Understanding the Paper 3 Challenge
Paper 3 consists of **two compulsory extended-response questions** (approx. 27–28 marks each) completed in 60 minutes. Unlike Paper 1 and Paper 2, Paper 3 is designed around **sustained inquiry into unfamiliar mathematical contexts**.
The Fundamental Architecture of Every Paper 3 Question:
- Part (a) & (b) [Entry Level]: Concrete, specific numerical examples (e.g. testing for $n = 1, 2, 3$). These are free foundation marks designed to ensure you understand the rules of the prompt.
- Part (c) & (d) [Pattern Detection]: Gathering numerical evidence and spotting an emerging relationship or algebraic recurrence relation.
- Part (e) & (f) [Conjecture & Proof]: Formulating a general theorem for case $n$ and proving it via mathematical induction, algebra, or calculus.
- Part (g) [Extension & Limitation]: Exploring boundary conditions, asymptotic behavior, or applying the result to an advanced scenario.
Stage 1: Deconstruct the Novel Definition (First 3 Mins)
Read the introductory stem twice. The IB examiner will define a new notation, sequence, or operation that does not appear in the standard textbook. Write down the first two small cases by hand to internalize the definition before typing into your calculator.
Stage 2: Tabulate & Calculate Systematically (Mins 4–12)
Always build a structured 3-column table on your rough sheet: $n$, intermediate steps, and final evaluated value. Use your GDC spreadsheet or sequence mode (`Seq`) to generate higher-order values quickly to confirm your manual calculation.
Stage 3: The Generalization Leap (Mins 13–22)
When the prompt asks you to *"Conjecture an expression for $T_n$ in terms of $n$"*, look for classic algebraic patterns: geometric multipliers ($2^n, 3^n$), factorial products ($n!$), or polynomial degrees. State your conjecture clearly using proper notation.
Stage 4: Rigorous Justification & Limits (Mins 23–28)
If proving by mathematical induction, follow the complete 4-step formal structure (Base case, Inductive hypothesis, Inductive step, Conclusion). If evaluating large-scale behavior, take limits as $n \to \infty$ or test the determinant condition.
Chebyshev Polynomials & Roots ($T_n(x)$)
Scenario: Chebyshev polynomials are defined by the recurrence relation $T_0(x) = 1$, $T_1(x) = x$, and $T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x)$ for $n \ge 1$.
- Find $T_2(x)$ and $T_3(x)$.
- Show that $T_n(\cos\theta) = \cos(n\theta)$ for $n = 0, 1, 2$.
- Using mathematical induction, prove that $T_n(\cos\theta) = \cos(n\theta)$ for all $n \in \mathbb{N}$.
- Deduce all real roots of $T_n(x) = 0$ in terms of cosine values.
Markscheme & Walkthrough:
(a) $T_2(x) = 2x(T_1) - T_0 = 2x(x) - 1 = 2x^2 - 1$. [M1][A1]
$T_3(x) = 2x(T_2) - T_1 = 2x(2x^2 - 1) - x = 4x^3 - 2x - x = 4x^3 - 3x$. [A1]
(b) $T_0(\cos\theta) = 1 = \cos(0\theta)$ (True).
$T_1(\cos\theta) = \cos\theta = \cos(1\theta)$ (True).
$T_2(\cos\theta) = 2\cos^2\theta - 1 = \cos(2\theta)$ by double angle identity (True). [M1][A1]
(c) **Induction Step:** Assume true for $n = k-1$ and $n = k$: $T_{k-1}(\cos\theta) = \cos((k-1)\theta)$ and $T_k(\cos\theta) = \cos(k\theta)$. [M1]
$T_{k+1}(\cos\theta) = 2\cos\theta\cos(k\theta) - \cos((k-1)\theta)$. [M1]
Using product-to-sum identity: $2\cos\theta\cos(k\theta) = \cos((k+1)\theta) + \cos((k-1)\theta)$. [M1]
$T_{k+1}(\cos\theta) = \cos((k+1)\theta) + \cos((k-1)\theta) - \cos((k-1)\theta) = \cos((k+1)\theta)$. [A1]
Conclude with standard IB induction statement. [R1]
(d) Set $T_n(x) = 0 \iff \cos(n\theta) = 0 \iff n\theta = \frac{(2k - 1)\pi}{2}$. [M1]
$\theta_k = \frac{(2k - 1)\pi}{2n}$ for $k = 1, 2, \dots, n$. The roots are $x_k = \cos\left(\frac{2k - 1}{2n}\pi\right)$. [A2]
Epidemic Isolation Network & Steady-State Spread
Scenario: A quarantined facility models individual states as Susceptible ($S$), Quarantined ($Q$), and Cleared ($C$). Daily transition rates depend on an intervention parameter $\alpha \in [0, 1]$:
$$\mathbf{T} = \begin{pmatrix} 1 - \alpha & \alpha & 0 \\ 0.2 & 0.5 & 0.3 \\ 0.05 & 0 & 0.95 \end{pmatrix}$$- Show that $\mathbf{T}$ is a regular Markov chain for any $\alpha \in (0, 1)$.
