Mathematics 9709,22 Oct Nov 2024 | Cambridge AS - A Level Past Papers With Mark Scheme

1. Logarithmic Transformations & Exponential Equations | Expressing \\(5(6x - 8)^7\\) in the form \\(y = kx\\) using logarithms and finding \\(k\\).

2. Differentiation & Gradient Analysis | Finding the derivative of \\(f(x) = 4\\sin^3 x\\), computing its inverse derivative, and determining the gradient of a given curve.

3. Exponential & Logarithmic Equations | Sketching graphs of \\(y = 1 + e^{2x}\\) and \\(y = 3x - 4\\), finding intersection points, and solving using an iterative method.

4. Polynomials & Factorization | Finding unknown coefficients in \\(p(x) = ax^3 + bx^2 + ax - 8\\) given a factor and remainder conditions, and proving a single real root exists.

5. Definite Integrals & Area Approximation | Using the trapezium rule for area estimation, finding exact areas using integration, and determining estimation accuracy.

6. Trigonometric Identities & Equations | Expressing \\(\\sin 4\\theta + \\sin(60^\\circ - \\theta)\\) in the form \\(R\\sin(\\theta - \\alpha)\\) and solving for the smallest positive \\(\\theta\\).