Let n be an odd integer. Which algebraic deduction directly proves that n^2 - 1 is always divisible by 4?

Letting n = 2k + 1 for integer k gives (2k + 1)^2 - 1 = 4k^2 + 4k = 4(k^2 + k), which has a factor of 4.

Writing n^2 - 1 = (n - 1)(n + 1), where both factors are odd, so their product is divisible by 4.

Letting n = 2k for integer k gives (2k)^2 - 1 = 4k^2 - 1, which leaves a remainder of 3.

Testing n = 3 gives 3^2 - 1 = 8, which is divisible by 4, so it holds for all integers.

The graph of y = f(x) is mapped to y = f(2x) + 3. Which sequence of transformations describes this mapping?

A horizontal stretch with scale factor 1/2, followed by a translation of 3 units upwards.

A horizontal translation of 2 units to the right, followed by a vertical stretch with scale factor 3.

A vertical stretch with scale factor 2, followed by a translation of 3 units upwards.

A horizontal stretch with scale factor 2, followed by a translation of 3 units downwards.

Free IB Diploma practice

Free IB Maths AA SL Quiz: 31 MCQs with Answers

Practise IB Diploma Mathematics: Analysis and Approaches SL for school tests and exams. Answer a question and it is marked straight away, with a short explanation. No sign-up needed; log in and your score is saved to your account.

Topic 3 - Geometry and Trigonometry: 3.4 Unit circle, definitions of sin θ, cos θ, tan θ, exact values, Pythagorean identity (cos^2 θ + sin^2 θ = 1), double angle formulas for sin 2θ and cos 2θ
Topic 1 - Number and Algebra: 1.2 Arithmetic sequences and series, nth term, sum of n terms
Topic 1 - Number and Algebra: 1.3 Geometric sequences and series, nth term, sum of finite and infinite convergent series
Topic 1 - Number and Algebra: 1.4 Financial applications of sequences and series (compound interest, annual depreciation)
Topic 1 - Number and Algebra: 1.5 Laws of exponents with integer and rational exponents, introduction to logarithms, change of base formula
Topic 1 - Number and Algebra: 1.6 Simple proofs and algebraic manipulation (direct proof)
Topic 1 - Number and Algebra: 1.7 The binomial theorem for positive integer powers n
Topic 2 - Functions: 2.1 Concept of a function, domain, range, function notation f(x)

31questions

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Bring the marked paper to a mentor and work on the topics it flags, at our Adarsh Nagar centre or online. See IB tuition in Jaipur or maths classes.

What is in this IB Diploma Mathematics: Analysis and Approaches SL quiz

This is a free 31-question multiple-choice quiz on IB Diploma Mathematics: Analysis and Approaches SL, put together and checked by Reportcards, A Learning Centre in Jaipur. It covers IB Diploma Mathematics: Analysis and Approaches SL, in a non-calculator multiple-choice style. You can take it without an account. Log in with a free account and your score is saved, with points and badges; otherwise it stays on the device you answer on.

The 31 questions are spread across these topics, so a weak area shows up as a pattern rather than as one unlucky question:

