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Algebra 2

Level · High school · second-year algebra Chapters · 8 Explorers · 3 interactive Feeds into · Pre-Calculus · SAT · ACT
1

Functions & Their Graphs

📌 Functions & Their Graphs A2  ·  Composition  ·  Inverse  ·  Domain & range  ·  Piecewise functions

Algebra 2 is really the study of functions as objects you can combine, invert and transform. Fluency with notation, domain and range underpins every later chapter.

● Definition 1.1 — Function

A function assigns to each input exactly one output. We write \( y=f(x) \); the set of allowed inputs is the domain and the set of resulting outputs is the range.

Composition
\( (f\circ g)(x)=f\big(g(x)\big) \)
Inverse
swap \( x \) and \( y \), then solve for \( y \)
Domain rule
no division by 0; no even root of a negative
Inverse check
\( f\big(f^{-1}(x)\big)=x \)
Worked example 1.A

Given \( f(x)=2x+1 \) and \( g(x)=x^{2} \), find \( f\big(g(2)\big) \).

Work inside-out: \( g(2)=4 \), then \( f(4)=2(4)+1 \):

\[ f\big(g(2)\big)=9. \]

Worked example 1.B

Find the inverse of \( f(x)=2x-6 \).

Write \( y=2x-6 \), swap and solve: \( x=2y-6 \Rightarrow y=\tfrac{x+6}{2} \):

\[ f^{-1}(x)=\frac{x+6}{2}. \]

If \(f(x)=x^2-1\) and \(g(x)=\sqrt{x+2}\), find (a) \(f(g(3))\), (b) the domain of \(g\circ f\).
(a) \(g(3)=\sqrt5\); \(f(\sqrt5)=5-1=4\). (b) \(g(f(x))=\sqrt{x^2-1+2}=\sqrt{x^2+1}\); always real, domain: all reals.
📝 Chapter Quiz
2

Quadratics & Complex Numbers

📌 Quadratics & Complex Numbers A2  ·  Completing the square  ·  Discriminant  ·  Complex roots \(a\pm bi\)

Every quadratic can be solved — and when the discriminant is negative, the solutions live in the complex numbers. This chapter ties graphing, factoring and the imaginary unit together.

● Definition 2.1 — The imaginary unit

Define \( i=\sqrt{-1} \), so \( i^{2}=-1 \). A complex number has the form \( a+bi \) with \( a,b \) real; add and multiply just as with binomials, replacing \( i^{2} \) by \( -1 \).

Quadratic formula
\( x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a} \)
Discriminant
\( \Delta=b^{2}-4ac \) decides the root type
Vertex
\( x=-\dfrac{b}{2a} \)
Powers of i
\( i^{2}=-1,\ i^{3}=-i,\ i^{4}=1 \)
Worked example 2.A

Describe the roots of \( x^{2}+2x+5=0 \).

Discriminant \( \Delta=2^{2}-4(1)(5)=-16<0 \), so there are no real roots — two complex conjugates. Solving:

\[ x=\frac{-2\pm\sqrt{-16}}{2}=-1\pm 2i. \]

Worked example 2.B

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

\[ 6-2i+3i-i^{2}=6+i+1=7+i. \]

Solve \(x^2-4x+13=0\) and express roots in the form \(a\pm bi\).
Discriminant \(=16-52=-36\). \(x=\dfrac{4\pm\sqrt{-36}}{2}=\dfrac{4\pm6i}{2}=2\pm3i\).
📝 Chapter Quiz
3

Polynomial Functions

📌 Polynomial Functions A2  ·  Factor theorem  ·  Remainder theorem  ·  Rational roots  ·  End behaviour

Beyond quadratics lie cubics, quartics and higher. Two questions dominate: where are the zeros, and what does the graph do at its ends? The leading term answers the second on its own.