- Find the steady-state distribution when $\alpha = 0.4$.
- Determine the threshold value of $\alpha$ such that less than $5\%$ of the population remains in Quarantine in the long term.
Markscheme & Walkthrough:
(a) Show that $\mathbf{T}^2$ has all strictly positive entries for $\alpha > 0$. Since every state can reach every other state in 2 steps, $\mathbf{T}$ is regular. [M1][R1]
(b) Set $\mathbf{\pi}\mathbf{T} = \mathbf{\pi}$ with $\pi_S + \pi_Q + \pi_C = 1$. When $\alpha = 0.4$: [M1]
Use row vectors because each row of $\mathbf T$ sums to $1$. The steady-state equations give $\pi_Q=2\alpha\pi_S$ and $\pi_C=6\pi_Q$. Normalize to obtain $\boldsymbol\pi=\frac{1}{1+14\alpha}(1,2\alpha,12\alpha)$. At $\alpha=0.4$, $\boldsymbol\pi=(\frac5{33},\frac4{33},\frac{24}{33})\approx(0.152,0.121,0.727)$. [A2]
(c) Solve $\pi_Q < 0.05$ algebraically or by graphing $\pi_Q(\alpha)$ on GDC. [M1]
Solve $\frac{2\alpha}{1+14\alpha}<0.05$. The denominator is positive on $[0,1]$, so $2\alpha<0.05+0.7\alpha$, giving $0\le\alpha<\frac1{26}\approx0.0385$. The equality boundary gives exactly $5\%$ and is excluded. For a regular chain use $0<\alpha<\frac1{26}$; $\alpha=0$ is a separate absorbing case with steady state $(1,0,0)$. [A2]
Top 5 GDC Superpowers for Paper 3
| 1. Sequence Mode (`Seq`) | Switch graphing mode from `Function` to `Sequence`. Enter recurrence relations like $u(n) = 2u(n-1) + 3$ to generate 50 terms instantaneously in the Table view (`[2nd] [TABLE]`). |
| 2. Matrix Powers ($A^{50}$) | Store the transition matrix in `[A]`. Compute `[A]^50` or `[A]^100` to instantly observe the numerical steady-state vector in seconds. |
| 3. Polynomial Root Finder | Use `PolySmlt2` app (TI-84) or `polyRoots()` (TI-Nspire) to find all real and complex roots of degree 3, 4, and 5 equations without manual factorization. |
| 4. Numerical Derivative `nDeriv` | Check your analytic derivative conjectures by plotting $Y_1 = f'(x)$ (analytic) and $Y_2 = \text{nDeriv}(f(x), x, x)$. Agreement provides a useful numerical check, but does not prove an identity. Check the domain and justify the derivative analytically; a finite plot can miss errors or exceptional points. |
| 5. List Operations (`seq()`) | Generate lists via `seq(2^N - 1, N, 1, 10)` to compute finite sums and discover series patterns directly on the home screen. |
The 6 Deadliest Paper 3 Pitfalls
- Pitfall 1: Getting stuck on Part (b) and refusing to move forward: In Paper 3, later subparts frequently say *"Given that $T_n = \dots$, show that..."*. You can earn all subsequent marks even if you failed to derive the earlier equation!
- Pitfall 2: Treating conjectures as proofs: Stating *"I notice that the numbers double, so it must be $2^n$"* only earns the conjecture mark. The *proof* requires algebraic deduction or formal mathematical induction.
- Pitfall 3: Poor time allocation: You have approximately **28 minutes per question** (plus 4 minutes reading time). If you spend 40 minutes on Question 1, you will inevitably drop 20+ easy entry marks on Question 2.
- Pitfall 4: Neglecting the initial definition constraints: If the prompt states $n \ge 3$, testing $n = 1$ will produce an undefined expression or a false pattern.
- Pitfall 5: Incomplete Induction Conclusions: Forgetting the standard closing sentence: *"Since true for $n=1$, and truth for $n=k$ implies truth for $n=k+1$, by the principle of mathematical induction it is true for all $n \in \mathbb{Z}^+$"*.
- Pitfall 6: Unformatted GDC outputs: Never write calculator-specific notation like `nDeriv` or `e^-4` in your final exam script. Use proper mathematical notation $\frac{dy}{dx}$ and $e^{-4}$.