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EST 2 Math

Provider · Academic Assessment Ltd. Format · EST II · Subject Test Chapters · 8 Level · Advanced Mathematics
i

Test format & strategy

🇬🇧 EST 2 Math — Domain Weightings (44 questions, 80 min)
Domain % of Test Key Topics
Advanced Algebra ~20% Logarithms, exponentials, polynomial & rational equations, systems
Functions ~20% Composite, inverse, logarithmic, transformations, piecewise
Sequences, Series & Complex Numbers ~10% AP/GP, sigma notation, complex arithmetic
Geometry — Lines, Triangles & Polygons ~10% Angle theorems, similarity, congruence, area
Trigonometry ~15% Unit circle, identities, graphs, law of sines/cosines
Circles & Coordinate Geometry ~10% Circle equations, tangents, conics, coordinate proofs
Solid Geometry & Similarity ~5% 3D surface area & volume, scale factors
Statistics & Probability ~10% Counting, combinatorics, probability, data analysis

The EST II Mathematics subject test is a rigorous examination for students in international and American-diploma programmes who wish to demonstrate advanced mathematical competency for university admissions. It covers algebra, geometry, trigonometry, and data analysis at a depth beyond the core EST I test.

What to expect

EST II Math is taken as a standalone subject test on the EST II examination day. It contains both multiple-choice and short constructed-response questions. The content aligns closely with pre-calculus and advanced secondary mathematics.

Level
Advanced secondary / pre-calculus
Topics
Algebra · Geometry · Trig · Stats
Format
MCQ + constructed response
Depth
Beyond EST I
▲ Strategy tip

EST II Math rewards fluency with algebraic manipulation and geometric reasoning. Practice moving quickly between forms (factored, vertex, standard) for quadratics, and know your trigonometric identities cold.

📝 Chapter Quiz
1

Advanced Algebra

🇬🇧 Advanced Algebra ~20%  ·  Logarithms & exponentials  ·  Polynomial long division  ·  Rational equations  ·  Nonlinear systems

Beyond linear equations, the EST II tests rational expressions, absolute value equations, and advanced polynomial manipulation.

Rational expression
\( \dfrac{P(x)}{Q(x)},\ Q\neq 0 \)
Absolute value
\( |x-a|=b \Rightarrow x=a\pm b \)
Remainder theorem
\( f(k) = \) remainder of \( f\div(x-k) \)
Factor theorem
\( f(k)=0 \Leftrightarrow (x-k)\mid f(x) \)
Worked example 1.A

Simplify \( \dfrac{x^2-9}{x^2-x-6} \). \[ \frac{(x-3)(x+3)}{(x-3)(x+2)}=\frac{x+3}{x+2},\quad x\neq 3. \]

Systems and inequalities

Graphical solutions of systems of inequalities define a feasible region. To find vertices, solve pairs of boundary lines simultaneously. EST II questions may ask for the maximum or minimum of a linear expression over the feasible region — an introduction to linear programming.

● Advanced Algebra

Polynomial long division: divide \(p(x)\) by \(d(x)\) to get quotient \(q(x)\) and remainder \(r(x)\). Rational equations: multiply through by LCD, then solve, checking for extraneous solutions. Logarithmic equations: use log laws to combine terms, then convert to exponential form.

Worked example — Rational equation with extraneous solution

Solve \(\dfrac{x}{x-2} + \dfrac{1}{x+2} = \dfrac{8}{x^2-4}\).

LCD \(= (x-2)(x+2)\). Multiply through: \(x(x+2)+(x-2) = 8\).

\(x^2+2x+x-2 = 8 \Rightarrow x^2+3x-10 = 0 \Rightarrow (x+5)(x-2)=0\).

\(x=2\) is extraneous (denominator = 0). Solution: \(x = -5\).

Solve \(\log_2(x+4) - \log_2 x = 3\).
\(\log_2\dfrac{x+4}{x} = 3 \Rightarrow \dfrac{x+4}{x} = 8 \Rightarrow x+4 = 8x \Rightarrow x = \dfrac{4}{7}\).
📝 Chapter Quiz
2

Functions: Composite, Inverse & Logarithmic

🇬🇧 Functions ~20%  ·  Composite & inverse  ·  Logarithmic functions  ·  Transformations  ·  Domain & range

EST II places heavy emphasis on function behaviour, including composite and inverse functions, domain and range analysis, and the properties of logarithms.

