IB Maths AA HL Topic 3: Geometry and trigonometry questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

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Question 1

Paper 2, calculatorEasy5 marks

A simplified model for the water depth at a harbour is

h(t)=2.4+1.6\sin\left(\frac{\pi}{6}(t-2)\right),

where $h$ is measured in metres and $t$ is the number of hours after midnight.

(a)

State the amplitude and the midline height of the model.

[2]

(b)

Determine the period of the model.

[1]

(c)

Calculate the water depth at $t=5$

[2]

Question 2

Paper 2, calculatorHard6 marks

The function $f$ is defined by

f(\theta)=a\sin\theta+b\cos\theta,\qquad 0\leq\theta\leq\pi,

where $a,b\in\mathbb R$ Figure 1 shows the graph of $y=f(\theta)$

(a)

Determine the values of $a$ and $b$

[3]

(b)

For $0<\theta<\pi$ $\theta\ne\frac{\pi}{2}$ define

H(\theta)=f(\theta)f(\pi-\theta)+\tan\theta+\tan(\pi-\theta).

Determine whether $H$ attains a maximum. If it does not, state its supremum.

[3]

A circuit model represents the normalized voltage at phase $\phi$ by $v(\phi)=\sin\phi$ A controller operates at phase $\theta$ where $0<\theta<\frac{\pi}{2}$ During calibration, the normalized voltage at phase $2\theta$ is $\frac12$

(a)

Starting from the compound-angle identities, derive the double-angle identities for $\sin 2x$ $\cos 2x$ and $\tan 2x$

[3]

(b)

Determine all possible exact values of $\tan\theta$

[3]

Question 4

Paper 2, calculatorHard6 marks

For $x,y,z,k\in\mathbb{R}$ consider the planes

P_1:\ x-z=1,\qquad P_2:\ y+z=2,\qquad Q_k:\ x+ky+(1-k)z=3.

(a)

Determine a direction vector of the line of intersection of $P_1$ and $P_2$

[1]

(b)

The acute angle between this line and $Q_k$ is $30^\circ$ Determine all possible values of $k$ and, for each value, the common solution of the three plane equations.

[5]

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