About this IB Math practice test

Whether you're on the AA (Analysis & Approaches) or AI (Applications & Interpretation) track, both share a large core syllabus of calculus, geometry, and probability. These problems cover that shared ground, with worked solutions that show how to lay out full analytical working for Paper 1 and technology-based reasoning for Paper 2.

IB Mathematics Practice Test sample questions

These starter questions help you launch a math mock test quickly. Swap them with your own worksheet, notebook, or textbook questions any time.

1. 1. Find the sum of the arithmetic sequence: 3, 7, 11, ... up to 15 terms.

2. 2. Solve: log₂(x + 3) = 4

3. 3. Differentiate f(x) = x³e^x using the product rule.

4. 4. Find ∫(3x² - 2x + 1) dx.

5. 5. A geometric sequence has first term 4 and common ratio 0.5. Find the sum to infinity.

6. 6. Solve the equation 2sin(x) + 1 = 0 for 0 ≤ x ≤ 2π.

7. 7. X ~ B(12, 0.4). Find P(X = 5).

8. 8. Find the equation of the tangent to y = x² - 4x + 3 at x = 3.

9. 9. A vector has components (3, -4). Find its magnitude and unit vector.

10. 10. Expand (2x - 1)⁴ using the binomial theorem, giving the first three terms.

11. 11. The probability that it rains is 0.3. If it rains, P(late) = 0.7; if no rain, P(late) = 0.2. Find P(late).

12. 12. Find the inverse function of f(x) = 3x - 5.

13. 13. The area between y = x² and y = x is bounded by x = 0 and x = 1. Find the exact area.

14. 14. Solve: e^(2x) - 5e^x + 6 = 0

15. 15. Using the cosine rule, find angle A in a triangle with sides a = 7, b = 5, c = 8.

16. 16. Solve the equation log₃(x) + log₃(x - 2) = 1.

17. 17. A curve satisfies the differential equation dy/dx = 2xy. Given that y = 3 when x = 0, find y in terms of x.

18. 18. (AA HL) Simplify (3 + 2i)(1 - 4i), giving your answer in the form a + bi.

19. 19. (AA HL) Prove by mathematical induction that 1 + 3 + 5 + ... + (2n - 1) = n² for all positive integers n.

20. 20. Find the domain and range of the function f(x) = √(9 - x²).

21. 21. Given f(x) = 2x + 1 and g(x) = x² - 3, find an expression for (g ∘ f)(x) and evaluate (g ∘ f)(2).

22. 22. A bacteria population is modelled by P(t) = 200e^(0.15t), where P is the population and t is the time in hours. Find, to 2 decimal places, the time taken for the population to reach 1000.

23. 23. Find the equations of the vertical and horizontal asymptotes of the graph of f(x) = (3x + 2)/(x - 4).

24. 24. Solve the equation 2cos²(x) - 1 = 0 for 0 ≤ x ≤ 2π.

25. 25. Points A(1, 2, 3) and B(4, -1, 5) are given. Find the vector AB and its magnitude, giving your answer as an exact value.

26. 26. (AI) In a Voronoi diagram with two sites P(0, 0) and Q(6, 0), find the equation of the boundary edge (perpendicular bisector) that separates the two regions.

27. 27. The random variable X is normally distributed with mean 50 and standard deviation 4. Find P(46 < X < 58).

28. 28. The regression line of y on x for a set of bivariate data is y = 2.5x + 3. Use this line to predict the value of y when x = 8.

29. 29. (HL) A chi-squared test for independence between two categorical variables gives a test statistic of χ² = 8.92 with 3 degrees of freedom. The critical value at the 5% significance level is 7.815. State, with a reason, whether the null hypothesis of independence should be rejected.

30. 30. Use the quotient rule to differentiate f(x) = (2x + 1)/(x² + 3).

Answer key & quick explanations

Short answers for the sample questions above. Use this to self-check before generating a fresh AI-built mock test.

1. 1. Arithmetic sequence sum, a=3, d=4, n=15

   S₁₅ = 465

   The sequence has first term a=3 and common difference d=4. Using Sₙ = (n/2)[2a+(n-1)d], S₁₅ = (15/2)[2(3)+14(4)] = 7.5×(6+56) = 7.5×62 = 465.

2. 2. Logarithmic equation log₂(x+3)=4

   x = 13

   Rewriting in exponential form gives x+3 = 2⁴ = 16, so x = 16-3 = 13. Checking the domain, x+3=16>0, so the solution is valid.

3. 3. Product rule differentiation of x³eˣ

   f'(x) = x²eˣ(x+3)

   Using the product rule with u=x³ (u'=3x²) and v=eˣ (v'=eˣ), f'(x) = u'v+uv' = 3x²eˣ + x³eˣ. Factoring out x²eˣ gives f'(x) = x²eˣ(x+3).

4. 4. Indefinite integral of 3x²-2x+1

   x³ - x² + x + C

   Integrating term by term using ∫xⁿdx = x^(n+1)/(n+1): ∫3x²dx=x³, ∫-2x dx=-x², ∫1 dx=x. Adding the constant of integration gives x³-x²+x+C.