Composition
\( (f\circ g)(x)=f(g(x)) \)
Inverse
\( f(f^{-1}(x))=x \)
Log product
\( \log_b(mn)=\log_b m+\log_b n \)
Change of base
\( \log_b a=\dfrac{\ln a}{\ln b} \)
Worked example 2.A — Inverse

Find \( f^{-1}(x) \) for \( f(x)=3x-2 \). \[ y=3x-2 \implies x=\frac{y+2}{3} \implies f^{-1}(x)=\frac{x+2}{3}. \]

Worked example 2.B — Logarithm

Solve \( \log_2(x+3)=5 \). \[ x+3=2^5=32 \implies x=29. \]

⊙ Explorer · A function and its inverseinteractive
f(x₀) = 2^x₀2.000
f⁻¹(f(x₀))1.000
Given \(f(x) = e^x\) and \(g(x) = \ln(x+1)\), find the domain of \(g(f(x))\) and simplify.
\(g(f(x)) = \ln(e^x+1)\). Since \(e^x > 0\) for all \(x\), \(e^x+1 > 1 > 0\). Domain: all reals. Cannot simplify further.
📝 Chapter Quiz
3

Sequences, Series & Complex Numbers

🇬🇧 Sequences, Series & Complex Numbers ~10%  ·  Arithmetic & geometric sequences  ·  Sigma notation  ·  Complex arithmetic

EST II tests arithmetic and geometric sequences, sigma notation, and arithmetic with complex numbers.

Arithmetic \( a_n \)
\( a_1+(n-1)d \)
Geometric \( a_n \)
\( a_1\cdot r^{n-1} \)
Arithmetic sum
\( \dfrac{n}{2}(a_1+a_n) \)
Complex product
\( (a+bi)(c+di) \) — FOIL, \(i^2=-1\)
Worked example 3.A

Find the 10th term of the arithmetic sequence \( 3, 7, 11, \ldots \) \[ a_{10}=3+(10-1)\cdot 4=3+36=39. \]

Geometric series

The sum of a finite geometric series with first term \( a_1 \), ratio \( r \), and \( n \) terms is \( S_n=\dfrac{a_1(1-r^n)}{1-r} \). For \( |r|<1 \), the infinite geometric series converges to \( S=\dfrac{a_1}{1-r} \).

● Sequences, Series & Complex Numbers

Arithmetic series sum: \(S_n = \dfrac{n}{2}(a_1+a_n)\). Geometric series sum: \(S_n = \dfrac{a_1(1-r^n)}{1-r}\); sum to infinity (\(|r|<1\)): \(S_\infty = \dfrac{a_1}{1-r}\). Complex: \((a+bi)+(c+di)=(a+c)+(b+d)i\); \((a+bi)(c+di)=(ac-bd)+(ad+bc)i\).

Worked example — Sum to infinity

Find the sum to infinity of \(12 + 4 + \dfrac{4}{3} + \cdots\)

\(r = 1/3\). \(S_\infty = \dfrac{12}{1-1/3} = \dfrac{12}{2/3} = 18\).

The sum of the first \(n\) terms of an arithmetic sequence is \(S_n = 2n^2 + 3n\). Find the 10th term.
\(a_n = S_n - S_{n-1} = (2n^2+3n) - (2(n-1)^2+3(n-1)) = 4n+1\). So \(a_{10} = 41\).
📝 Chapter Quiz
4

Lines, Triangles & Polygons

🇬🇧 Lines, Triangles & Polygons ~10%  ·  Angle theorems  ·  Triangle congruence/similarity  ·  Area of polygons

Advanced geometry on the EST II includes angle theorems, triangle congruence and similarity, and properties of quadrilaterals and regular polygons.

