1. Mathematical Induction & Sequences | Proving by induction that a given sequence satisfies a closed-form expression.

2. Vectors & Planes | Finding the equation of a plane given a line and a parallel vector, and calculating the acute angle between a line and a plane.

3. Polynomials & Root Properties | Computing sums of root products and evaluating higher powers of roots for a given polynomial equation.

4. Matrix Transformations & Invariant Lines | Performing matrix multiplication, identifying invariant lines, and determining a transformation matrix.

5. Summation & Series Convergence | Finding a general summation formula, determining convergence conditions, and evaluating an infinite series.

6. Curve Analysis & Asymptotes | Examining asymptotes, finding stationary points, sketching curves, and solving inequalities involving rational functions.