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ACT Math 2

Provider · ACT Inc. Format · Subject Test · Advanced Chapters · 8 Level · Pre-calculus & beyond
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Test format & strategy

🗻 ACT Math Advanced — Score Range 28–36 Topics
Reporting Category % of Test Key Topics
Advanced Number Theory ~8% Sequences, integer properties, modular arithmetic, complex numbers
Advanced Algebra ~15% Polynomial systems, rational equations, logarithms, exponentials
Advanced Functions ~15% Piecewise, recursive, parametric, function composition chains
Advanced Trigonometry ~10% Radians, unit circle, identities, inverse trig, law of sines/cosines
Pre-Calculus Concepts ~12% Limits (informal), sequences/series, conic sections, polar basics
Advanced Coordinate Geometry ~10% Conic equations, transformations, distance in 3D
Advanced Plane Geometry ~10% Circle theorems, similarity, 3D solids, geometric proof
Statistics, Counting & Probability ~10% Permutations, combinations, expected value, normal dist., regression

ACT Math 2 is an advanced subject-level mathematics course for students who have completed standard ACT Math preparation and wish to extend their skills into pre-calculus territory — the topics that appear in the hardest ACT questions and on ACT subject tests.

What to expect

This course covers the top-difficulty ACT Math topics: advanced algebra, pre-calculus functions, matrices, advanced trigonometry, and non-routine geometry. It is also ideal preparation for the ACT Subject Test in Mathematics, which tests material through pre-calculus.

Prerequisite
ACT Math (standard course)
Level
Pre-calculus and beyond
Ideal for
Score 28+ target students
Calculator
permitted throughout
📝 Chapter Quiz
1

Advanced Number Theory

🗻 Advanced Number Theory ~8%  ·  Integer properties  ·  Sequences & series  ·  Modular arithmetic  ·  Complex number operations

Number theory questions on hard ACT items test divisibility rules, prime factorisation, GCD and LCM, modular arithmetic, and properties of integers.

GCD
greatest common divisor
LCM
\( \text{LCM}(a,b)=\dfrac{ab}{\gcd(a,b)} \)
Modular arithmetic
\( a \equiv r \pmod{n} \)
Prime factorisation
unique decomposition theorem
Worked example 1.A

Find LCM(12, 18). \[ 12=2^2\cdot 3,\quad 18=2\cdot 3^2.\quad \text{LCM}=2^2\cdot 3^2=36. \]

Complex numbers revisited

On advanced ACT items, complex numbers appear in equation-solving contexts: \( x^2+4=0 \Rightarrow x=\pm 2i \). The modulus of \( a+bi \) is \( \sqrt{a^2+b^2} \).

● Sequences & Modular Arithmetic

An arithmetic sequence: \(a_n = a_1 + (n-1)d\). A geometric sequence: \(a_n = a_1 r^{n-1}\). Modular arithmetic: \(a \equiv b \pmod{m}\) means \(m \mid (a-b)\). Useful for finding last digits (mod 10) or remainders.

Worked example — Last digit of a power

What is the units digit of \(7^{2026}\)?

Powers of 7 cycle: \(7^1=7, 7^2=49, 7^3=343, 7^4=2401, 7^5=\ldots7\). Cycle length 4. \(2026 \div 4\) remainder \(2\). So units digit = same as \(7^2\) = 9.

The sum of an arithmetic sequence is 150, with first term 5 and last term 25. How many terms are there?
\(S = \dfrac{n}{2}(a_1+a_n) = \dfrac{n}{2}(30) = 150 \Rightarrow n = 10\).
📝 Chapter Quiz
2

Advanced Algebra & Systems

🗻 Advanced Algebra ~15%  ·  Polynomial systems  ·  Rational equations  ·  Logarithms & exponentials  ·  Absolute value in systems

Advanced ACT algebra includes matrices, non-linear systems, and applications of polynomial and rational functions.

