xy^{n-1}+\binom{n}{n}y^nbinomial expansion
|x|=x\ \text{if}\ x\ge0,\quad|x|=-x\ \text{if}\ x<0absolute value

How the exam works with the sheet

TASC's external assessment specifications set a written exam of three hours with an extra 15 minutes of preparation time:

ItemDetail
TimeThree hours, plus 15 minutes of preparation time for notes on the note paper provided
Marks180 in total, at approximately one mark per minute
LayoutSections in a single question booklet, answered in any order
Approved materialsThe current Mathematics Specialised Information Sheet and a TASC-approved calculator
CalculatorsAllowed in all sections unless a question says otherwise

The specifications are dated February 2022, so check your own exam instructions for the current layout. Compare the sheet with the QCE Specialist sheet guide, the WACE Specialist sheet guide and the SACE Specialist sheet guide.

What is not on the TASC Specialised information sheet

  • Statistics and probability. There is no distribution, confidence interval or sampling formula.
  • Mechanics and differential equations. Nothing on motion, simple harmonic motion or solving differential equations.
  • Vectors beyond lines and planes. No dot product, cross product or projection formulas.
  • Numerical methods. No trapezoidal rule, Simpson's rule or Euler's method.
  • Roots of complex numbers and conjugate rules. De Moivre's theorem is given, but not the root formula or the conjugate properties.
  • Proof by induction. No template is given.
  • Hyperbolas. Only the circle and the ellipse are printed.

How to use the sheet in the exam

  1. Use the preparation time to find the page for calculus and the page for series and matrices.
  2. Write the line you use, then substitute and show the working.
  3. For sequences, decide first whether the problem is arithmetic or geometric, then pick the nth term or sum.
  4. Check the integration by parts pattern: the sheet writes f(x)g'(x), so g' is the factor you integrate.

Example 1: a Maclaurin approximation

Use the first four terms of the Maclaurin series of eˣ to estimate e^0.5. Every derivative of eˣ at 0 is 1, so the series line gives:

e^{0.5}\approx1+0.5+\frac{0.5^2}{2!}+\frac{0.5^3}{3!}=1+0.5+0.125+0.0208=1.6458

The exact value is 1.6487, so the approximation is within 0.003.

Example 2: a sum to infinity

A ball drops 5 m and rebounds to 60% of its previous height each time. The heights form a geometric sequence with a = 5 and r = 0.6, and since |r| < 1 the sum to infinity line applies:

S_\infty=\frac{a}{1-r}=\frac{5}{1-0.6}=12.5

That is the total of the falling heights, in metres. Practise sequences with our Sequences and Series Calculator.

Example 3: integration by parts

Find ∫ x cos x dx. The sheet writes the rule as ∫ f(x)g'(x) dx, so take f(x) = x and g'(x) = cos x, which makes g(x) = sin x:

\int x\cos x\ dx=x\sin x-\int\sin x\ dx=x\sin x+\cos x+c

The integral of sin x comes from the integral table.

Practise with the sheet open

Our Maths Extension 2 and Specialist Maths daily paper is a short timed paper of 8 questions worth 25 to 35 marks. It is not a TASC paper. The Specialist Maths mechanics paper covers applications that the sheet does not. Learn calculus patterns in the Calculus course.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. They show which line of the sheet each question wanted. We do not publish official answers, past paper answers or leaked papers.

Common questions

Do you get a formula sheet in TCE Mathematics Specialised?
Yes. TASC lists the Mathematics Specialised Information Sheet as approved material in the external exam.
Can I use a calculator in the TCE Specialised exam?
Yes. TASC's specifications allow a TASC-approved calculator in all sections unless a question says otherwise.
How long is the exam?
Three hours, plus 15 minutes of preparation time.
Is the Maclaurin series on the sheet?
Yes, in the sequences and series section.
Is the quadratic formula on the sheet?
Yes, with the index and log laws on the last page.

Revision cards

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

Where to go next

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Australia · TCETCE Maths Specialised: Level 4 course, TASC exam, awardsTCE Maths Specialised (MTS415118) explained: five externally assessed criteria, the TASC 3-hour exam with five booklets, awards EA to PA and the 2026 date.