Triangle angle sum
\( 180^\circ \)
Polygon angle sum
\( (n-2)\times 180^\circ \)
Area of triangle
\( \tfrac{1}{2}bh \)
Law of cosines
\( c^2=a^2+b^2-2ab\cos C \)
Worked example 4.A

In triangle \( ABC \), \( a=7 \), \( b=10 \), \( C=60^\circ \). Find \( c \). \[ c^2=49+100-2(7)(10)\cos 60^\circ=149-70=79,\quad c=\sqrt{79}. \]

● Lines, Triangles & Polygons

Parallel lines cut by a transversal: alternate interior angles are equal; co-interior (same-side) angles are supplementary. Triangle congruence: SSS, SAS, ASA, AAS, RHS. Similarity: AA, SAS, SSS — corresponding sides proportional. Exterior angle = sum of two non-adjacent interior angles.

Worked example — Exterior angle theorem

In triangle \(ABC\), \(\angle A = 48°\) and \(\angle B = 63°\). Find the exterior angle at \(C\).

Exterior angle \(= \angle A + \angle B = 48° + 63° = 111°\).

Two sides of a triangle are 7 cm and 10 cm with an included angle of 90°. Find the hypotenuse and all angles.
\(c = \sqrt{49+100} = \sqrt{149} \approx 12.2\) cm. \(\tan A = 7/10 \Rightarrow A \approx 35.0°\). \(B \approx 55.0°\).
📝 Chapter Quiz
5

Trigonometry

🇬🇧 Trigonometry ~15%  ·  Unit circle & radians  ·  Trig identities  ·  Law of sines/cosines  ·  Graphs

EST II trigonometry goes beyond right-triangle ratios to include the unit circle, radian measure, graphing, and trigonometric identities.

Pythagorean identity
\( \sin^2\theta+\cos^2\theta=1 \)
Double angle sin
\( \sin 2\theta=2\sin\theta\cos\theta \)
Period of sin/cos
\( 2\pi \)
Radian–degree
\( 180^\circ=\pi \)
Worked example 5.A

Simplify \( \dfrac{\sin^2\theta}{1-\cos\theta} \). \[ \frac{1-\cos^2\theta}{1-\cos\theta}=\frac{(1-\cos\theta)(1+\cos\theta)}{1-\cos\theta}=1+\cos\theta. \]

⊙ Explorer · The unit circle, in radiansinteractive
cos θ0.707
sin θ0.707
tan θ1.000

EST II works in radians: a full turn is \(2\pi\approx6.28\), not \(360°\). \(P=(\cos\theta,\sin\theta)\) as above.

Graphs of trigonometric functions

The function \( y=A\sin(Bx+C)+D \) has amplitude \( |A| \), period \( \dfrac{2\pi}{B} \), phase shift \( -C/B \), and vertical shift \( D \). Knowing how to read these parameters from a graph or equation is a core EST II skill.

● Advanced Trigonometry

Unit circle: \((\cos\theta, \sin\theta)\). Key identities: \(\tan\theta = \sin\theta/\cos\theta\); \(\sec\theta = 1/\cos\theta\). Double angle: \(\sin 2\theta = 2\sin\theta\cos\theta\); \(\cos 2\theta = 1-2\sin^2\theta\). Law of cosines: \(c^2 = a^2+b^2-2ab\cos C\). Area: \(\tfrac{1}{2}ab\sin C\).

Worked example — Non-right triangle (Law of Sines)

In triangle \(ABC\): \(a=8\), \(B=35°\), \(C=75°\). Find side \(b\).

\(A = 180°-35°-75° = 70°\). By sine rule: \(\dfrac{b}{\sin B} = \dfrac{a}{\sin A} \Rightarrow b = \dfrac{8\sin 35°}{\sin 70°} \approx \dfrac{4.589}{0.9397} \approx 4.88\).