▲ Reference — End behaviour & the factor theorem

The end behaviour of a polynomial is governed entirely by its leading term \( a x^{n} \): the sign of \( a \) and the parity of \( n \). And \( (x-r) \) is a factor exactly when \( r \) is a zero (\( f(r)=0 \)).

Change the degree and the leading coefficient below. Watch how the arms of the graph flip with the sign of \( a \), and how an even degree sends both arms the same way while an odd degree sends them opposite ways.

⊙ Explorer · Degree, sign & end behaviourinteractive
as x → +∞—
as x → −∞—
turning points—
Remainder theorem
dividing by \( (x-r) \) leaves remainder \( f(r) \)
Factor theorem
\( f(r)=0 \iff (x-r) \) is a factor
Zeros
a degree-\( n \) polynomial has \( n \) zeros (with multiplicity)
Worked example 3.A

Find the zeros of \( f(x)=x^{3}-9x \).

Factor out \( x \), then a difference of squares:

\[ x(x^{2}-9)=x(x-3)(x+3)=0 \ \Rightarrow\ x=0,\ 3,\ -3. \]

Worked example 3.B

Find the remainder when \( x^{2}+3x-4 \) is divided by \( (x-1) \).

By the remainder theorem, evaluate at \( x=1 \):

\[ 1+3-4=0. \]

Show that \((x-2)\) is a factor of \(p(x)=x^3-3x^2+4\), then fully factorise \(p(x)\).
\(p(2)=8-12+4=0\) ✓. Divide: \(p(x)=(x-2)(x^2-x-2)=(x-2)(x-2)(x+1)=(x-2)^2(x+1)\).
📝 Chapter Quiz
4

Rational Expressions

📌 Rational Expressions A2  ·  Asymptotes  ·  Holes  ·  Graphing rational functions  ·  Partial fractions

A rational function is a ratio of polynomials. The new feature is the asymptote — a line the graph approaches but never meets — produced by values that make the denominator zero.

● Definition 4.1 — Asymptotes

For \( y=\dfrac{1}{x-h}+k \): a vertical asymptote at \( x=h \) (denominator zero) and a horizontal asymptote at \( y=k \) (the end-behaviour value).

Slide the asymptotes around. The two amber dashed lines are \( x=h \) and \( y=k \); the curve hugs them in every corner.

⊙ Explorer · Rational function & asymptotesinteractive
vertical asymptote—
horizontal asymptote—

y = 1 / (x − 1)

Worked example 4.A

Simplify \( \dfrac{x^{2}-9}{x-3} \).

Factor and cancel (for \( x\neq 3 \)):

\[ \frac{(x-3)(x+3)}{x-3}=x+3. \]

Worked example 4.B

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

Then \( \dfrac{1}{x}=\dfrac12 \), so:

\[ x=2. \]

▲ Rational expressions and equations

A rational expression is a ratio of polynomials. Simplify by factoring and cancelling common factors. Multiply LCD to solve rational equations; check for extraneous solutions.

Simplify
Factor numerator and denominator; cancel common factors
Add/subtract
Find LCD; rewrite each fraction; combine numerators
Multiply
Multiply numerators and denominators; simplify
Solve equation
Multiply both sides by LCD; solve; check for excluded values
Worked example

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

\[ \frac{(x-2)(x+2)}{(x-2)(x+1)}=\frac{x+2}{x+1},\quad x\neq 2,-1. \]

Sketch \(y=\dfrac{2x+1}{x-3}\), showing all asymptotes and intercepts.
Vertical asymptote: \(x=3\). Horizontal asymptote: \(y=2\) (leading coefficients). \(y\)-intercept: \(y=-1/3\). \(x\)-intercept: \(x=-1/2\).
📝 Chapter Quiz
5

Exponential & Logarithmic Functions

📌 Exponential & Logarithmic Functions A2  ·  Laws of logs  ·  Solving \(e^x=k\)  ·  Exponential growth/decay

Exponential functions model growth and decay; logarithms are their inverses, and the tool that lets you solve for an exponent. The log laws turn products into sums.

