Overview
This exam assesses students' ability to apply algebraic methods, including solving equations, simplifying expressions, working with polynomials, and using logarithms to solve real-life problems.
Question One: Algebraic Manipulation and Polynomials
- Convert x² + 8x − 5 to the form (x + p)² + q (completing the square).
- Find the discriminant of 3x² + 2 = 5x (use b² − 4ac).
- Express 1/(t + t − 5) as a single fraction over t² − 13t.
- Given f(x) = 3x³ + ax² + bx + c with roots at x = 2, 13, 4, find a, b, c using Vieta’s formulas.
- Simplify (x/32 + x/12 − x/12 − x/12) / (x/32 − x/12) to x + 1/(x(x − 1)).
Question Two: Equations and Application
- Solve the equation 3x − 14 = 5/x.
- Simplify (9x² + 30x + 25)/(5x + 3x²).
- Find k so that the roots of x² − 2x/(4x − 1) = (k − 1)/(k + 1) are equal in magnitude but opposite in sign.
- Calculate the area of a circle (letter O) in terms of x for a logo with circle radius r.
- If the rectangle's area is 10 cm², find its length x.
Question Three: Logarithms and Quadratics
- Solve for x in x = log₅ 625.
- Solve 9^(2x+3) = 127 for x.
- Given log_b(x) = 2 and log_3b(y) = 2, express y in terms of x.
- Solve 3x² − 4kx + k² = 0 for x in terms of k using the quadratic formula.
- Show with d = 10 log₁₀(P/P₀) that a cooling fan (38 dB) is >6× as intense as a heat pump (30 dB).
Key Terms & Definitions
- Discriminant — In ax² + bx + c = 0, the discriminant is b² − 4ac, indicating the nature of roots.
- Completing the Square — Rewriting a quadratic as (x + p)² + q.
- Vieta’s Formulas — Relates polynomial roots to its coefficients.
- Logarithm — The power to which a base must be raised to yield a number.
- Decibel (dB) — Logarithmic unit measuring sound intensity, calculated with d = 10 log₁₀(P/P₀).
Action Items / Next Steps
- Attempt all questions, showing full working.
- Use Formulae Sheet L2MATHF for reference.
- Ensure all answers and workings are in the provided spaces.
- Review algebraic methods, especially quadratics, logarithms, and simplifying expressions.