Matrix addition
add corresponding entries
Matrix product \( AB \)
row × column dot products
2×2 determinant
\( ad-bc \)
System via matrices
\( A\mathbf{x}=\mathbf{b} \)
Worked example 2.A

Solve the non-linear system: \( y=x^2 \) and \( y=x+2 \). \[ x^2=x+2 \implies x^2-x-2=0 \implies (x-2)(x+1)=0 \implies x=2 \text{ or } x=-1. \]

● Logarithms & Exponential Equations

Key log laws: \(\log_b(xy) = \log_b x + \log_b y\); \(\log_b(x/y) = \log_b x - \log_b y\); \(\log_b(x^n) = n\log_b x\). To solve \(a^{f(x)} = b\): take \(\log\) of both sides. Change of base: \(\log_b x = \dfrac{\ln x}{\ln b}\).

Worked example — Solving a logarithmic equation

Solve \(\log_2(x+3) + \log_2(x-1) = 5\).

\(\log_2[(x+3)(x-1)] = 5 \Rightarrow (x+3)(x-1) = 32 \Rightarrow x^2+2x-35 = 0 \Rightarrow (x+7)(x-5) = 0\).

\(x = 5\) (reject \(x=-7\) since \(x-1 > 0\) required). Answer: \(x=5\).

Solve for \(x\): \(3^{x+1} = 5^x\). Give your answer in terms of logarithms.
Take \(\ln\): \((x+1)\ln 3 = x\ln 5 \Rightarrow x\ln 3 + \ln 3 = x\ln 5 \Rightarrow x = \dfrac{\ln 3}{\ln 5 - \ln 3} = \dfrac{\ln 3}{\ln(5/3)}\).
📝 Chapter Quiz
3

Advanced Functions

🗻 Advanced Functions ~15%  ·  Piecewise & recursive  ·  Parametric  ·  Composition chains  ·  Inverse functions

Pre-calculus functions — rational, piecewise, exponential, and logarithmic — with composition, inverses, and detailed graph analysis.

Rational function
\( \dfrac{P(x)}{Q(x)} \) — vertical asymptotes where \(Q=0\)
Horizontal asymptote
compare degrees of \( P \) and \( Q \)
Piecewise
different rules on different intervals
Even / odd
\( f(-x)=f(x) \) / \( f(-x)=-f(x) \)
Worked example 3.A

Find the horizontal asymptote of \( f(x)=\dfrac{3x^2+1}{x^2-4} \).

Both numerator and denominator have degree 2; ratio of leading coefficients: \( y=\dfrac{3}{1}=3 \).

⊙ Explorer · Rational function asymptotesinteractive
vertical asymptotex = 1
horizontal asymptotey = 0

\(f(x)=\dfrac{2}{x-p}+q\) — the denominator vanishing at \(x=p\) makes the vertical asymptote; as \(x\to\pm\infty\), \(f(x)\to q\).

● Piecewise & Recursive Functions

A piecewise function uses different rules for different intervals. A recursive function is defined by its previous terms: e.g. \(a_1 = 2\), \(a_n = 3a_{n-1} - 1\). Parametric equations express \(x\) and \(y\) as separate functions of a parameter \(t\).

Worked example — Parametric to Cartesian

Eliminate \(t\) from \(x = t+2\), \(y = t^2 - 1\).

From \(x = t+2\): \(t = x-2\). Substitute: \(y = (x-2)^2 - 1 = x^2 - 4x + 3\). Parabola.

A function is defined piecewise: \(f(x) = x^2\) for \(x < 0\), and \(f(x) = 2x+1\) for \(x \geq 0\). Find \(f(-3) + f(2)\).
\(f(-3) = (-3)^2 = 9\). \(f(2) = 2(2)+1 = 5\). Sum \(= 14\).
📝 Chapter Quiz
4

Advanced Trigonometry

🗻 Advanced Trigonometry ~10%  ·  Radians & unit circle  ·  Trig identities  ·  Law of sines/cosines  ·  Inverse trig

Beyond SOHCAHTOA — the unit circle, inverse trig functions, sum/difference identities, and solving trig equations over a given interval.