Topic 3 - Geometry and Trigonometry: 3.4 Unit circle, definitions of sin θ, cos θ, tan θ, exact values, Pythagorean identity (cos^2 θ + sin^2 θ = 1), double angle formulas for sin 2θ and cos 2θ · 2 questions
Topic 1 - Number and Algebra: 1.2 Arithmetic sequences and series, nth term, sum of n terms · 1 question
Topic 1 - Number and Algebra: 1.3 Geometric sequences and series, nth term, sum of finite and infinite convergent series · 1 question
Topic 1 - Number and Algebra: 1.4 Financial applications of sequences and series (compound interest, annual depreciation) · 1 question
Topic 1 - Number and Algebra: 1.5 Laws of exponents with integer and rational exponents, introduction to logarithms, change of base formula · 1 question
Topic 1 - Number and Algebra: 1.6 Simple proofs and algebraic manipulation (direct proof) · 1 question
Topic 1 - Number and Algebra: 1.7 The binomial theorem for positive integer powers n · 1 question
Topic 2 - Functions: 2.1 Concept of a function, domain, range, function notation f(x) · 1 question
Topic 2 - Functions: 2.2 Graphs of functions, features (intercepts, extrema), composite functions, identity function, inverse functions f^(-1) · 1 question
Topic 2 - Functions: 2.3 Transformations of graphs: f(x) + a, f(x + a), af(x), f(ax), -f(x), f(-x) · 1 question
Topic 2 - Functions: 2.4 Rational functions of the form f(x) = (ax + b)/(cx + d), equations of vertical and horizontal asymptotes · 1 question
Topic 2 - Functions: 2.5 Quadratic functions: standard, vertex, and factored forms; axis of symmetry; discriminant and nature of roots · 1 question
Topic 2 - Functions: 2.6 Exponential functions and graphs (base e and other bases), logarithmic functions and graphs · 1 question
Topic 2 - Functions: 2.7 Solving equations analytically and graphically using technology · 1 question
Topic 3 - Geometry and Trigonometry: 3.1 3D space: coordinates, midpoint, distance formula, volume and surface area of standard 3D solids · 1 question
Topic 3 - Geometry and Trigonometry: 3.2 Right-angled trigonometry, sine rule, cosine rule, area of triangle (1/2 ab sin C) · 1 question
Topic 3 - Geometry and Trigonometry: 3.3 Radian measure, arc length, sector area · 1 question
Topic 3 - Geometry and Trigonometry: 3.5 Circular functions: graphs of sin x, cos x, tan x, amplitude, period, phase shift · 1 question
Topic 3 - Geometry and Trigonometry: 3.6 Solving trigonometric equations in a finite interval · 1 question
Topic 4 - Statistics and Probability: 4.3 Measures of central tendency (mean, median, mode) and dispersion (range, IQR, standard deviation, variance) · 1 question
Topic 4 - Statistics and Probability: 4.4 Linear correlation and linear regression: scatter diagrams, Pearson's product-moment correlation coefficient r, line of regression of y on x · 1 question
Topic 4 - Statistics and Probability: 4.5 Concepts of probability: trial, outcome, sample space, complementary events, Venn diagrams, tree diagrams · 1 question
Topic 4 - Statistics and Probability: 4.6 Conditional probability P(A|B), independent events, mutually exclusive events · 1 question
Topic 4 - Statistics and Probability: 4.8 Binomial distribution B(n, p), mean and variance · 1 question
Topic 4 - Statistics and Probability: 4.9 Normal distribution, properties, standardized normal variable z, finding probabilities and inverse normal calculations · 1 question
Topic 5 - Calculus: 5.1 Introduction to limits and derivative as rate of change / gradient of a curve · 1 question
Topic 5 - Calculus: 5.2 Increasing and decreasing functions, derivative of f(x) = ax^n · 1 question
Topic 5 - Calculus: 5.4 Differentiation of sin x, cos x, e^x, ln x, and rules: constant multiple, sum/difference, product, quotient, and chain rule · 1 question
Topic 5 - Calculus: 5.7 Indefinite integration as anti-differentiation, standard integrals (x^n, 1/x, sin x, cos x, e^x) · 1 question
Topic 5 - Calculus: 5.8 Definite integrals and analytical evaluation, area between a curve and the x-axis, area between two curves · 1 question

How the marking works

Each question is marked the moment it is answered, and the explanation appears straight away while the question is still fresh. A test that only gives a score at the end teaches a student what they scored; a test that explains each answer teaches them why. At the end you get a marked paper listing every question that was missed, with a topic-by-topic breakdown of strengths and focus areas, so the next revision session has somewhere obvious to start.

A question from this quiz

An arithmetic sequence has third term u_3 = 11 and eighth term u_8 = 31. What is the sum of the first 10 terms, S_10?

Answer: 210

The common difference is d = (31 - 11) / 5 = 4, giving first term u_1 = 11 - 2(4) = 3. Using the sum formula yields S_10 = (10 / 2)(2(3) + 9(4)) = 5(42) = 210.

How to use it at home

Sit the quiz once cold, without notes, and keep the marked paper. Revise only the topics the paper flags, then take it again a week later: the second score is the one worth talking about, because it measures what changed rather than what was already known. The marked paper is often more useful than the score itself, since it names the topic rather than saying a student is "weak in IB Diploma Mathematics: Analysis and Approaches SL".

Related on this site: the IB MYP Maths Quiz (Years 4–5), all free quizzes, IB tuition.

Questions parents ask

Is this IB Maths AA SL Quiz free?

Yes. It is completely free and works without an account. Log in with a free account if you want your scores saved, with points, streaks and badges; otherwise the score stays on your own device.

How long does it take?

Most students finish the 31 questions in about 20 minutes. You can stop part way and resume later on the same device, because your progress is saved in the browser.

Which class and board is it for?

It is written for IB DP AA SL students following IB Diploma, covering IB Diploma Mathematics: Analysis and Approaches SL, in a non-calculator multiple-choice style. Any student revising the same topics can use it.

Are the answers explained?

Yes. Every question is marked as soon as it is answered and each answer carries a short explanation, and the marked paper at the end lists every question that was missed.