IGCSE Math Exam Questions

Algebraic Expressions and Calculations

• To evaluate (5(7 + 3)), first simplify the expression inside the parentheses: (7 + 3 = 10). Thus, (5 \times 10 = 50).

Simplification

• The expression (3x^2 - 2x - 5) is already in its simplest form as a quadratic expression.

Derivatives

• The derivative of the function (f(x) = 2x^3 - x^2 + 3) is computed as follows:
  • (f'(x) = 6x^2 - 2x).

Roots of Quadratic Equations

• For the equation (2x^2 - 5x + 3 = 0), use the formula for the sum of the roots given by (-\frac{b}{a}):
  • Here, (a = 2) and (b = -5), so the sum of the roots is (\frac{5}{2}).

Higher Order Derivatives

• To find the second derivative (f''(x)) of the function (f(x) = x^4 - 6x^2 + 9):
  • First, calculate the first derivative: (f'(x) = 4x^3 - 12x).
  • Then, find the second derivative: (f''(x) = 12x^2 - 12).

Solving Inequalities

• For the inequality (2x^2 - 8x + 7 < 0), rewrite it as (x^2 - 4x + \frac{7}{2} < 0).
  • Find the roots using the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}):
    • Here, (a = 2), (b = -8), and (c = 7).
  • The expression may take negative values between the roots if they exist within the interval determined by the quadratic's graph.