Sum formula sin
\( \sin(A+B)=\sin A\cos B+\cos A\sin B \)
Sum formula cos
\( \cos(A+B)=\cos A\cos B-\sin A\sin B \)
Law of sines
\( \dfrac{a}{\sin A}=\dfrac{b}{\sin B} \)
Inverse trig
\( \sin^{-1},\cos^{-1},\tan^{-1} \)
Worked example 4.A

Find all solutions to \( 2\sin\theta-1=0 \) on \( [0,2\pi) \). \[ \sin\theta=\tfrac{1}{2} \implies \theta=\tfrac{\pi}{6} \text{ or } \theta=\tfrac{5\pi}{6}. \]

⊙ Explorer · Law of Cosinesinteractive
c² = a² + b² − 2ab cos C8.89
● Law of Sines, Cosines & Identities

Law of Sines: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\). Law of Cosines: \(c^2 = a^2+b^2-2ab\cos C\). Key identities: \(\sin^2\theta+\cos^2\theta=1\); \(\sin 2\theta = 2\sin\theta\cos\theta\); \(\cos 2\theta = \cos^2\theta - \sin^2\theta\).

Worked example — Law of Cosines

A triangle has sides \(a=7, b=10\) and included angle \(C=60°\). Find side \(c\).

\(c^2 = 49 + 100 - 2(7)(10)\cos 60° = 149 - 70 = 79\). \(c = \sqrt{79} \approx 8.89\).

Prove that \(\dfrac{1-\cos 2\theta}{\sin 2\theta} = \tan\theta\).
\(\dfrac{1-(1-2\sin^2\theta)}{2\sin\theta\cos\theta} = \dfrac{2\sin^2\theta}{2\sin\theta\cos\theta} = \dfrac{\sin\theta}{\cos\theta} = \tan\theta\). ✓
📝 Chapter Quiz
5

Pre-Calculus Concepts

🗻 Pre-Calculus Concepts ~12%  ·  Limits (informal)  ·  Sequences & series  ·  Conic sections  ·  Polar coordinates

Limits, sequences, series, and polynomial behaviour give students the conceptual scaffolding for calculus — and appear on the hardest ACT questions.

Limit intuition
\( \lim_{x\to a}f(x) \) — value \( f \) approaches
Geometric series
\( \dfrac{a_1}{1-r},\ |r|<1 \)
Binomial theorem
\( (a+b)^n=\sum\binom{n}{k}a^{n-k}b^k \)
Parametric
\( x=f(t),\ y=g(t) \)
Worked example 5.A

Evaluate \( \displaystyle\sum_{n=1}^{\infty}3\left(\tfrac{1}{2}\right)^{n-1} \). \[ S=\frac{3}{1-\tfrac{1}{2}}=\frac{3}{\tfrac{1}{2}}=6. \]

● Limits & Conic Sections

Informal limit: \(\lim_{x \to a}f(x) = L\) means \(f(x)\) approaches \(L\) as \(x\) approaches \(a\). Conic sections — standard forms: circle \((x-h)^2+(y-k)^2=r^2\); ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\); hyperbola \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\); parabola \(y=ax^2+bx+c\).

Worked example — Identify and graph a conic

Identify the conic: \(4x^2 + 9y^2 = 36\).

Divide by 36: \(\dfrac{x^2}{9}+\dfrac{y^2}{4}=1\). This is an ellipse with \(a=3\) along \(x\)-axis and \(b=2\) along \(y\)-axis, centre \((0,0)\).

Find \(\lim_{x \to 3}\dfrac{x^2-9}{x-3}\).
Factor: \(\dfrac{(x-3)(x+3)}{x-3} = x+3\). As \(x \to 3\): limit \(= 6\).
📝 Chapter Quiz
6

Advanced Coordinate Geometry

🗻 Advanced Coordinate Geometry ~10%  ·  Conic equations  ·  Rotations & transformations  ·  Distance in 3D  ·  Locus of points

Conics in standard and general form, transformations, and polar coordinates are tested on advanced ACT and ACT subject items.

Parabola (vertical)
\( (x-h)^2=4p(y-k) \)
Ellipse
\( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \)
Hyperbola
\( \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 \)
Polar
\( x=r\cos\theta,\ y=r\sin\theta \)
Worked example 6.A

Convert the polar point \( \left(4,\dfrac{\pi}{3}\right) \) to Cartesian. \[ x=4\cos\tfrac{\pi}{3}=4\cdot\tfrac{1}{2}=2,\quad y=4\sin\tfrac{\pi}{3}=4\cdot\tfrac{\sqrt3}{2}=2\sqrt3. \]

● Conics in Standard Form & 3D Distance

Completing the square converts a general conic to standard form. 3D distance: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\). A rotation of axes by \(\theta\) transforms \(x = X\cos\theta - Y\sin\theta\), \(y = X\sin\theta + Y\cos\theta\).

