IBDP Maths AI HL Predicted Paper 2 - IB Maths AI Predicted Papers

Question 1

A small ball $P$ is projected from point $A$ with speed $39,\text{m s}^{-1}$ at an angle $\theta$ to the horizontal, where $\sin\theta=\frac{5}{13}$ and $\cos\theta=\frac{12}{13}$ . Point $A$ is $20,\text{m}$ above horizontal ground. The ball first lands at point $C$ on the ground. Let $g=9.8,\text{m s}^{-2}$ . Take the horizontal through $A$ as the $x$ -axis (positive to the right) and the vertical through $A$ as the $y$ -axis (positive upwards). $B$ is the point on the ground vertically below $A$ .

Part a

[2]

Resolve the initial velocity into horizontal and vertical components.

Part b

[5]

Show that the vertical position satisfies $y(t)=20+15t-\frac{1}{2}gt^{2}$ . Hence find the time of flight to $C$ , giving your answer to $3$ significant figures.

Part c

[3]

Find the horizontal distance $BC$ . Give your answer to $3$ significant figures.

Part d

[3]

Find the maximum height above the ground reached by $P$ .

Part e

[5]

Find the speed and the angle to the horizontal of $P$ on impact at $C$ (angle below the horizontal). Give both to $3$ significant figures.

[18]

Question 2

An inverted conical tank has semi-vertical angle $30^\circ$ and height $50,\text{cm}$ . It is initially full of water. The outflow is modelled by $\frac{dV}{dt}=-2h$ , where $V,(\text{cm}^3)$ is the volume of water and $h,(\text{cm})$ is the depth.

Part a

[4]

Express $V$ as a function of $h$ .

Part b

[5]

Using $\frac{dV}{dt}=\left(\frac{dV}{dh}\right)\left(\frac{dh}{dt}\right)$ , derive a differential equation for $h(t)$ and hence find $\frac{dh}{dt}$ in terms of $h$ .

Part c

[6]

Solve your differential equation to obtain $h(t)$ , given that $h(0)=50$ .

Part d

[5]

Find (i) the time taken for the tank to empty; (ii) the value of $\frac{dh}{dt}$ when $h=30,\text{cm}$ . State appropriate units.

[20]

Question 3

Consider the region $R$ bounded by the curve $y=\frac{1}{\sqrt{3x^{2}-3x+1}}$ , the $x$ -axis, and the vertical lines $x=\frac{1}{2}$ and $x=1$ . Distances are in centimetres.

Part a

[3]

Show that $3x^{2}-3x+1>0$ for all real $x$ .

Part b

[9]

The pendant is formed when $R$ is rotated $2\pi$ radians about the $x$ -axis. Show that
$V=\pi\int_{x=\frac{1}{2}}^{x=1}\left[\frac{1}{3x^{2}-3x+1}\right],dx$ ,
and evaluate $V$ exactly.

Part c

[4]

Using your GDC, find the surface area $S=2\pi\int_{x=\frac{1}{2}}^{x=1}y\sqrt{1+(y')^{2}},dx$ , giving your answer to $3$ significant figures.

[16]

Question 4

Two curves intersect on the $y$ -axis and a shaded region is bounded by the curves and the $x$ -axis. The curves are $y=\sqrt{2x+9}$ and $y=4e^{-2x}-1$ .

Part a

[2]

Verify that the curves meet at $(0,3)$ .

Part b

[4]

Find the $x$ -intercepts of each curve.

Part c

[8]

Write $A$ , the total shaded area, as a sum of definite integrals and evaluate $A$ exactly.

Part d

[4]

Using your GDC, plot both curves on the interval $-5\le x\le 2$ with an appropriate viewing window, and state which curve lies above the other on $0\le x\le \ln 2$ .

[18]

Question 5

The velocity–time graph “Velocity-Time Graph for Two Cars_OCR.png” shows two cars $A$ and $B$ moving along a straight, level road. At $t=0,\text{s}$ , car $B$ is travelling at $12,\text{m s}^{-1}$ and is overtaken by car $A$ travelling at $20,\text{m s}^{-1}$ . From $t=0$ to $t=30,\text{s}$ , $A$ decelerates uniformly to $8,\text{m s}^{-1}$ , then continues at $8,\text{m s}^{-1}$ thereafter.

Part a

[3]

Find the deceleration of car $A$ in $\text{m s}^{-2}$ .

Part b

[5]

Find the distance travelled by each car in the first $30,\text{s}$ .

Part c

[5]

Determine the time after $t=0$ when $B$ next overtakes $A$ (i.e., the next time their positions are equal).

Part d

[5]

Find the distance from the start at the instant of overtaking in part (c).

[18]

Question 6

A system switches between two states $S_1$ and $S_2$ each minute. The transition matrix is
$T=\begin{pmatrix}0.8&0.3\0.2&0.7\end{pmatrix}$ ,
where $T_{ij}$ is the probability of moving from state $j$ to state $i$ in one minute.

Part a

[4]

Starting in $S_1$ with certainty, write the initial state vector $s_0$ and find $s_1$ and $s_2$ .

Part b

[6]

Find the steady-state vector $s_\infty$ exactly.

Part c

[6]

Show that $T$ has eigenvalues $\lambda_1=1$ and $\lambda_2=0.5$ . Find a corresponding eigenvector for each eigenvalue.

Part d

[4]

Explain why $T^{n}\to\begin{pmatrix}0.6&0.6\0.4&0.4\end{pmatrix}$ as $n\to\infty$ , and interpret this result in context.

[20]

IBDP Maths AI HL Predicted Paper 2 set 1

Questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

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