IB Mathematics: Analysis and Approaches Higher Level Paper 1 (May 2026)

Đề thi chính thức IB Diploma Program May 2026 - Non-Calculator Examination

IB Mathematics
Past Papers
IB Math AA HL
Exam Papers
Author

NKK-Education Hub

Published

May 15, 2026

Note📝 Thông Tin Đề Thi IB Math AA HL Paper 1 (May 2026)
  • Thời gian làm bài: 2 giờ (120 phút)
  • Tổng điểm: 110 marks
  • Quy định: Không sử dụng máy tính bỏ túi (Non-Calculator Paper).
  • Cấu trúc: Section A (9 câu hỏi ngắn, 68 điểm) & Section B (3 câu hỏi mở rộng, 52 điểm).

Section A

Question 1 [Maximum mark: 5]

Consider the function \(f(x)=3(x-2)(x-q)\), where \(x \in \mathbb{R}\) and \(q\) is a real constant.

The axis of symmetry of the graph of \(f\) has equation \(x=4\).

  1. Show that \(q=6\). [2 marks]

  2. Find the coordinates of the vertex of the graph of \(f\). [2 marks]

  3. Hence, write down the range of \(f\). [1 mark]

Question 2 [Maximum mark: 5]

A particle P is travelling along a straight line, with displacement measured relative to a point O on the line. The velocity, \(v \text{ m s}^{-1}\), of the particle at time \(t\) seconds is given by

\[v=\frac{1}{t+1}+e^{-t}\]

where \(t \ge 0\).

  1. Find \(\int\left(\frac{1}{t+1}+e^{-t}\right)dt\). [2 marks]

Initially the particle is at O.

  1. Find an expression for the displacement, \(s\) metres, in terms of \(t\). [3 marks]

Question 3 [Maximum mark: 6]

Consider a geometric sequence, \(u_n\), with common ratio \(r\), where each term in the sequence is positive.

It is given that \(u_6=1.6 \times 10^5\) and \(u_8=6.4 \times 10^5\).

  1. Find the value of \(r\). [3 marks]

  2. Find the value of \(u_1\), giving your answer in the form \(a \times 10^k\), where \(1 \le a < 10\) and \(k \in \mathbb{Z}\). [3 marks]

Question 4 [Maximum mark: 7]

  1. Show that \(\log_{25}(81x^2)=\log_5{9}+\log_5{x}\) for \(x>0\). [4 marks]

  2. Hence, solve the equation \(\log_5{\sqrt[3]{x}}=\log_{25}(81x^2)-\log_5{4}\) where \(x>0\). [3 marks]

Question 5 [Maximum mark: 9]

Consider two events, A and B.

It is given that \(P(A' \cap B)=\frac{1}{6}\), \(P(A \cap B)=\frac{7-x}{12}\) and \(P(A \cap B')=\frac{5+x}{24}\), where \(0 \le x \le 7\).

  1. Show that \(P(A)=\frac{19-x}{24}\). [3 marks]

It is given that A and B are independent.

  1. Find the possible values of \(x\). [6 marks]

Question 6 [Maximum mark: 4]

Consider a piece of paper in the shape of a circle with centre O and radius \(r\) cm, where \(r>6\). A second circle with centre O and radius 6 cm is drawn inside the first and shaded.

The points A and B are on the circumference of the larger circle such that the acute angle \(A\hat{O}B=\frac{\pi}{3}\). The paper is cut along the lines AO and BO and the sector AOB with a central angle of \(\frac{\pi}{3}\) is removed.

After removing the sector, it is now given that

\[\frac{\text{the area of the unshaded region}}{\text{the area of the shaded region}}=\frac{2}{3}\]

Find the value of \(r\), giving your answer in the form \(\sqrt{k}\), where \(k \in \mathbb{Z}\).

Question 7 [Maximum mark: 6]

Reyn is investigating the fertility of a group of adult female foxes. He has developed a clinical test that aims to identify whether or not a given fox is pregnant, and he wants to evaluate the reliability of this test.

Reyn knows that:

  • 40% of the foxes in the group are pregnant
  • if a fox is pregnant, the probability that it will test positive is 90%
  • if a fox is not pregnant, the probability it will test positive is 20%.

  1. Find the probability that a randomly selected fox from the group will test positive. [3 marks]

  2. Find the probability that a randomly selected fox from the group is pregnant, given that it tested positive. [3 marks]

Question 8 [Maximum mark: 8]

Consider the polynomial \(p(z)=3z^5-7z^4+cz^3+dz^2+ez-26\) where \(z \in \mathbb{C}\) and \(c, d, e \in \mathbb{R}\).