● Definition 5.1 — Logarithm

\( \log_{a} b=c \) means exactly \( a^{c}=b \) (with \( a>0,\ a\neq 1 \)). So a logarithm is an exponent.

Product
\( \log_a(xy)=\log_a x+\log_a y \)
Quotient
\( \log_a\!\tfrac{x}{y}=\log_a x-\log_a y \)
Power
\( \log_a(x^{k})=k\log_a x \)
Change of base
\( \log_a x=\dfrac{\ln x}{\ln a} \)
Worked example 5.A

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

Write 16 as a power of 2: \( 2^{x}=2^{4} \):

\[ x=4. \]

Worked example 5.B

Condense \( 2\log x+\log y \) to a single logarithm.

Power rule then product rule:

\[ 2\log x+\log y=\log\!\big(x^{2}y\big). \]

Solve \(2^{x+1}=5^{x-1}\), giving your answer to 3 significant figures.
Take logs: \((x+1)\ln2=(x-1)\ln5\). \(x\ln2-x\ln5=-\ln5-\ln2\). \(x=\dfrac{-\ln10}{\ln2-\ln5}=\dfrac{-\ln10}{\ln(2/5)}\approx2.48\).
📝 Chapter Quiz
6

Radicals & Rational Exponents

📌 Radicals & Rational Exponents A2  ·  Simplifying radicals  ·  Rationalising  ·  Solving radical equations

Roots and fractional exponents are two notations for the same idea. Converting between them turns awkward radical problems into routine exponent arithmetic.

● Definition 6.1 — Rational exponent

\( a^{m/n}=\sqrt[n]{a^{m}}=\big(\sqrt[n]{a}\big)^{m} \). The denominator is the root; the numerator is the power.

Product
\( \sqrt{a}\,\sqrt{b}=\sqrt{ab} \)
Quotient
\( \dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}} \)
Rationalising
multiply by \( \dfrac{\sqrt{b}}{\sqrt{b}} \)
Worked example 6.A

Evaluate \( 8^{2/3} \).

Take the cube root first, then square:

\[ \big(\sqrt[3]{8}\big)^{2}=2^{2}=4. \]

Worked example 6.B

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

Square both sides: \( x+3=25 \):

\[ x=22. \]

Solve \(\sqrt{3x+1}=x-1\), checking for extraneous solutions.
Square both sides: \(3x+1=(x-1)^2=x^2-2x+1\Rightarrow x^2-5x=0\Rightarrow x=0\) or \(x=5\). Check \(x=0\): \(1=-1\) ✗ (extraneous). Check \(x=5\): \(4=4\) ✓. Solution: \(x=5\).
📝 Chapter Quiz
7

Sequences & Series

📌 Sequences & Series A2  ·  Arithmetic & geometric  ·  Sigma notation  ·  Binomial theorem

Arithmetic sequences add a constant; geometric sequences multiply by one. Summing their terms gives series — including the surprising fact that an infinite geometric series can have a finite total.

● Definition 7.1 — Arithmetic vs geometric

An arithmetic sequence has a common difference \( d \); a geometric sequence has a common ratio \( r \). Their \( n \)th terms are \( a_1+(n-1)d \) and \( a_1 r^{\,n-1} \).

