IB Maths AA SL 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 1, no calculatorMedium6 marks

A square-based right pyramid models a roof structure. The side length of its horizontal base is $4\text{ m}$ The apex $S$ is vertically above the centre $O$ of the base, and $SO=2\sqrt{2}\text{ m}$ Let $A$ be a vertex of the base.

(a)

Determine the distance $OA$

[2]

(b)

Hence determine the distance $SA$

[2]

(c)

Determine the angle between $SA$ and the plane of the base. Give your answer in degrees.

[2]

Question 2

Paper 2, calculatorMedium6 marks

In triangle $ABC$ angles $A,B,C$ are measured in degrees. Using the usual notation for sides opposite the corresponding angles, $A=30^\circ$ $a=6\text{ cm}$ and $b=9\text{ cm}$

(a)

Show that $\sin B=0.75$

[2]

(b)

Hence determine all possible values of $B$ and the corresponding values of $C$ correct to the nearest $0.1^\circ$ Explain why the sine rule permits two possibilities here.

[4]

Question 3

Paper 1, no calculatorHard6 marks

A Ferris-wheel passenger’s height above the ground, $h$ metres, $t$ minutes after the ride begins is modelled by

h(t)=a\sin\bigl(b(t+c)\bigr)+d,

where $a,b>0$ and $0\leq c<\frac{2\pi}{b}$

The passenger’s lowest and highest heights are $3$ m and $15$ m respectively. The wheel reaches its highest point at $t=2$ its lowest point at $t=8$ and its next highest point at $t=14$

(a)

Write down the amplitude of $h$

[1]

(b)

Use the model and observations above.

(i)

Show that

h(t)=6\sin\left(\frac{\pi}{6}(t+1)\right)+9.

[3]

(ii)

Hence determine all values of $t$ where $0\leq t\leq 28$ for which the passenger’s height is $12$ m.

[2]

Question 4

Paper 2, calculatorMedium6 marks

For $0\leq x<2\pi$ let $f(x)=2\sin^2x+3\cos x-1$

(a)

Solve $f(x)=0$ analytically. Give all solutions in the interval to 3 significant figures.

[4]

(b)

Hence, use a GDC to check graphically that these are the only solutions in the stated interval.

[2]

Question 5

Paper 1, no calculatorMedium6 marks

Figure 1 shows a point $P$ on the unit circle. The angle $\theta$ is measured anticlockwise from the positive $x$ -axis to $OP$ and $\theta$ lies in quadrant II.

(a)

Determine $\cos\theta$ and $\tan\theta$

[3]

(b)

Hence, determine the exact values of $\sin 2\theta$ and $\cos 2\theta$

[3]

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