Paper 1 Final Exam Predictions & Review Guide

1

Function (Comprehensive)

Graph Transformations: Master shifts $f(x \pm a)$, $f(x) \pm b$, reflections across axes $f(-x)$, $-f(x)$, and analytical stretching/compressing $af(x)$, $f(kx)$.
Trigonometric Properties: Calculate periods using $T = \frac{2\pi}{\vert \omega \vert}$ and identify max/min values without a calculator.
Odd & Even Functions: Confirm $f(-x) = -f(x)$ (origin symmetry) and $f(-x) = f(x)$ (y-axis symmetry).
Absolute Value Integrations: Sketch $\vert f(x) \vert$ and $f(\vert x \vert)$ coupled with $e^x$ or $\ln x$.
Asymptotes: Identify vertical ($x = a$), horizontal ($y = L$), and slant ($y = mx + c$) asymptotes. Master rational function degree comparison for classification.

📘 Asymptote Classification — Rational Functions $f(x) = \frac{P(x)}{Q(x)}$

Let $\deg P = m$, $\deg Q = n$:

$m < n$: HA at $y = 0$ (horizontal asymptote at $y = 0$).
$m = n$: HA at $y = \frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$.
$m = n + 1$: No HA. SA at $y = (\text{quotient from division})$ — a linear oblique asymptote.
$m > n + 1$: No HA, no SA. End behavior follows a higher-degree polynomial (curve asymptote).
VA: Occurs at real roots of $Q(x) = 0$ that are not cancelled by $P(x) = 0$.

⚠️ Special Function Asymptotes (Paper 1 Traps):

Exponential $y = e^x$: HA at $y = 0$ (as $x \to -\infty$). No VA.
Exponential $y = e^{-x}$: HA at $y = 0$ (as $x \to +\infty$). No VA.
Logarithm $y = \ln x$: VA at $x = 0$. No HA ($\ln x \to \infty$ as $x \to \infty$).
Inverse $y = \frac{1}{x}$: VA at $x = 0$, HA at $y = 0$ (both directions).
Arctan $y = \arctan x$: HA at $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$. No VA.

🔗 Cross-Topic (Sequences): Function outputs can form an arithmetic or geometric progression.

AP: $u_n = u_1 + (n-1)d$, $S_n = \frac{n}{2}\big(2u_1 + (n-1)d\big) = \frac{n}{2}(u_1 + u_n)$
GP: $u_n = u_1 \cdot r^{\,n-1}$, $S_n = \frac{u_1(r^n - 1)}{r - 1} = \frac{u_1(1 - r^n)}{1 - r}$, $S_\infty = \frac{u_1}{1 - r}$ for $\vert r \vert < 1$

⚠️ Critical Checks:

Inverse Functions: Ensure domain is strictly one-to-one (pass Horizontal Line Test). Domain $\leftrightarrow$ Range swaps for $f^{-1}(x)$.
Composite Functions: Check that Range of inner $g(x)$ fits entirely inside Domain of outer $f(x)$. Practice $f(g(x))$ vs $g(f(x))$.

2

Logarithmic & Exponential

Core Log Laws: Apply $\ln(xy) = \ln x + \ln y$, $\ln\!\left(\frac{x}{y}\right) = \ln x - \ln y$, and $\ln(x^k) = k \ln x$.
Special Boundaries: Instinctively use $\ln e = 1$, $\ln 1 = 0$, $\log_a a = 1$, $\log_a 1 = 0$.
Equation Solving: Track quadratic-form equations (e.g., $a^{2x} + b\,a^x + c = 0$) using substitution $t = a^x$.
Exponential & Log Graphs: $y = e^x$ has horizontal asymptote $y = 0$; $y = \ln x$ has vertical asymptote $x = 0$. They are inverses.

⚠️ Change-of-Base Rule (IB Booklet Notation): $$ \log_a b = \frac{\log_c b}{\log_c a} $$

In practice: $\log_a b = \frac{\ln b}{\ln a}$. Base stays on the bottom!

