A-Level Mathematics - Medium Difficulty Core Concepts

Front

What is the geometric interpretation of the discriminant Δ = b² - 4ac?

Back

The discriminant determines the nature of the roots of a quadratic equation ax² + bx + c = 0. If Δ > 0, the parabola intersects the x-axis at two distinct points (two real roots). If Δ = 0, the vertex touches the x-axis (one repeated root). If Δ < 0, the parabola does not touch the x-axis (complex roots).

Front

Describe the transformation y = f(2x - 4) in terms of order of operations.

Back

Apply the horizontal stretch by scale factor \(\tfrac12\) parallel to the x-axis first: \(y=f(2x)\). Then translate 2 units to the right: \(y=f(2(x-2))=f(2x-4)\).

Front

Define a many-to-one function and explain why it has no inverse function.

Back

A many-to-one function is a mapping where multiple elements in the domain map to the same element in the range (e.g., y = x²). An inverse function f⁻¹(x) exists only if the function is one-to-one, ensuring every output has a unique input. To find the inverse of a many-to-one function, you must restrict the domain (e.g., y = x² for x ≥ 0).

Front

Explain the domain restriction for the inverse of f(x) = sin(x).

Back

For f(x) = sin(x) to have an inverse function f⁻¹(x) = arcsin(x), it must be one-to-one. The principal domain is restricted to -π/2 ≤ x ≤ π/2. This ensures the function passes the horizontal line test, resulting in the principal value range of -1 ≤ y ≤ 1.

Front

Calculate the length of arc 's' and area of sector 'A' for a central angle θ (radians).

Back

Use the radian formulas derived from the circumference C = 2πr. **Arc Length:** s = rθ **Sector Area:** A = ½r²θ These formulas are valid only when θ is expressed in radians.