Three of the roots of \(p(z)\) are \(\frac{1}{3}\), \(a-i\), \(2+bi\), where \(a, b \in \mathbb{R}\), \(b>1\).

  1. By considering the sum and the product of the roots of \(p(z)\),
    1. show that \(a=-1\);
    2. find the value of \(b\). [5 marks]
  2. Show that \(p(z)=(Az+B)(z^2+Cz+D)(z^2+Ez+F)\), where \(A, B, C, D, E, F\) are integers to be determined. [3 marks]

Question 9 [Maximum mark: 8]

Consider non-zero vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) in three-dimensional space.

The non-zero vector \(\mathbf{v}\) is defined as \(\mathbf{v}=(\mathbf{a} \cdot \mathbf{c})\mathbf{b}-(\mathbf{a} \cdot \mathbf{b})\mathbf{c}\).

  1. Show that \(\mathbf{v} \cdot \mathbf{a}=0\). [2 marks]

It is given that \(\mathbf{b} \times \mathbf{c}\) and \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})\) are non-zero vectors.

    1. Justify why \(\mathbf{b} \cdot (\mathbf{b} \times \mathbf{c})=0\).
    2. Show that \(\mathbf{v} \cdot (\mathbf{b} \times \mathbf{c})=0\). [4 marks]
  2. Hence, state the geometrical relationship between \(\mathbf{v}\) and \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})\), briefly justifying your answer. [2 marks]

Section B

Question 10 [Maximum mark: 15]

Consider the function \(f(x)=ax^3+bx^2+cx+d\), where \(x \in \mathbb{R}\) and where \(a, b, c\) and \(d\) are real constants.

The graph of \(f\) has a point of inflexion at \((-2,-2c+d+24)\).

  1. Show that \(a=\frac{3}{2}\) and \(b=9\). [6 marks]

It is given that the tangents to the graph of \(f\) at \(x=-3\) and \(x=k\) are horizontal.

    1. Show that \(c=\frac{27}{2}\).
    2. Find the value of \(k\).
    3. State whether \(f\) has a local maximum or a local minimum at \(x=k\), justifying your answer. [7 marks]

The graph of \(f\) intersects the y-axis at the point P.

  1. Show that the tangent to the graph of \(f\) at \(x=-3\) passes through P. [2 marks]

Question 11 [Maximum mark: 19]

  1. Prove the identity \(\cot{2\theta} \equiv \frac{1}{2}\left(\cot{\theta}-\frac{1}{\cot{\theta}}\right)\). [3 marks]

  2. Hence, solve the equation \(\cot{2\theta}=\frac{3}{2}\cot{\theta}(\cot^2{\theta}-1)\) for \(0 < \theta < \pi\) where \(\theta \ne \frac{\pi}{2}\). [6 marks]

  3. By differentiating both sides of the identity in part (a), show that \(4\csc^2{2\theta} \equiv \csc^2{\theta}+\sec^2{\theta}\). [5 marks]

Consider the right-angled triangle \(\Delta PQR\), where \(R\hat{P}Q=\frac{\pi}{2}\), \(PR=\csc{\theta}\) and \(QR=2\csc{2\theta}\), where \(0 < \theta < \frac{\pi}{2}\) and all lengths are given in metres.

QR has length \(L\) metres and \(\Delta PQR\) has area \(A\) square metres.

  1. Calculate the ratio \(L:A\) in the form \(n:1\) where \(n \in \mathbb{Z}^+\). [5 marks]

Question 12 [Maximum mark: 18]

Rowan is investigating the spread of an infection through a population of rabbits. He models \(N\), the number (in thousands) of infected rabbits in the population, at time \(t\) days.

Rowan models the spread of the infection by the differential equation

\[\frac{dN}{dt}=k(-3+4N-N^2)\]

where \(t \ge 0\) and \(k \in \mathbb{R}^+\).

Initially, 1500 rabbits are infected.

  1. Express \(\frac{1}{-3+4x-x^2}\) in the form \(\frac{A}{x-1}+\frac{B}{3-x}\), where \(A, B \in \mathbb{R}\). [3 marks]

  2. Hence, show that \(\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln{p}\), where \(p \in \mathbb{R}^+\) is a constant to be determined. [8 marks]

  3. Find an expression for \(N\) in the form \(N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}\) where \(a, b, c \in \mathbb{Z}\). [5 marks]

According to Rowan’s model, the number of infected rabbits approaches a limit, \(L\), over the long term.

  1. Find the value of \(L\). [2 marks]