Arithmetic sum
\( S_n=\tfrac{n}{2}\big(a_1+a_n\big) \)
Geometric sum
\( S_n=a_1\dfrac{1-r^{n}}{1-r} \)
Infinite geometric
\( S=\dfrac{a_1}{1-r}\ \ (|r|<1) \)
Worked example 7.A

Find the 10th term of the arithmetic sequence with \( a_1=2 \), \( d=3 \).

\[ a_{10}=2+(10-1)(3)=29. \]

Worked example 7.B

Find the sum of \( 8+4+2+1+\cdots \)

Geometric with \( a_1=8 \), \( r=\tfrac12 \):

\[ S=\frac{8}{1-\tfrac12}=16. \]

Find the coefficient of \(x^2\) in the expansion of \((3-2x)^5\).
\(\binom{5}{2}(3)^3(-2x)^2=10\times27\times4x^2=1080x^2\). Coefficient: \(1080\).
📝 Chapter Quiz
8

Conic Sections

📌 Conic Sections A2  ·  Parabola  ·  Ellipse  ·  Hyperbola  ·  Standard forms & transformations

Slice a cone and the cross-section is a circle, ellipse, parabola or hyperbola. Each has a standard equation you can read a graph straight off of.

▲ Reference — The four conics

Circle \( x^{2}+y^{2}=r^{2} \); ellipse \( \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \); parabola \( y=\dfrac{x^{2}}{4p} \); hyperbola \( \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1 \). A plus sign closes the curve; a minus sign opens it.

Switch between the four and stretch the defining parameter. Notice how the ellipse becomes a circle when its two radii match, and how the hyperbola's minus sign splits it into two branches.

⊙ Explorer · The four conic sectionsinteractive

Equation: x²/16 + y²/9 = 1

Worked example 8.A

Give the centre and radius of \( (x-2)^{2}+(y+3)^{2}=16 \).

Standard circle form \( (x-h)^{2}+(y-k)^{2}=r^{2} \):

\[ \text{centre } (2,-3),\qquad r=4. \]

Worked example 8.B

Find the vertices of \( \dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1 \).

The \( x^{2} \) term is positive, so the branches open left/right with \( a=4 \):

\[ (\pm 4,\,0). \]

▲ Conic sections

Conics are curves formed by intersecting a plane with a double cone: circles, ellipses, parabolas, and hyperbolas.

Circle
\( (x-h)^2+(y-k)^2=r^2 \), centre \( (h,k) \), radius \( r \)
Ellipse
\( \dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1 \)
Parabola
\( y=a(x-h)^2+k \) or \( x=a(y-k)^2+h \)
Hyperbola
\( \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 \); asymptotes \( y=\pm\dfrac{b}{a}x \)
Worked example

Identify \( 4x^2+9y^2=36 \) and find its semi-axes.

\[ \frac{x^2}{9}+\frac{y^2}{4}=1 \implies a=3,\;b=2.\text{ Ellipse.} \]

Write \(9x^2+4y^2-36x+8y+4=0\) in standard form and identify the conic.
Complete the square: \(9(x-2)^2+4(y+1)^2=36\Rightarrow\dfrac{(x-2)^2}{4}+\dfrac{(y+1)^2}{9}=1\). Ellipse, centre \((2,-1)\), \(a=3\), \(b=2\).
📝 Chapter Quiz
9

Trigonometric Functions

📌 Trigonometric Functions A2  ·  Unit circle  ·  Graphs of \(\sin/\cos/\tan\)  ·  Inverse trig  ·  Identities

The unit circle extends sine, cosine and tangent to all angles and produces periodic graphs.

▲ Reference — Unit circle & graphs

On the unit circle a point is \( (\cos\theta,\sin\theta) \). Radians: \( 180^{\circ}=\pi \). The graphs of \( \sin x \) and \( \cos x \) have period \( 2\pi \); \( \tan x \) has period \( \pi \).