Worked example — Complete the square to find a circle

Write \(x^2+y^2-6x+4y-3=0\) in standard form.

\((x^2-6x+9)+(y^2+4y+4) = 3+9+4 = 16\). Centre \((3,-2)\), radius \(4\).

Find the distance between points \(A(1,2,3)\) and \(B(4,6,3)\).
\(d = \sqrt{(4-1)^2+(6-2)^2+(3-3)^2} = \sqrt{9+16+0} = \sqrt{25} = 5\).
📝 Chapter Quiz
7

Advanced Plane Geometry

🗻 Advanced Plane Geometry ~10%  ·  Circle theorems  ·  Similarity & congruence proofs  ·  3D solids  ·  Geometric inequalities

Non-routine geometry questions involve multiple theorems applied together — circle inscribed in a triangle, overlapping regions, and proof-based reasoning.

Inscribed angle
\( \tfrac{1}{2}\times \) central angle
Power of a point
\( PA\cdot PB=PC\cdot PD \)
Heron's formula
\( \sqrt{s(s-a)(s-b)(s-c)} \)
Regular polygon area
\( \tfrac{1}{4}ns^2\cot(\pi/n) \)
Worked example 7.A

Find the area of a triangle with sides 7, 8, 9. \[ s=12,\quad A=\sqrt{12\cdot5\cdot4\cdot3}=\sqrt{720}=12\sqrt5. \]

⊙ Explorer · Inscribed angle theoreminteractive
central angle ∠AOB140.0°
inscribed angle ∠ACB70.0°

C sits anywhere on the major arc — drag the slider and the inscribed angle stays exactly half the central angle, every time.

● Circle Theorems & 3D Solids

Key circle theorems: (1) Angle at centre = 2 × angle at circumference. (2) Angles in same segment are equal. (3) Opposite angles in cyclic quadrilateral sum to 180°. (4) Tangent ⊥ radius. Surface area of sphere: \(4\pi r^2\); volume: \(\dfrac{4}{3}\pi r^3\). Cone volume: \(\dfrac{1}{3}\pi r^2 h\).

Worked example — Circle theorem

O is the centre of a circle. Angle \(AOB = 140°\). Find the angle \(ACB\) where \(C\) is on the major arc.

Angle at circumference on minor arc \(= 140°/2 = 70°\). \(C\) is on major arc, so \(\angle ACB = 180° - 70° = 110°\) (opposite angles in cyclic quad sum to \(180°\)).

A cone has base radius 6 cm and slant height 10 cm. Find its total surface area.
Height: \(h = \sqrt{100-36} = 8\) cm. TSA \(= \pi r^2 + \pi r l = 36\pi + 60\pi = 96\pi \approx 301.6\) cm\(^2\).
📝 Chapter Quiz
8

Statistics, Counting & Probability

🗻 Statistics, Counting & Probability ~10%  ·  Permutations & combinations  ·  Expected value  ·  Normal distribution  ·  Regression & residuals

Advanced ACT Math includes combinations, permutations, compound probability, and normal distribution concepts.

Permutations
\( _nP_r=\dfrac{n!}{(n-r)!} \)
Combinations
\( \binom{n}{r}=\dfrac{n!}{r!(n-r)!} \)
Bayes (simple)
\( P(A|B)=\dfrac{P(B|A)P(A)}{P(B)} \)
Expected value
\( E=\sum x_i P(x_i) \)
Worked example 8.A

How many 3-letter arrangements (order matters) can be made from the word MATH? \[ _4P_3=\frac{4!}{1!}=24. \]

● Counting, Expected Value & Normal Distribution

Fundamental counting principle: \(m \times n\) ways for two independent events. Expected value: \(E(X) = \sum x_i P(x_i)\). Standard normal: \(z = \dfrac{x-\mu}{\sigma}\); about 68% of data within \(1\sigma\), 95% within \(2\sigma\), 99.7% within \(3\sigma\).

