DP Mathematics AA HL – Practice Set
Note: This worksheet excludes Integration, Statistics, and Probability.
🔢 Section A: Algebra & Functions
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Solve for \( x \in \mathbb{R} \):
\( \log_2(x^2 - 5x + 6) = 1 \) -
Given \( f(x) = \dfrac{2x + 3}{x - 1} \):
(a) Find the inverse function \( f^{-1}(x) \)
(b) State the domain and range of \( f(x) \) -
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
- Given \( u_n = 2^n + 3n \), find the value of \( \sum_{n=1}^{6} u_n \).
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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
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Solve for \( \theta \in [0^\circ, 360^\circ] \):
\( 2\sin\theta - \sqrt{3} = 0 \) -
Prove the identity:
\( \dfrac{1 + \tan^2x}{1 - \tan^2x} = \sec(2x) \) -
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
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Differentiate:
\( f(x) = \dfrac{x^2 \sin x}{e^x} \) - Given \( f'(x) = 6x^2 - 18x \), find the stationary points and determine their nature.
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Find the derivative of:
\( f(x) = \ln\left(\sqrt{x^2 + 4}\right) \)
🧭 Section E: Vectors
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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 \( \vec{b} \) -
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
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Solve for \( z \in \mathbb{C} \):
\( z^2 + (3 - 2i)z + (4 + i) = 0 \) - 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.
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