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

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IBDP Maths AI HL Predicted Paper 2 sample

Questions

Question 1

Question 2

Question 3

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