Free IB Maths AA SL Paper 1 Practice Exam (PDF)

This free IB Mathematics: Analysis and Approaches SL practice exam mirrors Paper 1: no calculator allowed, a Section A of short questions and a Section B extended-response question on calculus. It covers sequences, functions, logarithms, trigonometry, differentiation, and probability — and every answer resolves to an exact value, just like the real paper. Each part has a markscheme-style worked solution so you can self-mark. View it online below or download the print-ready PDF; no account needed.

1. Question 16 marks · Arithmetic sequences & series
   An arithmetic sequence has third term $u_3 = 11$ and seventh term $u_7 = 23$.
   (a)

   Find the first term $u_1$ and the common difference $d$.

   [4]
   (b)

   Find $S_{20}$, the sum of the first 20 terms.

   [2]
2. Question 25 marks · Composite & inverse functions
   The functions $f$ and $g$ are defined by $f(x) = 2x - 3$ for $x \in \mathbb{R}$, and $g(x) = \frac{x + 1}{x - 2}$ for $x \neq 2$.
   $f(x) = 2x - 3, \qquad g(x) = \frac{x + 1}{x - 2}, \quad x \neq 2$
   (a)

   Find $(g \circ f)(3)$.

   [2]
   (b)

   Find $g^{-1}(x)$, stating its domain.

   [3]
3. Question 36 marks · Exponents & logarithms
   This question is about exponents and logarithms.
   (a)

   Solve $3^{2x - 1} = 27^{x - 2}$.

   [3]
   (b)

   Show that $2\log_3 6 - \log_3 4 = 2$.

   [3]
4. Question 45 marks · Trigonometry: exact values
   This question is about trigonometry with exact values.
   (a)

   Solve $2\cos x = \sqrt{3}$ for $0 \leq x \leq 2\pi$.

   [3]
   (b)

   In triangle $ABC$, $AB = 6$ cm, $AC = 8$ cm and $B\hat{A}C = \frac{\pi}{3}$. Find the exact area of the triangle.

   [2]
5. Question 56 marks · Differentiation: tangents & normals
   The curve $y = f(x)$ where $f(x) = x^3 - 3x + 1$ passes through the point $P(2, 3)$.
   $f(x) = x^3 - 3x + 1$
   (a)

   Find $f'(x)$.

   [2]
   (b)

   Find the equation of the tangent to the curve at $P$.

   [2]
   (c)

   Find the equation of the normal to the curve at $P$.

   [2]
6. Question 66 marks · Conditional probability & independence
   The probability that it rains on a given morning is $\frac{1}{4}$. If it rains, the probability that Mia cycles to school is $\frac{1}{5}$; if it does not rain, the probability that she cycles is $\frac{3}{5}$. Let $R$ be the event "it rains" and $C$ the event "Mia cycles". A tree diagram may help.
   (a)

   Find $P(C)$, the probability that Mia cycles to school.

   [2]
   (b)

   Given that Mia cycles to school, find the probability that it rained.

   [2]
   (c)

   Determine, with a reason, whether the events $R$ and $C$ are independent.

   [2]
7. Question 714 marks · Calculus of a cubic curve
   The function $f$ is defined by $f(x) = x^3 - 6x^2 + 9x$ for $x \in \mathbb{R}$. The curve $y = f(x)$ is studied throughout this question.
   $f(x) = x^3 - 6x^2 + 9x$
   (a)

   Find $f'(x)$, and hence find the coordinates of the two stationary points of the curve.

   [4]
   (b)

   Justify the nature of each stationary point.

   [2]
   (c)

   Find $f''(x)$, and hence find the coordinates of the point of inflexion.

   [2]
   (d)

   Find the exact area of the region enclosed by the curve and the $x$-axis.

   [4]
   (e)

   Find the equation of the tangent to the curve at the point of inflexion.

   [2]