Find all solutions of \(\cos 2\theta = \cos\theta\) for \(0° \leq \theta \leq 360°\).
\(2\cos^2\theta-1 = \cos\theta \Rightarrow 2\cos^2\theta-\cos\theta-1=0 \Rightarrow (2\cos\theta+1)(\cos\theta-1)=0\). \(\cos\theta=1 \Rightarrow \theta=0°\); \(\cos\theta=-1/2 \Rightarrow \theta=120°, 240°\).
📝 Chapter Quiz
6

Circles & Coordinate Geometry

🇬🇧 Circles & Coordinate Geometry ~10%  ·  Circle equation & properties  ·  Tangent lines  ·  Coordinate proofs

Circle theorems and coordinate geometry of conic sections are significant topics on the EST II Mathematics subject test.

Circle equation
\( (x-h)^2+(y-k)^2=r^2 \)
Arc length
\( r\theta \) (θ in radians)
Sector area
\( \tfrac{1}{2}r^2\theta \)
Inscribed angle
\( =\tfrac{1}{2}\times \text{central angle} \)
Worked example 6.A

Write the equation of a circle with centre \( (3,-2) \) and radius 5. \[ (x-3)^2+(y+2)^2=25. \]

⊙ Explorer · Circle equation, centre-radiusinteractive
equation(x-3)² + (y+2)² = 25

Ellipses and hyperbolas

An ellipse: \( \dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1 \). A hyperbola: \( \dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1 \). EST II may ask you to identify the type of conic from a standard-form equation and state key features like vertices, foci, or asymptotes.

● Circles & Coordinate Geometry

Circle equation: \((x-h)^2+(y-k)^2=r^2\). A tangent to a circle at point \(P\) is perpendicular to the radius at \(P\). Circle theorems: angle at centre = twice angle at circumference; angles in same segment are equal; tangent-radius angle = 90°. Chord perpendicular bisector passes through the centre.

Worked example — Tangent from external point

A circle has centre \(O(3,2)\) and radius 5. Point \(P(10,2)\) lies outside the circle. Find the length of the tangent from \(P\) to the circle.

\(OP = \sqrt{(10-3)^2+(2-2)^2} = 7\). Tangent length \(= \sqrt{OP^2 - r^2} = \sqrt{49-25} = \sqrt{24} = 2\sqrt{6}\).

Find the equation of the circle that passes through \((0,0)\), \((6,0)\) and \((0,8)\).
Let \((x-h)^2+(y-k)^2=r^2\). Using \((0,0)\): \(h^2+k^2=r^2\). Using \((6,0)\): \((6-h)^2+k^2=r^2\) gives \(h=3\). Using \((0,8)\): \(h^2+(8-k)^2=r^2\) gives \(k=4\). So \(r^2=25\). Equation: \((x-3)^2+(y-4)^2=25\).
📝 Chapter Quiz
7

Solid Geometry & Similarity

🇬🇧 Solid Geometry & Similarity ~5%  ·  Surface area & volume of 3D solids  ·  Scale factors for area/volume

Three-dimensional geometry and similarity ratios appear at the end of the EST II test, often combined with algebraic reasoning.

Sphere volume
\( \tfrac{4}{3}\pi r^3 \)
Cone volume
\( \tfrac{1}{3}\pi r^2 h \)
Cylinder volume
\( \pi r^2 h \)
Similarity ratio
lengths \( k \) : areas \( k^2 \) : volumes \( k^3 \)
Worked example 7.A

Two similar spheres have radii 3 and 6. Find the ratio of their volumes. \[ \text{Ratio of radii}=1:2,\quad\text{Volume ratio}=1^3:2^3=1:8. \]

● Solid Geometry & Similarity

Sphere: SA \(=4\pi r^2\), V \(=\tfrac{4}{3}\pi r^3\). Cylinder: SA \(=2\pi r^2+2\pi rh\), V \(=\pi r^2 h\). Cone: SA \(=\pi r^2+\pi rl\), V \(=\tfrac{1}{3}\pi r^2 h\). For similar solids with scale factor \(k\): area scales as \(k^2\), volume scales as \(k^3\).

Worked example — Volume of composite solid

A solid consists of a cylinder (radius 4 cm, height 6 cm) with a hemisphere on top. Find the total volume.

Cylinder: \(\pi(16)(6) = 96\pi\). Hemisphere: \(\tfrac{1}{2}\cdot\tfrac{4}{3}\pi(64) = \tfrac{128\pi}{3}\).

