MathematicsGrade 12English

IB DP Mathematics: Analysis & Approaches HL Paper 1 (Sample) 2023

In this sample paper, you'll face the exact mix of algebra, functions, and calculus that appears in real IB exams—the non-calculator section where one small error in your working can cost 2 marks. We've packed 15 questions across three core syllabus strands: polynomial and rational functions, exponential and logarithmic equations, and calculus fundamentals. You'll see mostly multiple-choice, plus a couple of true/false and fill-in-the-blank items designed to catch the shortcuts students take. By the end, you'll spot the common trap answers instantly and know which working steps the examiners actually mark.

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Question 1
Let $f(x) = 2x^3 - 3x^2 - 12x + 5$ . Which of the following statements is true about the graph of $f(x)$ ?
Question 2
The function $g(x) = \frac{x^2 - 4}{x^2 - 2x - 3}$ has a vertical asymptote at:
Question 3
What is the remainder when $p(x) = x^4 + 2x^3 - 5x + 7$ is divided by $(x - 2)$ ?
Question 4
Solve $3^{2x} = 27$ .
Question 5
Which of the following is equal to $\log_4 16$ ?
Question 6
True or False: $\log_5(25) + \log_5(4) = \log_5(100)$ .
Question 7
If $f(x) = e^{2x}$ , then $f'(x) =$
Question 8
Find the x-coordinate of the critical point of $h(x) = x^2 e^{-x}$ .
Question 9
A curve has equation $y = x^3 - 6x^2 + 9x$ . At how many points does the tangent to the curve have a gradient of 0?
Question 10
True or False: The function $f(x) = -x^4 + 8x^2 - 10$ has a local minimum at $x = 0$ .
Question 11
If $\frac{dy}{dx} = 3x^2 + 4x$ and $y(0) = 5$ , what is the value of $y(1)$ ?
Question 12
The curve $y = 2x^3 - 9x^2 + 12x - 1$ has an inflection point at $x =$
Question 13
Which statement correctly describes the end behaviour of $f(x) = -2x^5 + 3x^3 + 1$ ?
Question 14
Solve $\log_2(x - 1) + \log_2(x + 1) = 3$ for $x$ .