Unit circle
\( (\cos\theta,\sin\theta) \)
Radians
\( 180^{\circ}=\pi \)
Identity
\( \sin^{2}\theta+\cos^{2}\theta=1 \)
Period
\( 2\pi \) (sin, cos)
Worked example 9.A

Convert \( 90^{\circ} \) to radians.

\[ \frac{\pi}{2}. \]

Worked example — Proving a trig identity

Prove that \(\dfrac{\sin2\theta}{1+\cos2\theta}=\tan\theta\).

\[\text{LHS}=\frac{2\sin\theta\cos\theta}{1+(2\cos^2\theta-1)}=\frac{2\sin\theta\cos\theta}{2\cos^2\theta}=\frac{\sin\theta}{\cos\theta}=\tan\theta=\text{RHS}.\]

Find all solutions of \(2\cos^2\theta+\cos\theta-1=0\) for \(0°\leq\theta\leq360°\).
Factor: \((2\cos\theta-1)(\cos\theta+1)=0\Rightarrow\cos\theta=\tfrac{1}{2}\) or \(\cos\theta=-1\). Solutions: \(\theta=60°,300°,180°\).
📝 Chapter Quiz
10

Matrices & Systems

📌 Matrices & Systems A2  ·  Matrix operations  ·  Determinant  ·  Inverse  ·  Solving systems

A matrix is a rectangular array of numbers; matrices add, multiply, and solve linear systems.

▲ Reference — Matrix basics

Add matrices of the same size entrywise. For a \( 2\times2 \) matrix with rows \( (a,b) \) and \( (c,d) \), the determinant is \( ad-bc \). A system \( A\mathbf{x}=\mathbf{b} \) solves as \( \mathbf{x}=A^{-1}\mathbf{b} \) when the determinant is non-zero.

Order
rows × columns
Add
entrywise
Determinant
\( ad-bc \)
Solve
\( \mathbf{x}=A^{-1}\mathbf{b} \)
Worked example 10.A

Find the determinant of the matrix with rows \( (2,1) \) and \( (3,4) \).

\[ 2(4)-1(3)=5. \]

Worked example — Solving a 3×3 system

Use the inverse matrix method to solve \(2x+y=5\), \(x-3y=1\).

\[A=\begin{pmatrix}2&1\\1&-3\end{pmatrix},\;\det A=-6-1=-7,\;A^{-1}=\frac{1}{-7}\begin{pmatrix}-3&-1\\-1&2\end{pmatrix}.\]\[\begin{pmatrix}x\\y\end{pmatrix}=A^{-1}\begin{pmatrix}5\\1\end{pmatrix}=\frac{1}{-7}\begin{pmatrix}-16\\-3\end{pmatrix}=\begin{pmatrix}16/7\\3/7\end{pmatrix}.\]

Find the value(s) of \(k\) for which \(A=\begin{pmatrix}k&2\\3&k\end{pmatrix}\) is singular.
\(\det A=k^2-6=0\Rightarrow k=\pm\sqrt6\).
📝 Chapter Quiz
11

Probability & Statistics

📌 Probability & Statistics A2  ·  Counting  ·  Probability rules  ·  Normal distribution  ·  Data analysis

Counting principles and probability describe and predict uncertain outcomes.

▲ Reference — Counting & probability

Permutations \( {}_nP_r=\tfrac{n!}{(n-r)!} \); combinations \( {}_nC_r=\tfrac{n!}{r!\,(n-r)!} \). Probability \( =\tfrac{\text{favourable}}{\text{total}} \).

Factorial
\( n! \)
Permutation
\( \tfrac{n!}{(n-r)!} \)
Combination
\( \tfrac{n!}{r!(n-r)!} \)
Probability
\( \tfrac{\text{fav}}{\text{total}} \)
Worked example 11.A

Compute \( {}_5C_2 \).

\[ \frac{5!}{2!\,3!}=10. \]

Worked example — Normal distribution application

Heights are normally distributed with \(\mu=170\) cm and \(\sigma=10\) cm. Find the probability that a randomly chosen person has height between 160 and 185 cm.

\[P(160\leq X\leq185)=P(-1\leq Z\leq1.5)=\Phi(1.5)-\Phi(-1)=0.9332-0.1587=0.7745.\]

In a class of 200 students taking a test with mean 65 and SD 12 (normally distributed), how many students would you expect to score above 80?
\(P(X>80)=P(Z>1.25)=1-0.8944=0.1056\). Expected: \(200\times0.1056\approx21\) students.
📝 Chapter Quiz
§

Practice set

Fifty mixed problems spanning all eight chapters — functions through conic sections — and tagged by difficulty. Click any problem to reveal a full worked solution.