Worked example — Expected value

A game pays $8 if you roll a 6 on a fair die, and you lose $2 otherwise. What is the expected gain per roll?

\(E = 8 \cdot \dfrac{1}{6} + (-2) \cdot \dfrac{5}{6} = \dfrac{8-10}{6} = -\dfrac{2}{6} \approx -\$0.33\).

The game is unfair — you lose on average 33 cents per roll.

In a class of 30, 18 play football (F), 14 play basketball (B), and 6 play both. How many play neither sport?
\(|F \cup B| = 18+14-6 = 26\). Neither \(= 30-26 = 4\).
📝 Chapter Quiz
§

Practice set

Fifty advanced ACT Math problems — pre-algebra and number theory, advanced algebra, functions, trigonometry, coordinate geometry, plane geometry, and statistics — all graded by difficulty.

1

Simplify \( \dfrac{4!}{2!\cdot2!} \).

Basic
\[ \dfrac{24}{2\times2}=6. \]
2

Find \( \gcd(48,\,36) \).

Basic
\[ \gcd(48,36)=12. \]
3

Express \( 0.\overline{27} \) as a fraction.

Intermediate

Let \( x=0.\overline{27} \). Then \( 100x=27.\overline{27} \), so \( 99x=27 \).

\[ x=\dfrac{27}{99}=\dfrac{3}{11}. \]
4

A \( 12\% \) annual interest rate is applied to 5000. Find the amount after 2 years with compound interest.

Intermediate
\[ A=5000(1.12)^2=5000\times1.2544=6272. \]
5

Find the LCM of 12 and 18.

Basic
\[ \text{LCM}(12,18)=36. \]
6

Solve \( 3x^2 - 5x - 2 = 0 \).

Intermediate
\[ (3x+1)(x-2)=0 \Rightarrow x=2\text{ or }x=-\tfrac{1}{3}. \]
7

For what values of \( k \) does \( x^2-kx+9=0 \) have equal roots?

Advanced

Discriminant \( =k^2-36=0 \).

\[ k=\pm6. \]
8

Simplify \( \dfrac{x^2-x-6}{x^2-4} \).

Intermediate
\[ \dfrac{(x-3)(x+2)}{(x-2)(x+2)}=\dfrac{x-3}{x-2}. \]
9

Solve \( \sqrt{x+6}-\sqrt{x}=1 \).

Advanced

Square both sides after isolating: \( x+6=x+2\sqrt{x}+1 \Rightarrow 5=2\sqrt{x} \Rightarrow x=\tfrac{25}{4} \).

10

Solve the system: \( x^2-y=3 \) and \( x+y=1 \).

Advanced

\( y=1-x \). Then \( x^2-1+x=3\Rightarrow x^2+x-4=0 \Rightarrow x=\dfrac{-1\pm\sqrt{17}}{2} \).

11

If \( f(x)=|2x-3| \), find \( f(-1) \).

Basic
\[ |2(-1)-3|=|-5|=5. \]
12

Find the vertical asymptote of \( f(x)=\dfrac{3x}{x^2-4} \).

Intermediate
\[ x=2\text{ and }x=-2. \]
13

What is the horizontal asymptote of \( f(x)=\dfrac{3x^2}{x^2+1} \)?

Intermediate

Degrees equal; ratio of leading coefficients.

\[ y=3. \]
14

Is \( f(x)=x^4-2x^2 \) even, odd, or neither?

Intermediate

\( f(-x)=x^4-2x^2=f(x) \).

\[ \text{Even.} \]
15

Find the inverse of \( f(x)=\dfrac{2x-1}{x+3} \).

Advanced

Swap \( x,y \) and solve: \( x(y+3)=2y-1 \Rightarrow y(x-2)=-1-3x \).

\[ f^{-1}(x)=\dfrac{-3x-1}{x-2}. \]
16

Find the exact value of \( \sin 135^\circ \).

Intermediate
\[ \sin(180^\circ-45^\circ)=\sin45^\circ=\dfrac{\sqrt{2}}{2}. \]
17

Solve \( 2\sin\theta-1=0 \) for \( 0^\circ\le\theta<360^\circ \).