3,6

Polynomial / Binomial Theorem / Counting

Remainder & Factor Theorems: If divided by $(ax-b)$, remainder is $P\!\left(\frac{b}{a}\right)$. If $P\!\left(\frac{b}{a}\right) = 0$, then $(ax-b)$ is a factor.
Vieta's Formulas: For quadratic $ax^2 + bx + c = 0$: $\alpha + \beta = -\frac{b}{a}$, $\alpha\beta = \frac{c}{a}$. For cubic: $\alpha + \beta + \gamma = -\frac{b}{a}$, $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$, $\alpha\beta\gamma = -\frac{d}{a}$.

📘 Binomial Theorem (IB Formula Booklet)

For $n \in \mathbb{N}$:

$$(a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{r}a^{n-r}b^r + \dots + b^n$$

Or equivalently in summation form:

$$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{\,n-r} b^{\,r}.$$

General term (used to find a specific term):

$$T_{r+1} = \binom{n}{r} a^{\,n-r} b^{\,r}, \quad r = 0, 1, 2, \dots, n.$$

Binomial coefficient (combinations):

$$\binom{n}{r} = {}^nC_r = \frac{n!}{r!\,(n-r)!}.$$

Key properties: $\binom{n}{0} = \binom{n}{n} = 1$, $\binom{n}{1} = \binom{n}{n-1} = n$, $\binom{n}{r} = \binom{n}{n-r}$ (symmetry).
Pascal's triangle: Each entry is the sum of the two above: $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$.
Common Paper 1 trick: Find the coefficient of $x^k$ in expansions like $(2x - 3)^5$ or $(1 + 2x)^n(1 - x)^m$. Identify which $r$ gives the desired power.

📘 Permutations & Combinations (Counting)

Permutation (order matters): ${}^nP_r = \frac{n!}{(n-r)!}$.
Combination (order does not matter): ${}^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}$.
Tie-Together Method: Treat locked items as one block, then multiply by internal permutations.
Slotting / Gap Method: Arrange unrestricted items first, then place restricted items into the gaps.
Case-by-case: When direct counting is complex, break into mutually exclusive cases and sum.

5

Vector (Matrix Equations)

Gaussian Elimination: Perform row operations to bring the augmented matrix into Row Echelon Form (REF). $$ \begin{bmatrix} 1 & 1 & 1 & \vert & d_1 \\ 2 & 3 & 1 & \vert & d_2 \\ \vdots & \vdots & \vdots & \vert & \vdots \end{bmatrix} \implies x+y+z = ? $$

⚠️ Solution Classification (parametric constants like $k$):

Unique Solution: REF gives three pivot rows & no contradictions. Solve directly.
No Solution: A contradictory row appears: $[0 \; 0 \; 0 \; \vert \; k]$ where $k \neq 0$.
Infinite Solutions: A zero row appears: $[0 \; 0 \; 0 \; \vert \; 0]$. Assign a parameter (e.g., $z = t$) and express $x, y$ in terms of $t$.

7

Mathematical Induction

Step 1 — Base Case: Show explicitly that the statement holds for the initial value (typically $n = 1$). Verify both sides numerically.
Step 2 — Inductive Hypothesis: Formally record: "Assume true for $n = k$". This is the assumption you will use to prove $n = k + 1$.
Step 3 — Inductive Step: Prove the statement for $n = k + 1$ using the hypothesis. Clearly highlight where Step 2 is substituted.
Step 4 — Conclusion: Append the formal inductive wrap-up: "Therefore, by mathematical induction, the statement is true for all $n \in \mathbb{Z}^+$."

⚠️ Common Paper 1 Induction Types:

Summation formulas: Prove $\sum_{r=1}^n f(r) = g(n)$.
Divisibility: Prove $4^n - 1$ is divisible by 3 for all $n \in \mathbb{Z}^+$.
Inequalities: Prove $2^n > n^2$ for $n \ge 5$.
Recurrence relations: Given $u_{n+1} = f(u_n)$, prove closed form $u_n = g(n)$.

8,9

Limit & Implicit Differentiation

Chain Rule execution: Always append $\frac{dy}{dx}$ when differentiating a $y$-variable term. E.g., $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$.
Product & Quotient Rules in Implicit: For $xy$: $\frac{d}{dx}(xy) = y + x\frac{dy}{dx}$. For $\frac{x}{y}$: $\frac{d}{dx}\!\left(\frac{x}{y}\right) = \frac{y - x\frac{dy}{dx}}{y^2}$.
L'Hôpital's Rule: Identify $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms, then differentiate numerator and denominator separately. May need repeated application.