1

If \( f(x)=2x+3 \), find \( f(4) \).

Basic
\[ 2(4)+3=11. \]
2

If \( f(x)=x^{2}-1 \), find \( f(-3) \).

Basic
\[ (-3)^{2}-1=8. \]
3

State the domain of \( f(x)=\sqrt{x-2} \).

Intermediate

The radicand must be \( \ge 0 \):

\[ x\ge 2. \]
4

If \( f(x)=2x+1 \) and \( g(x)=x^{2} \), find \( f\big(g(2)\big) \).

Intermediate

\( g(2)=4 \), then \( f(4)=9 \):

\[ 9. \]
5

Find the inverse of \( f(x)=2x-6 \).

Intermediate
\[ f^{-1}(x)=\frac{x+6}{2}. \]
6

If \( f(x)=x^{2} \) and \( g(x)=x+3 \), find \( (g\circ f)(x) \).

Advanced

\( g\big(f(x)\big)=g\big(x^{2}\big)=x^{2}+3 \):

\[ x^{2}+3. \]
7

Solve \( x^{2}=49 \).

Basic
\[ x=\pm 7. \]
8

Solve \( x^{2}+6x+5=0 \).

Intermediate

\( (x+1)(x+5)=0 \):

\[ x=-1\ \text{or}\ x=-5. \]
9

Find the vertex of \( y=x^{2}-4x+1 \).

Intermediate

\( x=-\tfrac{b}{2a}=2 \), then \( y=4-8+1=-3 \):

\[ (2,-3). \]
10

Find the discriminant of \( x^{2}+2x+5 \) and state the nature of the roots.

Intermediate

\( \Delta=4-20=-16<0 \):

\[ \text{no real roots (two complex)}. \]
11

Simplify \( (3+2i)+(1-5i) \).

Intermediate
\[ 4-3i. \]
12

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

Intermediate

\( 6+i-i^{2}=6+i+1 \):

\[ 7+i. \]
13

Solve \( x^{2}+4=0 \).

Advanced

\( x^{2}=-4 \):

\[ x=\pm 2i. \]
14

Solve \( 2x^{2}-4x-3=0 \) with the quadratic formula.

Advanced
\[ x=\frac{4\pm\sqrt{16+24}}{4}=\frac{2\pm\sqrt{10}}{2}. \]
15

State the degree of \( 4x^{3}-2x^{5}+7 \).

Basic
\[ 5. \]
16

Describe the end behaviour of \( y=-x^{3} \) as \( x\to+\infty \).

Intermediate
\[ y\to-\infty. \]
17

Factor \( x^{3}-9x \) completely.

Intermediate
\[ x(x-3)(x+3). \]
18

Find the remainder when \( x^{2}+3x-4 \) is divided by \( (x-1) \).

Intermediate

Evaluate at \( x=1 \): \( 1+3-4 \):

\[ 0. \]
19

Find the zeros of \( (x-2)(x+1)(x-5) \).

Advanced
\[ x=2,\ -1,\ 5. \]
20

The cubic \( x^{3}-6x^{2}+11x-6 \) factors as \( (x-1)(x-2)(x-3) \). State its roots.

Advanced
\[ x=1,\ 2,\ 3. \]
21

Simplify \( \dfrac{x^{2}-9}{x-3} \).

Basic

Factor and cancel (\( x\neq 3 \)):

\[ x+3. \]
22

Simplify \( \dfrac{x}{2}+\dfrac{x}{3} \).

Intermediate
\[ \frac{3x+2x}{6}=\frac{5x}{6}. \]
23

State the vertical asymptote of \( y=\dfrac{1}{x-4} \).

