Mock Test (AAHL, AASL) || Functions, Sequences, Algebra, Trigonometry, Differentiation, Vectors, Complex Numbers

DP Mathematics AA HL – Practice Set

Note: This worksheet excludes Integration, Statistics, and Probability.

Section A: Algebra & Functions

  1. Solve for $x \in \mathbb{R}$:

    $ \log_2(x^2 - 5x + 6) = 1 $
  2. Given $ f(x) = \dfrac{2x + 3}{x - 1} $:
    (a) Find the inverse function $ f^{-1}(x) $
    (b) State the domain and range of $ f(x) $
  3. Let $ f(x) = ax^3 + bx^2 + cx + d $, with a local maximum at $ x = 1 $ and an inflection point at $ x = 2 $.
    Given $ f(1) = 4 $, find $ a, b, c, d $.

Section B: Sequences & Series

  1. Given $ u_n = 2^n + 3n $, find the value of $ \sum_{n=1}^{6} u_n $.
  2. The first three terms of a geometric sequence are $ x, 2x - 3, 4x - 6 $.
    Find the value of $ x $ and the common ratio $ r $.

Section C: Trigonometry

  1. Solve for $ \theta \in [0^\circ, 360^\circ] $:

    $ 2\sin\theta - \sqrt{3} = 0 $
  2. Prove the identity:

    $ \dfrac{1 + \tan^2x}{1 - \tan^2x} = \sec(2x) $
  3. The angle of elevation from the ground to a tower is $ 30^\circ $.
    After walking 50 m closer, the angle becomes $ 45^\circ $.
    Find the height of the tower.

Section D: Differentiation

  1. Differentiate:

    $ f(x) = \dfrac{x^2 \sin x}{e^x} $
  2. Given $ f'(x) = 6x^2 - 18x $, find the stationary points and determine their nature.
  3. Find the derivative of:

    $ f(x) = \ln\left(\sqrt{x^2 + 4}\right) $

Section E: Vectors

  1. Let $ \vec{a} = (3, -1, 2), \vec{b} = (1, 4, -2) $:
    (a) Find $ \vec{a} \cdot \vec{b} $
    (b) Find the angle between vectors $ \vec{a} $ and $ \vec{b} $
  2. A line passes through point $ A(1, 2, -1) $ with direction vector $ \vec{d} = (2, -1, 3) $.
    Find the Cartesian equation of the line.

Section F: Complex Numbers

  1. Solve for $ z \in \mathbb{C} $:

    $ z^2 + (3 - 2i)z + (4 + i) = 0 $
  2. Express $ \dfrac{3 + 4i}{1 - 2i} $ in the form $ a + bi $, where $ a, b \in \mathbb{R} $.

Instructions

  • Show all working clearly and neatly.
  • Use GDC for verification only—full reasoning must be shown.
  • Use correct mathematical notation at all times.