⚠️ Board Notation Warning: The board sometimes maps $\frac{dx}{dy}$. Verify carefully whether the problem asks for $\frac{dy}{dx}$ or $\frac{dx}{dy}$.

Standard derivatives to recall (IB Booklet):
$(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \sec^2 x$,
$(e^x)' = e^x$, $(\ln x)' = \frac{1}{x}$, $(x^n)' = nx^{n-1}$.

10

Vector (3D Geometry)

Dot Product (Scalar): $\vec{a} \cdot \vec{b} = \vert\vec{a}\vert\vert\vec{b}\vert\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$. Use for angles and perpendicularity ($\vec{a} \cdot \vec{b} = 0$).
Cross Product (Vector): $\vec{a} \times \vec{b} = \vert\vec{a}\vert\vert\vec{b}\vert\sin\theta \,\hat{n}$. Yields a vector normal to both $\vec{a}$ and $\vec{b}$. Modulus $\vert\vec{a} \times \vec{b}\vert$ = area of parallelogram.
Vector Equation of a Line: $L: \vec{r} = \vec{a} + \lambda \vec{b}$, where $\vec{a}$ is a position vector and $\vec{b}$ is the direction vector.
Vector Equation of a Plane: $\Pi: \vec{r} \cdot \hat{n} = d$ or $\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}$.
Line/Plane Intersections: Master line-line crossing (parallel, intersect, skew), point-to-plane distance, and line-plane intersection (substitute param eq into plane eq).

📘 Key Formulas (IB Booklet):

Distance from point $A$ to plane $\Pi$: $\frac{\vert \overrightarrow{AB} \cdot \hat{n} \vert}{\vert \hat{n} \vert}$, where $B$ is a point on the plane.
Angle between two lines: $\cos\theta = \frac{\vec{b}_1 \cdot \vec{b}_2}{\vert\vec{b}_1\vert\vert\vec{b}_2\vert}$.
Angle between line and plane: $\sin\theta = \frac{\vec{b} \cdot \hat{n}}{\vert\vec{b}\vert\vert\hat{n}\vert}$.

11

Complex Number

Cartesian Form: $z = a + bi$. Conjugate: $\bar{z} = a - bi$. Modulus: $\vert z \vert = \sqrt{a^2 + b^2}$.
Argand Diagram: Real axis (horizontal), Imaginary axis (vertical). Plot $a + bi$ as point $(a,b)$. Conjugate is reflection across real axis.
Polar & Euler Forms: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$, where $r = \vert z \vert$ and $\theta = \arg(z)$.
De Moivre's Theorem: $z^n = r^n(\cos n\theta + i\sin n\theta) = r^n e^{in\theta}$. Essential for high powers and roots.
Complex Roots: Solving $z^n = a + bi$ yields $n$ distinct roots, equally spaced around a circle of radius $\sqrt[n]{r}$.

📘 n-th Roots of Unity — Solving $z^n = 1$:

Roots: $z_k = e^{i\frac{2\pi k}{n}}$ for $k = 0, 1, 2, \dots, n-1$.
Positioned as vertices of a regular $n$-gon on the Argand diagram.
Sum of all $n$ roots $= 0$ (unless $n = 1$).

12

Differentiation & Value Trend

Basic Derivatives (IB Booklet): $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \sec^2 x$, $(e^x)' = e^x$, $(\ln x)' = \frac{1}{x}$, $(x^n)' = nx^{n-1}$.
Product Rule: $(uv)' = u'v + uv'$. Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$.
Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Essential for composite functions.

📘 Value Trend Analysis (Curve Behavior):

First Derivative (monotonicity): $f'(x) > 0$ ⟶ increasing; $f'(x) < 0$ ⟶ decreasing. $f'(x) = 0$ isolates stationary points.
Second Derivative (concavity): $f''(x) > 0$ ⟶ concave up (accelerating); $f''(x) < 0$ ⟶ concave down (decelerating). $f''(x) = 0$ marks inflection points.
Stationary Points: $f'(x) = 0$. Use $f''(x)$: if $f''(x) > 0$ ⟶ local minimum; $f''(x) < 0$ ⟶ local maximum; $f''(x) = 0$ ⟶ test further.
End Behavior: Evaluate $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ for long-range asymptotic trends.