Intermediate
\[ x=4. \]
24

State the horizontal asymptote of \( y=\dfrac{2x}{x+1} \).

Intermediate

Ratio of leading coefficients:

\[ y=2. \]
25

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

Advanced

\( \dfrac{1}{x}=\dfrac12 \):

\[ x=2. \]
26

Simplify \( \dfrac{x+1}{x^{2}-1} \).

Advanced

\( x^{2}-1=(x-1)(x+1) \), cancel \( (x+1) \):

\[ \frac{1}{x-1}. \]
27

Evaluate \( \log_{2}8 \).

Basic

\( 2^{3}=8 \):

\[ 3. \]
28

Evaluate \( \log_{10}1000 \).

Basic
\[ 3. \]
29

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

Intermediate
\[ 2^{x}=2^{4}\ \Rightarrow\ x=4. \]
30

Solve \( \log_{3}x=2 \).

Intermediate

\( x=3^{2} \):

\[ x=9. \]
31

Evaluate \( \log 50+\log 2 \) (base 10).

Intermediate

Product rule: \( \log(50\times 2)=\log 100 \):

\[ 2. \]
32

Evaluate \( \ln e^{5} \).

Intermediate
\[ 5. \]
33

Solve \( 3\cdot 2^{x}=24 \).

Advanced

\( 2^{x}=8=2^{3} \):

\[ x=3. \]
34

Condense \( 2\log x+\log y \).

Advanced
\[ \log\!\big(x^{2}y\big). \]
35

Simplify \( \sqrt{50} \).

Basic
\[ \sqrt{25\cdot 2}=5\sqrt{2}. \]
36

Evaluate \( 8^{2/3} \).

Intermediate
\[ \big(\sqrt[3]{8}\big)^{2}=4. \]
37

Simplify \( \sqrt{12}+\sqrt{27} \).

Intermediate
\[ 2\sqrt{3}+3\sqrt{3}=5\sqrt{3}. \]
38

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

Intermediate

Square: \( x+3=25 \):

\[ x=22. \]
39

Rationalise \( \dfrac{1}{\sqrt{2}} \).

Advanced
\[ \frac{1}{\sqrt2}\cdot\frac{\sqrt2}{\sqrt2}=\frac{\sqrt2}{2}. \]
40

Evaluate \( 16^{3/4} \).

Advanced
\[ \big(\sqrt[4]{16}\big)^{3}=2^{3}=8. \]
41

Write the next term of \( 3,\,7,\,11,\,15,\ldots \)

Basic

Common difference 4:

\[ 19. \]
42

Find the 10th term of the arithmetic sequence with \( a_1=2 \), \( d=3 \).

Intermediate
\[ 2+9(3)=29. \]
43

Find the common ratio of \( 2,\,6,\,18,\,54,\ldots \)

Intermediate
\[ r=3. \]
44

Find the sum \( 1+2+4+8+16 \).

Intermediate
\[ 31. \]
45

Find the sum of the infinite series \( 8+4+2+1+\cdots \)

Advanced

Geometric, \( r=\tfrac12 \):

\[ S=\frac{8}{1-\tfrac12}=16. \]
46

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

Advanced
\[ 3+5+7+9=24. \]
47

Give the centre and radius of \( x^{2}+y^{2}=25 \).

Basic
\[ \text{centre } (0,0),\quad r=5. \]
48

Give the centre and radius of \( (x-2)^{2}+(y+3)^{2}=16 \).

Intermediate
\[ \text{centre } (2,-3),\quad r=4. \]
49

Is \( \dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1 \) an ellipse or a hyperbola?

Intermediate

The two terms are added:

\[ \text{ellipse}. \]
50

Find the vertices of \( \dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1 \).

Advanced

Opens left/right with \( a=4 \):

\[ (\pm 4,\,0). \]
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