5. 5. Sum to infinity of geometric sequence, a=4, r=0.5

   S∞ = 8

   Since |r|=0.5<1 the series converges. Using S∞ = a/(1-r) = 4/(1-0.5) = 4/0.5 = 8.

6. 6. Trigonometric equation 2sin(x)+1=0

   x = 7π/6, 11π/6

   Rearranging gives sin(x) = -1/2. The reference angle is π/6, and since sine is negative in the third and fourth quadrants, the solutions in [0,2π] are x=π+π/6=7π/6 and x=2π-π/6=11π/6.

7. 7. Binomial probability X~B(12,0.4), P(X=5)

   P(X=5) ≈ 0.227

   Using P(X=k) = C(n,k)pᵏ(1-p)ⁿ⁻ᵏ with n=12, k=5, p=0.4: P(X=5) = C(12,5)(0.4)⁵(0.6)⁷ = 792 × 0.01024 × 0.0279936 ≈ 0.227.

8. 8. Tangent line to y=x²-4x+3 at x=3

   y = 2x - 6

   The derivative is y'=2x-4, so at x=3 the gradient is 2(3)-4=2. The point on the curve is (3, 3²-4(3)+3)=(3,0). Using y-y₁=m(x-x₁): y-0=2(x-3), giving y=2x-6.

9. 9. Magnitude and unit vector of (3,-4)

   |v| = 5; unit vector = (3/5, -4/5)

   The magnitude is |v|=√(3²+(-4)²)=√(9+16)=√25=5. The unit vector is found by dividing each component by the magnitude: (3/5, -4/5).

10. 10. Binomial expansion of (2x-1)⁴, first three terms

    16x⁴ - 32x³ + 24x²

    Using the binomial theorem (2x-1)⁴=Σ C(4,k)(2x)^(4-k)(-1)^k. The first three terms (k=0,1,2) are C(4,0)(2x)⁴=16x⁴, C(4,1)(2x)³(-1)=-32x³, and C(4,2)(2x)²(1)=24x².

11. 11. Conditional probability tree, P(late)

    P(late) = 0.35

    Using the law of total probability, P(late) = P(rain)×P(late|rain) + P(no rain)×P(late|no rain) = (0.3×0.7)+(0.7×0.2) = 0.21+0.14 = 0.35.

12. 12. Inverse function of f(x)=3x-5

    f⁻¹(x) = (x+5)/3

    Let y=3x-5. Solving for x: x=(y+5)/3. Swapping x and y and relabeling gives f⁻¹(x)=(x+5)/3.

13. 13. Area between y=x² and y=x on [0,1]

    Area = 1/6

    On [0,1], x≥x², so the area is ∫₀¹(x-x²)dx = [x²/2 - x³/3]₀¹ = (1/2-1/3) - 0 = 1/6.

14. 14. Exponential equation e^(2x)-5eˣ+6=0

    x = ln 2 or x = ln 3

    Substituting u=eˣ gives u²-5u+6=0, which factors as (u-2)(u-3)=0, so u=2 or u=3. Since eˣ=u, x=ln2 or x=ln3.

15. 15. Cosine rule, find angle A (a=7,b=5,c=8)

    A = 60°

    Using the cosine rule a²=b²+c²-2bc·cosA, rearranged as cosA=(b²+c²-a²)/(2bc) = (25+64-49)/(2×5×8) = 40/80 = 0.5. So A = cos⁻¹(0.5) = 60°.

16. 16. Logarithmic equation log₃(x)+log₃(x-2)=1

    x = 3

    Combining logs gives log₃[x(x-2)]=1, so x(x-2)=3¹=3, i.e. x²-2x-3=0, which factors as (x-3)(x+1)=0, giving x=3 or x=-1. Since log₃(x-2) requires x>2, x=-1 is rejected, leaving x=3.

17. 17. (HL) Separable differential equation dy/dx=2xy

    y = 3e^(x²)

    Separating variables gives (1/y)dy = 2x dx. Integrating both sides: ln|y| = x² + C. Substituting y=3 when x=0 gives C=ln3, so ln|y| = x²+ln3, which gives y = 3e^(x²).

18. 18. (AA HL) Multiplying complex numbers (3+2i)(1-4i)

    11 - 10i

    Expanding using the distributive law: (3+2i)(1-4i) = 3(1) + 3(-4i) + 2i(1) + 2i(-4i) = 3 -12i + 2i -8i². Since i²=-1, -8i²=8, so combining real and imaginary parts gives (3+8) + (-12+2)i = 11 - 10i.

19. 19. (AA HL) Proof by induction: sum of first n odd numbers = n²

    True for all positive integers n (proven by induction)

    Base case n=1: LHS=1, RHS=1²=1, so true. Assume true for n=k: 1+3+...+(2k-1)=k². For n=k+1: k² + (2(k+1)-1) = k²+2k+1 = (k+1)², which matches the RHS for n=k+1. Since the base case holds and the statement holds for n=k+1 whenever it holds for n=k, by the principle of mathematical induction it is true for all positive integers n.