Total \(= 96\pi + \tfrac{128\pi}{3} = \dfrac{288\pi+128\pi}{3} = \dfrac{416\pi}{3} \approx 435.6\) cm\(^3\).

Two similar cones have surface areas in the ratio \(4:9\). If the smaller cone has volume \(16\pi\) cm\(^3\), find the volume of the larger.
Area ratio \(= k^2 = 4/9\), so \(k = 2/3\). Volume ratio \(= k^3 = 8/27\). Larger volume \(= 16\pi \times 27/8 = 54\pi\) cm\(^3\).
📝 Chapter Quiz
8

Statistics & Probability

🇬🇧 Statistics & Probability ~10%  ·  Permutations & combinations  ·  Probability rules  ·  Data distributions

Data analysis and probability on the EST II go beyond basic means and medians to include distributions, graphical representations, and counting methods.

Permutations
\( _nP_r=\dfrac{n!}{(n-r)!} \)
Combinations
\( \binom{n}{r}=\dfrac{n!}{r!(n-r)!} \)
Conditional probability
\( P(A|B)=\dfrac{P(A\cap B)}{P(B)} \)
Standard deviation
measures spread from mean
Worked example 8.A

In how many ways can 3 students be chosen from a group of 8? \[ \binom{8}{3}=\frac{8!}{3!\cdot 5!}=\frac{8\times7\times6}{6}=56. \]

● Statistics, Counting & Probability

Permutations (order matters): \(^nP_r = \dfrac{n!}{(n-r)!}\). Combinations (order irrelevant): \(^nC_r = \dfrac{n!}{r!(n-r)!}\). Conditional probability: \(P(A|B) = \dfrac{P(A \cap B)}{P(B)}\). Independent events: \(P(A \cap B) = P(A) \cdot P(B)\). Normal distribution: ~68% within \(1\sigma\), ~95% within \(2\sigma\).

Worked example — Combinations with restrictions

From 4 boys and 5 girls, choose a team of 4 with at least 2 girls. How many ways?

Exactly 2G: \(\binom{5}{2}\binom{4}{2}=10 \times 6=60\). Exactly 3G: \(\binom{5}{3}\binom{4}{1}=10 \times 4=40\). Exactly 4G: \(\binom{5}{4}\binom{4}{0}=5\). Total: \(105\).

Events \(A\) and \(B\) are such that \(P(A)=0.6\), \(P(B)=0.5\), \(P(A|B)=0.4\). Find \(P(A \cup B)\).
\(P(A \cap B) = P(A|B)\cdot P(B) = 0.4 \times 0.5 = 0.2\). \(P(A \cup B) = 0.6+0.5-0.2 = 0.9\).
📝 Chapter Quiz
§

Practice set

Fifty EST II Mathematics problems spanning advanced algebra, functions, sequences, trigonometry, geometry, circles, solids, and statistics — graded by difficulty.

1

Factor completely: \( x^3 - 4x \).

Intermediate
\[ x(x^2-4)=x(x-2)(x+2). \]
2

Simplify \( \dfrac{x^2-9}{x^2-x-6} \).

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

When \( f(x)=x^3-2x^2+x-5 \) is divided by \( (x-2) \), find the remainder.

Intermediate
\[ f(2)=8-8+2-5=-3. \]
4

Solve \( |3x-1|=8 \).

Intermediate
\[ 3x-1=8\Rightarrow x=3;\quad 3x-1=-8\Rightarrow x=-\tfrac{7}{3}. \]
5

Is \( (x-1) \) a factor of \( x^3 - 3x + 2 \)?

Advanced

\( f(1)=1-3+2=0 \), so yes.

\[ \text{Yes — }(x-1)\text{ is a factor.} \]
6

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

Intermediate
\[ (2x-5)(x+1)=0 \Rightarrow x=\tfrac{5}{2}\text{ or }x=-1. \]
7

Solve the system \( x^2+y=5 \), \( x+y=3 \).

Advanced

\( y=3-x \). Substitute: \( x^2+3-x=5\Rightarrow x^2-x-2=0\Rightarrow x=2\text{ or }x=-1 \).

\[ (2,1)\text{ and }(-1,4). \]
8

Solve \( \dfrac{2x}{x-1}=3 \).

