HL AA · Practice Paper 2 · Interactive

IB Maths Analysis & Approaches HL — Interactive Practice Paper 2

Paper 2 is the calculator-required paper — rewards students who can wield the GDC efficiently for systems, vector geometry, probability and calculus. Two modes: instant per-part feedback while you learn, or full timed exam simulation.

29 marks · 5 questions · GDC required · systems, vectors, probability, calculus
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Question 1

5 marksHLGDC required
Systems of Equations — consistent / inconsistent

Consider the following system of three linear equations in three unknowns:

$$\begin{aligned} x + 2y - z &= 5 \\ 3x - y + 2z &= 8 \\ 4x + y + z &= 15 \end{aligned}$$

(a) Use your GDC (matrix RREF) to determine whether the system is consistent or inconsistent. Type consistent or inconsistent. [2]

Casio fx-CG50: MENU A (Equation) → Simultaneous → 3 unknowns.

(b) Find the values of $x$, $y$, $z$. Give as a comma-separated triple. [3]

(a) Coefficient matrix has determinant $\det\begin{pmatrix}1&2&-1\\3&-1&2\\4&1&1\end{pmatrix} = -14 \ne 0$, so a unique solution exists — the system is consistent.
(b) Row-reducing the augmented matrix on the GDC yields the unique solution $\mathbf{(x, y, z) = (3, 2, 2)}$.

Question 2

6 marksHLGDC required
Vectors — angle between a line and a plane

The line $L$ has vector equation $$\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$$ and the plane $\Pi$ has Cartesian equation $3x - 4y = 10$.

(a) Write down the direction vector of $L$ and a normal vector of $\Pi$. Give each as a comma-separated triple. [2]

(b) Find the angle $\theta$ between $L$ and $\Pi$, in degrees, correct to one decimal place. [4]

$\sin\theta = \dfrac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$ (line–plane angle uses sine, not cosine).
(a) $\mathbf{d}_L = (1, 2, 2)$ and $\mathbf{n}_\Pi = (3, -4, 0)$.
(b) $\mathbf{d} \cdot \mathbf{n} = 3 - 8 + 0 = -5$.
$|\mathbf{d}| = 3$, $|\mathbf{n}| = 5$.
$\sin\theta = \dfrac{|-5|}{3 \times 5} = \dfrac{1}{3}$.
$\theta = \arcsin(1/3) \approx \mathbf{19.5°}$. This is the acute angle between the line and the plane (IB convention).

Question 3

6 marksHLGDC required
Conditional Probability — independence and $P(G|T)$

At a pet shop, $G$ is the event that a customer buys a goldfish and $T$ is the event that a customer buys a tortoise. It is given that $P(G) = 0.3$, $P(T) = 0.4$, and $P(G' \cap T') = 0.42$.

(a) Find $P(G \cup T)$. [1]

(b) Find $P(G \cap T)$. [2]

(c) Are $G$ and $T$ independent? Type yes or no. [1]

(d) Find $P(G | T)$. [2]

(a) $P(G \cup T) = 1 - P(G' \cap T') = 1 - 0.42 = \mathbf{0.58}$.
(b) $P(G \cap T) = P(G) + P(T) - P(G \cup T) = 0.3 + 0.4 - 0.58 = \mathbf{0.12}$.
(c) Check independence: $P(G) \cdot P(T) = 0.3 \times 0.4 = 0.12 = P(G \cap T)$. Since the product of marginals equals the joint, $G$ and $T$ are independent.
(d) $P(G | T) = \dfrac{P(G \cap T)}{P(T)} = \dfrac{0.12}{0.4} = \mathbf{0.3}$. Equivalently, because $G, T$ are independent, $P(G|T) = P(G) = 0.3$.

Question 4

6 marksHLGDC required
Advanced Differentiation — tangent to $y = x \cdot 3^x$

Consider the function $y = x \cdot 3^x$.

(a) Find $\dfrac{dy}{dx}$. Give in the form $3^x(1 + kx\ln 3)$ for some constant $k$. State $k$. [3]

Product rule: differentiate $x$ times $3^x$ separately, remembering $\dfrac{d}{dx}(3^x) = 3^x \ln 3$.

(b) Evaluate the gradient at $x = 1$. Give the exact form as $a(1 + b\ln 3)$ and enter $a + b$ (both integers). [1]

(c) Find the exact equation of the tangent line at $x = 1$, in the form $y = mx + c$. Give $c$ exactly. [2]

(a) Product rule: $\dfrac{dy}{dx} = 3^x + x \cdot 3^x \ln 3 = 3^x(1 + x\ln 3)$, so $k = \mathbf{1}$.
(b) At $x = 1$: $m = 3^1(1 + \ln 3) = 3(1 + \ln 3)$, so $a = 3, b = 3$ and $a + b = \mathbf{4}$.
(c) Point: $(1, 3)$. Tangent: $y - 3 = 3(1 + \ln 3)(x - 1) \Rightarrow y = 3(1 + \ln 3)x - 3\ln 3$, so $c = \mathbf{-3\ln 3}$.

Question 5

6 marksHLGDC required
Advanced Integration — volume of revolution about the $y$-axis

The region $R$ is enclosed by the curve $y = \ln x$, the $x$-axis, the $y$-axis, and the horizontal line $y = 2$. The region is rotated $360°$ about the $y$-axis.

(a) Express $x$ in terms of $y$. [1]

(b) Write down the definite integral for the volume $V$ using the disc method. Give as pi*int(f(y), a, b). [2]

(c) Evaluate the exact volume. Give in the form $\dfrac{\pi}{2}(e^k - 1)$ and state $k$. [3]

(a) From $y = \ln x$: $x = \mathbf{e^y}$.
(b) Disc about the $y$-axis: $V = \pi \displaystyle\int_0^2 x^2\,dy = \mathbf{\pi \int_0^2 e^{2y}\,dy}$.
(c) $\pi \displaystyle\int_0^2 e^{2y}\,dy = \pi\left[\tfrac{1}{2}e^{2y}\right]_0^2 = \tfrac{\pi}{2}(e^4 - e^0) = \dfrac{\pi}{2}(e^4 - 1)$, so $k = \mathbf{4}$.

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