20. 20. Domain and range of f(x)=√(9-x²)

    Domain: -3 ≤ x ≤ 3; Range: 0 ≤ f(x) ≤ 3

    For the square root to be defined, 9-x²≥0, i.e. x²≤9, giving -3≤x≤3 as the domain. The maximum value of 9-x² is 9 (at x=0) giving f(x)=3, and the minimum is 0 (at x=±3), so the range is [0,3].

21. 21. Composite function (g∘f)(x) and evaluation at x=2

    (g∘f)(x) = 4x²+4x-2; (g∘f)(2) = 22

    Substituting f(x)=2x+1 into g gives (g∘f)(x) = g(2x+1) = (2x+1)²-3 = 4x²+4x+1-3 = 4x²+4x-2. Evaluating at x=2: 4(2)²+4(2)-2 = 16+8-2 = 22.

22. 22. Exponential growth model, time to reach population 1000

    t ≈ 10.73 hours

    Setting 200e^(0.15t)=1000 gives e^(0.15t)=5. Taking natural logs: 0.15t = ln5, so t = ln5/0.15 ≈ 1.6094/0.15 ≈ 10.73 hours.

23. 23. Vertical and horizontal asymptotes of (3x+2)/(x-4)

    Vertical asymptote: x = 4; Horizontal asymptote: y = 3

    The vertical asymptote occurs where the denominator is zero: x-4=0, so x=4. For the horizontal asymptote, as x→±∞ the function approaches the ratio of the leading coefficients, 3x/x=3, so y=3.

24. 24. Trigonometric identity equation 2cos²(x)-1=0

    x = π/4, 3π/4, 5π/4, 7π/4

    Rearranging gives cos²x = 1/2, so cosx = ±√2/2. In the interval [0,2π], cosx=√2/2 at x=π/4 and x=7π/4, and cosx=-√2/2 at x=3π/4 and x=5π/4.

25. 25. Vector AB between two 3D points and its magnitude

    AB = (3, -3, 2); |AB| = √22

    The vector AB is found by subtracting position vectors: AB = B - A = (4-1, -1-2, 5-3) = (3,-3,2). The magnitude is |AB| = √(3²+(-3)²+2²) = √(9+9+4) = √22.

26. 26. (AI) Voronoi diagram boundary edge between two sites

    x = 3

    The boundary edge of a Voronoi diagram between two sites is the perpendicular bisector of the segment joining them. The midpoint of P(0,0) and Q(6,0) is (3,0), and since PQ lies along the x-axis, the perpendicular bisector is the vertical line x=3.

27. 27. Normal distribution probability P(46<X<58)

    P(46 < X < 58) ≈ 0.819

    Standardising using Z=(X-μ)/σ gives z-values of (46-50)/4=-1 and (58-50)/4=2. Using the standard normal distribution, P(Z<2)≈0.9772 and P(Z<-1)≈0.1587, so P(46<X<58) = 0.9772-0.1587 ≈ 0.819.

28. 28. Regression line prediction y=2.5x+3 at x=8

    y = 23

    Substituting x=8 into the regression equation y=2.5x+3 gives y=2.5(8)+3 = 20+3 = 23.

29. 29. (HL) Chi-squared test for independence, decision at 5% level

    Reject H0; there is evidence of association between the variables

    Since the test statistic χ²=8.92 exceeds the critical value of 7.815 at the 5% significance level with 3 degrees of freedom, the null hypothesis of independence is rejected. This indicates significant evidence of an association between the two categorical variables.

30. 30. Quotient rule differentiation of (2x+1)/(x²+3)

    f'(x) = (-2x² - 2x + 6)/(x²+3)²

    Using the quotient rule with u=2x+1 (u'=2) and v=x²+3 (v'=2x): f'(x) = (u'v-uv')/v² = [2(x²+3) - (2x+1)(2x)]/(x²+3)². Expanding the numerator: 2x²+6-4x²-2x = -2x²-2x+6, giving f'(x) = (-2x²-2x+6)/(x²+3)².

Curriculum Mapping & Learning Guide

Use this breakdown to identify which skills each question tests and guide post-test review.

IB Math units covered

1. Chapter 1: Number & Algebra: sequences (arithmetic, geometric), binomial theorem, logarithms
2. Chapter 2: Number & Algebra: complex numbers (AA HL), proof by induction (AA HL)
3. Chapter 3: Functions: domain/range, inverse, composite, transformations
4. Chapter 4: Functions: exponential, logarithmic, rational, modelling
5. Chapter 5: Geometry & Trigonometry: sine/cosine rule, unit circle, identities
6. Chapter 6: Geometry & Trigonometry: vectors, 3D geometry, Voronoi diagrams (AI)
7. Chapter 7: Statistics & Probability: binomial and normal distributions, hypothesis testing
8. Chapter 8: Statistics & Probability: regression, correlation, chi-squared test (HL)
9. Chapter 9: Calculus: differentiation (product, chain, quotient), integration, differential equations (HL)

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