Intermediate
\[ \sin\theta=\tfrac{1}{2} \Rightarrow \theta=30^\circ\text{ or }150^\circ. \]
18

Find the period of \( y=3\sin(2x) \).

Intermediate
\[ T=\dfrac{2\pi}{2}=\pi. \]
19

Simplify \( \dfrac{\tan\theta}{\sin\theta} \).

Advanced
\[ \dfrac{\sin\theta/\cos\theta}{\sin\theta}=\dfrac{1}{\cos\theta}=\sec\theta. \]
20

In triangle \( ABC \), \( a=8 \), \( A=30^\circ \), \( B=45^\circ \). Find \( b \).

Advanced
\[ \dfrac{b}{\sin45^\circ}=\dfrac{8}{\sin30^\circ} \Rightarrow b=8\sqrt{2}. \]
21

Find the amplitude and period of \( y=-4\cos\!\left(\dfrac{\pi x}{3}\right) \).

Intermediate
\[ \text{Amplitude}=4;\quad T=\dfrac{2\pi}{\pi/3}=6. \]
22

Convert \( 210^\circ \) to radians.

Intermediate
\[ 210\times\dfrac{\pi}{180}=\dfrac{7\pi}{6}. \]
23

Find the distance from \( (1,\,-2) \) to \( (5,\,1) \).

Basic
\[ \sqrt{16+9}=5. \]
24

Find the midpoint of the segment from \( (-3,\,4) \) to \( (7,\,-2) \).

Basic
\[ \left(\dfrac{-3+7}{2},\,\dfrac{4-2}{2}\right)=(2,\,1). \]
25

Find the equation of the ellipse with semi-axes \( a=5 \) and \( b=3 \).

Intermediate
\[ \dfrac{x^2}{25}+\dfrac{y^2}{9}=1. \]
26

Find the focus of the parabola \( y^2=12x \).

Advanced

\( 4p=12 \Rightarrow p=3 \).

\[ \text{Focus: }(3,\,0). \]
27

Find the asymptotes of \( \dfrac{x^2}{4}-\dfrac{y^2}{9}=1 \).

Advanced
\[ y=\pm\dfrac{3}{2}x. \]
28

A line passes through \( (2,3) \) and is parallel to \( 2x-y=5 \). Find its equation.

Intermediate

Slope \( =2 \); point-slope: \( y-3=2(x-2) \).

\[ y=2x-1. \]
29

Find the area enclosed by \( y=x^2 \) and \( y=x+2 \). (Use integration concept only.)

Advanced

Intersect at \( x=-1 \) and \( x=2 \).

\[ A=\int_{-1}^{2}(x+2-x^2)\,dx=\left[\tfrac{x^2}{2}+2x-\tfrac{x^3}{3}\right]_{-1}^{2}=\dfrac{9}{2}. \]
30

The vertices of a triangle are \( (0,0) \), \( (4,0) \), \( (0,3) \). Find the area.

Basic
\[ A=\tfrac{1}{2}\times4\times3=6. \]
31

Find the exterior angle of a regular hexagon.

Intermediate
\[ \dfrac{360^\circ}{6}=60^\circ. \]
32

Two parallel lines are cut by a transversal. Co-interior angles sum to what?

Basic
\[ 180^\circ. \]
33

Find the area of a regular hexagon with side 4.

Advanced
\[ A=\dfrac{3\sqrt{3}}{2}s^2=\dfrac{3\sqrt{3}}{2}\times16=24\sqrt{3}. \]
34

Two similar triangles have sides in ratio \( 3:5 \). Find the ratio of their areas.

Intermediate
\[ \left(\dfrac{3}{5}\right)^2=\dfrac{9}{25}. \]
35

A tangent from an external point touches the circle at \( T \). The distance from the external point to the centre is 13 and radius is 5. Find the tangent length.

Intermediate
\[ PT=\sqrt{13^2-5^2}=\sqrt{144}=12. \]
36

Find the volume of a cone with base radius 6 and height 8.

Intermediate
\[ V=\tfrac{1}{3}\pi r^2h=\tfrac{1}{3}\pi(36)(8)=96\pi. \]
37

Find the sum of interior angles of a polygon with 9 sides.