Advanced
\[ 2x=3(x-1)=3x-3 \Rightarrow x=3. \]
9

Solve \( \sqrt{2x+3}=5 \).

Intermediate
\[ 2x+3=25 \Rightarrow x=11. \]
10

Find the range of \( f(x)=x^2+3 \).

Intermediate
\[ [3,\infty). \]
11

Let \( f(x)=2x-1 \) and \( g(x)=x+3 \). Find \( (f\circ g)(x) \).

Intermediate
\[ f(g(x))=2(x+3)-1=2x+5. \]
12

Find \( f^{-1}(x) \) for \( f(x)=\dfrac{x+2}{3} \).

Intermediate
\[ y=\dfrac{x+2}{3}\Rightarrow x=3y-2 \Rightarrow f^{-1}(x)=3x-2. \]
13

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

Intermediate
\[ x-1=2^3=8 \Rightarrow x=9. \]
14

Simplify \( \log_5 125 - \log_5 25 \).

Intermediate
\[ \log_5\!\dfrac{125}{25}=\log_5 5=1. \]
15

Solve \( e^x = 10 \). Give exact form.

Advanced
\[ x=\ln 10. \]
16

Find the 15th term of the arithmetic sequence \( 4,\,7,\,10,\ldots \)

Intermediate
\[ a_{15}=4+(14)(3)=46. \]
17

Find the sum of the first 20 terms of \( 2+5+8+\cdots \)

Intermediate

\( a_{20}=2+19\times3=59 \).

\[ S_{20}=\dfrac{20}{2}(2+59)=610. \]
18

Find the 6th term of the geometric sequence \( 3,\,6,\,12,\ldots \)

Intermediate
\[ a_6=3\cdot2^5=96. \]
19

Find the sum to infinity of \( 8 + 4 + 2 + \cdots \)

Advanced
\[ S_\infty=\dfrac{8}{1-\tfrac{1}{2}}=16. \]
20

An arithmetic sequence has \( a_1=5 \) and common difference 4. Find the sum of the first 10 terms.

Advanced
\[ S_{10}=\dfrac{10}{2}(2\times5+9\times4)=5\times46=230. \]
21

Find the exact value of \( \sin 30^\circ + \cos 60^\circ \).

Basic
\[ \tfrac{1}{2}+\tfrac{1}{2}=1. \]
22

Simplify \( \sin^2\theta + \cos^2\theta \).

Basic
\[ 1. \]
23

Prove: \( \dfrac{\sin^2\theta}{1-\cos\theta}=1+\cos\theta \).

Advanced
\[ \dfrac{1-\cos^2\theta}{1-\cos\theta}=\dfrac{(1-\cos\theta)(1+\cos\theta)}{1-\cos\theta}=1+\cos\theta.\ \checkmark \]
24

In a right triangle, \( \sin\theta=\dfrac{5}{13} \). Find \( \cos\theta \).

Intermediate
\[ \cos\theta=\sqrt{1-\tfrac{25}{169}}=\dfrac{12}{13}. \]
25

Find the exact value of \( \tan 45^\circ + \sin 90^\circ \).

Intermediate
\[ 1+1=2. \]
26

In triangle \( ABC \), \( AB=6 \), \( BC=8 \), and angle \( B=90^\circ \). Find \( AC \).

Intermediate
\[ AC=\sqrt{36+64}=10. \]
27

Find the area of a triangle with sides 5, 7, and included angle \( 60^\circ \).

Advanced
\[ A=\tfrac{1}{2}\times5\times7\times\sin60^\circ=\dfrac{35\sqrt{3}}{4}. \]
28

Use the law of cosines to find \( c \) when \( a=5 \), \( b=7 \), \( C=90^\circ \).

Advanced
\[ c^2=25+49-0=74 \Rightarrow c=\sqrt{74}. \]
29

Find the equation of a circle centred at \( (2,-3) \) with radius 5.

Intermediate
\[ (x-2)^2+(y+3)^2=25. \]
30

Find the centre and radius of \( x^2+y^2-4x+6y-12=0 \).

