Mathematics

1. 1.

   Find and classify both stationary points of f(x) = x^3 - 3x^2 - 9x + 5.

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Stationary points have zero gradient. Differentiate, solve the quadratic, then substitute into the original function before classifying.
   2. f'(x) = 3x^2 - 6x - 9.
   3. 3(x - 3)(x + 1) = 0 gives x = 3 and x = -1.
   4. The points are (3, -22) and (-1, 10).
   5. f''(x) = 6x - 6: (-1, 10) is a local maximum; (3, -22) is a local minimum.

   Marking points

   • f'(x) = 3x^2 - 6x - 9.
   • 3(x - 3)(x + 1) = 0 gives x = 3 and x = -1.
   • The points are (3, -22) and (-1, 10).
   • f''(x) = 6x - 6: (-1, 10) is a local maximum; (3, -22) is a local minimum.

   Examiner tip: A zero first derivative finds a candidate, not its type. Use the second derivative or a sign change.

2. 2.

   Solve ln(x - 1) + ln(x + 1) = ln(8), stating the domain restriction.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Set the log domain before combining the logarithms. Both arguments must be strictly positive.
   2. Both logarithms require x > 1.
   3. ln((x - 1)(x + 1)) = ln(8), so x^2 - 1 = 8.
   4. x = 3; reject x = -3 because it is outside the domain.

   Marking points

   • Both logarithms require x > 1.
   • ln((x - 1)(x + 1)) = ln(8), so x^2 - 1 = 8.
   • x = 3; reject x = -3 because it is outside the domain.

   Examiner tip: Check solutions in the original logarithms; squaring can introduce an inadmissible root.

3. 3.

   X follows a binomial distribution with n = 5 and p = 0.4. Calculate P(X = 2) and state two assumptions of this model.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Choose which two trials succeed, then multiply the success and failure probabilities for that pattern.
   2. P(X = 2) = C(5, 2)(0.4)^2(0.6)^3.
   3. The probability is 0.3456.
   4. Trials are independent.
   5. Each trial has two outcomes and the same success probability.

   Marking points

   • P(X = 2) = C(5, 2)(0.4)^2(0.6)^3.
   • The probability is 0.3456.
   • Trials are independent.
   • Each trial has two outcomes and the same success probability.

   Examiner tip: Use exactly two successes, not at least two. Independence and constant p are modelling assumptions.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.