Applied Mathematics Problem-Solving Strategies

Paper 2 optional topics span mechanics, statistics, and discrete mathematics. This lesson provides targeted strategies for each of the five optional areas, focusing on the most frequently examined question types and the most common errors.

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Further Mechanics Strategies

Circular Motion

The most common topic in Further Mechanics exam questions. Key formulas:

Quantity	Formula
Centripetal acceleration	$a = \frac{v^2}{r} = r\omega^2$a=rv2​=rω2
Centripetal force	$F = \frac{mv^2}{r}$F=rmv2​
Angular velocity	$\omega = \frac{v}{r} = \frac{2\pi}{T}$ω=rv​=T2π​

Strategy for "motion in a vertical circle":

1. Draw a force diagram at the relevant point (top, bottom, or angle $\theta$θ).
2. Resolve forces toward the centre: $\text{resultant toward centre} = \frac{mv^2}{r}$resultant toward centre=rmv2​.
3. Use energy conservation between two positions.
4. At the "critical" point (usually the top), the minimum speed occurs when the normal reaction or tension is zero.

Exam Tip: At the top of a vertical circle, weight acts toward the centre. At the bottom, weight acts away from the centre. Many marks are lost by resolving forces in the wrong direction.

Elastic Strings and Hooke's Law

$T = \frac{\lambda x}{l}$T=lλx​

where $T$T = tension, $\lambda$λ = modulus of elasticity, $x$x = extension, $l$l = natural length.

Elastic potential energy: $EPE = \frac{\lambda x^2}{2l}$EPE=2lλx2​

Strategy: Use conservation of energy (KE + GPE + EPE = constant) rather than $F = ma$F=ma when possible. Energy methods avoid the need to find the variable acceleration.

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Further Statistics 1 Strategies

Chi-Squared Tests

The most common exam format: you are given observed frequencies and asked to test against a theoretical distribution.

Systematic approach:

1. State $H_0$H0​ and $H_1$H1​ clearly.
2. Estimate parameters from data if needed (state this — it affects degrees of freedom).
3. Calculate expected frequencies.
4. Combine classes if $E_i < 5$Ei​<5 (state what you combined and why).
5. Calculate $X^2 = \sum (O - E)^2 / E$X2=∑(O−E)2/E in a clear table.
6. State degrees of freedom: $\nu = \text{classes} - 1 - \text{parameters estimated}$ν=classes−1−parameters estimated.
7. Compare with the critical value and conclude in context.

Exam Tip: The most common error is incorrect degrees of freedom. Always state clearly: "There are $k$k classes and $p$p parameters were estimated, so $\nu = k - 1 - p$ν=k−1−p."

Continuous Random Variables

For PDF questions, the exam typically asks you to:

1. Find an unknown constant $k$k by setting $\int f(x) dx = 1$∫f(x)dx=1.
2. Find $E(X)$E(X), $E(X^2)$E(X2), and $\text{Var}(X)$Var(X).
3. Find the CDF by integrating the PDF.
4. Find the median by solving $F(m) = 0.5$F(m)=0.5.

Strategy: Lay out each integral separately with clear limits. For piecewise PDFs, handle each piece and check continuity at the boundaries.

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Further Statistics 2 Strategies

Hypothesis Testing

Questions typically involve:

1. A z-test or t-test for the mean.
2. Identifying whether $\sigma$σ is known (z) or unknown (t).
3. Calculating the test statistic and comparing with the critical value.
4. Stating the conclusion in context.

Strategy: Write a structured answer: (i) Hypotheses, (ii) Significance level, (iii) Test statistic formula and calculation, (iv) Critical value, (v) Decision, (vi) Conclusion in context.