Advanced

Complete the square: \( (x-2)^2+(y+3)^2=25 \).

\[ \text{Centre }(2,-3),\quad r=5. \]
31

An arc of a circle with radius 6 subtends an angle of \( 2 \) radians. Find the arc length.

Intermediate
\[ s=r\theta=6\times2=12. \]
32

A sector has radius 4 and angle \( \pi/3 \). Find its area.

Advanced
\[ A=\tfrac{1}{2}r^2\theta=\tfrac{1}{2}(16)\tfrac{\pi}{3}=\dfrac{8\pi}{3}. \]
33

An inscribed angle intercepts an arc of \( 80^\circ \). Find the angle.

Intermediate
\[ \text{Inscribed angle}=\tfrac{1}{2}\times80^\circ=40^\circ. \]
34

Find the volume of a sphere with radius 3.

Intermediate
\[ V=\tfrac{4}{3}\pi(3)^3=36\pi. \]
35

Find the surface area of a cone with radius 4 and slant height 5.

Advanced
\[ SA=\pi r(r+l)=\pi(4)(9)=36\pi. \]
36

Two similar cones have radii \( 3 \) and \( 6 \). Find the ratio of their volumes.

Advanced
\[ \left(\dfrac{3}{6}\right)^3=\dfrac{1}{8}. \]
37

Find the volume of a cylinder with radius 5 and height 9.

Intermediate
\[ V=\pi r^2 h=225\pi. \]
38

Five test scores are 70, 75, 80, 85, 90. Find the mean and range.

Basic
\[ \bar{x}=80;\quad \text{range}=90-70=20. \]
39

How many ways can 4 students be chosen from 10?

Intermediate
\[ \binom{10}{4}=\dfrac{10!}{4!\cdot6!}=210. \]
40

A bag contains 5 red and 3 blue balls. Two are drawn without replacement. Find \( P(\text{both red}) \).

Advanced
\[ \dfrac{5}{8}\times\dfrac{4}{7}=\dfrac{20}{56}=\dfrac{5}{14}. \]
41

Evaluate \( 5! \div 3! \).

Basic
\[ \dfrac{120}{6}=20. \]
42

Two events \( A \) and \( B \) are independent. \( P(A)=0.4 \), \( P(B)=0.5 \). Find \( P(A\cap B) \).

Intermediate
\[ P(A\cap B)=0.4\times0.5=0.2. \]
43

Expand \( (x+2)^4 \) using the binomial theorem. Find the coefficient of \( x^3 \).

Advanced
\[ \binom{4}{1}2^1=8. \]
44

A complex number is \( z=3+4i \). Find \( |z| \).

Intermediate
\[ |z|=\sqrt{9+16}=5. \]
45

Multiply \( (2+3i)(1-i) \).

Advanced
\[ 2-2i+3i-3i^2=2+i+3=5+i. \]
46

Express \( 0.3\overline{6} \) as a fraction. (\( 0.3666\ldots \))

Advanced

Let \( x=0.3\overline{6} \). Then \( 10x=3.\overline{6} \), \( 9x=3.3 \).

\[ x=\dfrac{3.3}{9}=\dfrac{11}{30}. \]
47

How many 3-digit numbers can be formed from \( \{1,2,3,4,5\} \) with no repetition?

Intermediate
\[ _5P_3=5\times4\times3=60. \]
48

Find \( \lim_{x\to 2}\dfrac{x^2-4}{x-2} \).

Advanced
\[ \dfrac{(x-2)(x+2)}{x-2}=x+2 \xrightarrow{x\to2} 4. \]
49

A geometric sequence has first term 6 and common ratio \( -\tfrac{1}{2} \). Find the 5th term.

Intermediate
\[ a_5=6\times\left(-\tfrac{1}{2}\right)^4=6\times\tfrac{1}{16}=\tfrac{3}{8}. \]
50

Solve \( 2^{3x}=16^{x-1} \).

Advanced

\( 2^{3x}=2^{4(x-1)} \Rightarrow 3x=4x-4 \).

\[ x=4. \]
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