Intermediate
\[ (9-2)\times180^\circ=1260^\circ. \]
38

The mean of 10 numbers is 15. One number is removed and the new mean is 14. Find the removed number.

Intermediate

Total was 150; new total \( =9\times14=126 \).

\[ 150-126=24. \]
39

A data set has standard deviation 3 and mean 20. All values increase by 5. Find the new mean and new standard deviation.

Advanced
\[ \text{New mean}=25;\quad\text{Std dev unchanged}=3. \]
40

A box has 3 red, 4 white, 5 blue cards. Find \( P(\text{not white}) \).

Intermediate
\[ P(\text{not white})=\dfrac{8}{12}=\dfrac{2}{3}. \]
41

Find \( \lim_{x\to\infty}\dfrac{3x^2+1}{x^2-2} \).

Advanced
\[ \lim_{x\to\infty}\dfrac{3+1/x^2}{1-2/x^2}=3. \]
42

How many arrangements of the letters CALCULUS exist?

Advanced

8 letters: C×2, A×1, L×3, U×2.

\[ \dfrac{8!}{2!\cdot1!\cdot3!\cdot2!}=1680. \]
43

Find the parametric point on \( x=2t-1 \), \( y=t^2 \) at \( t=3 \).

Intermediate
\[ (2(3)-1,\;3^2)=(5,\,9). \]
44

Convert the polar point \( \left(4,\,\dfrac{\pi}{3}\right) \) to Cartesian coordinates.

Intermediate
\[ x=4\cos\dfrac{\pi}{3}=2;\quad y=4\sin\dfrac{\pi}{3}=2\sqrt{3}. \]
45

Expand \( (a+b)^5 \) and find the coefficient of \( a^2b^3 \).

Advanced
\[ \binom{5}{3}=10. \]
46

A geometric series has \( a_1=5 \) and \( r=0.6 \). Find \( S_\infty \).

Intermediate
\[ S_\infty=\dfrac{5}{1-0.6}=12.5. \]
47

Evaluate \( \sum_{k=1}^{4}(2k-1) \).

Intermediate
\[ 1+3+5+7=16. \]
48

A box has 5 red and 3 blue balls. Find the probability that 3 drawn (without replacement) are all red.

Advanced
\[ \dfrac{5}{8}\times\dfrac{4}{7}\times\dfrac{3}{6}=\dfrac{60}{336}=\dfrac{5}{28}. \]
49

If \( P(A)=0.6 \) and \( P(B|A)=0.4 \), find \( P(A\cap B) \).

Advanced
\[ P(A\cap B)=P(B|A)\cdot P(A)=0.4\times0.6=0.24. \]
50

Using Bayes' theorem: \( P(A)=0.3 \), \( P(B|A)=0.8 \), \( P(B|\overline{A})=0.2 \). Find \( P(A|B) \).

Advanced

\( P(B)=0.3\times0.8+0.7\times0.2=0.38 \).

\[ P(A|B)=\dfrac{0.3\times0.8}{0.38}=\dfrac{0.24}{0.38}\approx0.632. \]
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Test generator

Build a randomised ACT Math 2 mini-test from the question bank, or generate a complete 60-question full-length paper structured exactly like the official ACT Mathematics section — 60 minutes, four answer choices per question.

Full official exam paper

Generates a complete 50-question advanced ACT Mathematics paper formatted like an official timed ACT exam — one section, 60 minutes, calculator permitted.

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Downloads

Take the ACT Math 2 material offline. The PDF prints from your browser; the Word packet contains editable OMML equations for classroom customisation.

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ACT Math 2 — notes & practice

All eight advanced chapters plus every worked solution — algebra, functions, trig, pre-calc, and statistics.

DOCX · on request

Native Word packet

Editable .docx with real Word equations (OMML) — all ACT Math 2 topics ready for classroom printing.

Official · Past Papers

Past Papers & Official Tests

Officially released exams and mark schemes from the examining body.

ACT Free Practice Tests Official ACT full-length practice tests ACT Prep Guide PDF Official ACT "Preparing for the ACT" PDF with full test

Worksheet Library

8 original ClipSAT worksheets with full answer keys, organized by unit — free to download and print.

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Mock Exam

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