Geometry and Trigonometry Examination Questions SL

On this page you can find examination questions from the topic of geometry and trigonometry

3-Dimensional Solids

Question 1

A glass is made up of a hemisphere and a cone.

Find the volume of the glass.

Give your answer to 3 significant figures


Hint

Full Solution

 

Question 2

The total surface area of a hemisphere is 1360 cm²

Find the radius.

Give your answer to 3 significant figures.


Hint

Full Solution

 

Question 3

a) A sphere has a radius of 10cm. Find the volume, giving your answer in terms of \(\large \pi\).

b) A cone has the same volume and the same radius as the sphere. Find the height of the cone.

c) Another sphere and cone have the same volume and the same radius, r. Find an equation for the height of the cone, h in terms of r.

Hint

Full Solution

 

Question 4

Three metal spheres have radii 1cm, 6cm and 8cm.

The spheres are melted down and made into one bigger sphere.

What is the radius of the single sphere?

Hint

Full Solution

 

Question 5

A cylindrical metal bar with height 12cm and diameter 12cm is melted down and made into spheres of diameter 3cm.

How many spheres will it make?

Hint

Full Solution

 

Question 6

A solid is made up of a cone and a cylinder.

The radius is 5cm, the height of the cone is 12cm and the height of the cylinder is 15cm.

Show that the total surface area of the solid is \(\large 240\pi\)


Hint

Full Solution

 

Radians, Arcs and Sectors

Question 1

The following diagram shows a circle with centre O and radius 12cm. A and B lie on the circumference of the circle and \(\large AÔB=50°\)

a) Find the area of the minor sector OAB

b) Find the area of the triangle AOB

c) Hence, find the area of the shaded segment


Hint

Full Solution

 

Question 2

The following diagram shows a circle with centre O and radius r cm

The area of the shaded sector OAB is \(\large \frac{40\pi}{3}\) cm²

The length of the minor arc AB is \(\large \frac{10\pi}{3}\) cm

a) Find the radius of the circle

b) Find the angle \(\large \theta\) , in radians


Hint

Full Solution

Question 3

The following diagram shows a circle with centre O and radius 5cm and another circle with centre P and radius r. The two circles overlap meeting at points A and B. \(\large AÔP=45°\) and \(\large A\hat{P}O=30°\)

a) Show that \(\large r=5\sqrt{2}\) cm

b) Hence, show that the shaded area bounded by the two circles is \(\large \frac{25}{12}(7\pi-6-6\sqrt3)\) cm²

Hint

Full Solution

 

Question 4

The following diagram shows a circle with centre O and radius r. A and B are points on the circumference of the circle and \(\large A\hat{O} B =\theta\) radians

The area of the green shaded region is three times greater than the area of the blue region.

a) Show that \(\large \sin \theta=\frac{4\theta-2\pi}{3}\)

b) Find the value of \(\large \theta\) , giving your answer correct to 3 significant figures.

Hint

Full Solution

Question 5

The following diagram shows a circle with centre O and radius r. Points A and B lie on the circumference of the circle and \(\large A\hat{O}B=\theta\) radians. The tangents to the circle A and B intersect at C.

a) Show that \(\large AC=r\tan (\frac{\theta}{2})\)

b) Hence, find the value of \(\large \theta\) when the two shaded regions have an equal area.

Hint

Full Solution

Right-angled Trigonometry

Question 1

A, B and C are points on horizontal ground.

C is due West of B. A is due South of B. AB = 60m

A flagpole stands vertically at B.

From A, the angle of elevation of the top of the flagpole is 11°.

From B, the angle of elevation of the top of the flagpole is 15°.

Calculate the distance AC giving your answer to 3 significant figures.

Hint

Full Solution

Question 2

The diagram shows a cuboid ABCDEFGH. AB = 8 cm, AE = 6 cm and BC = 15 cm.

a) Find the length of AC.

Give your answer correct to 3 significant figures

b) Find the size of the angle between the line EC and the plane ABCD.

Give your answer correct to 1 decimal place.

Hint

Full Solution

 

Question 3

ABCDEF is a prism in which the triangle BCF is the cross section.

BC = 12cm, EF = 16cm, angle CBF = 30° and angle FCB = 90°

The angle AF makes with the plane ABCD is \(\large \theta\).

Show that \(\large \tan\theta=\frac{\sqrt{3}}{5}\)

Hint

Full Solution

 

Question 4

ABCDEFG is a triangular prism.

AB = 12cm, AE = 8cm, EF = 18cm.

Angle BAE = 90°

G is the midpoint of BC.

Calculate the angle between EG and the plane ABCD.

Give your answer correct to 1 decimal place.

Hint

Full Solution

 

Question 5

ABCDEF is a prism.

AB = AE = BE = 6cm. BC = 10cm

Calculate

a) the length EC

b) the angle AEC

c) the angle between EC and the plane ABCD

Give lengths to 3 significant figures and angles to 1 decimal place.

Hint

Full Solution

Sine and Cosine Rule

Question 1

The following diagram shows a quadrilateral ABCD.

AB = 7cm , AD = 5cm ∠DAB=120° , ∠DBC=45° , ∠BCD=60°

BD = \(\sqrt{a}\)

CD = \(\sqrt{b}\)

\(a,b \in \mathbb{Z}\)

Find a and b

Hint

Full Solution

Question 2

The following diagram shows a quadrilateral ABCD.

AD = x – 1 , BD = x + 1 , DC = 2x and \(\angle CDA\) = 120°

The sum of the area of triangle ADC and triangle BDC is \(4 \sqrt{3}\)

Find x

Hint

Full Solution

Question 3

In a triangle ABC, AB = 8cm, BC = a, AC = b and \(\angle BAC\) = 30°

a) Show that \(b^2-8\sqrt{3}b+64-a^2=0\)

b) Hence find the possible values of a (in cm) for which the triangle has two possible solutions.

Hint

Full Solution

Unit Circle

Question 1

Given that \(cosx=-\frac{\sqrt{7}}{3}\) and \(\frac{\pi}{2}\le x\le \pi\) , find the possible values of sinx and tanx

Hint

Full Solution

Question 2

If \(tanx=\frac{12}{5}\) and \(\pi\le x\le \frac{3\pi}{2}\) , find the value of cosx

Hint

Full Solution

Trigonometric Graphs

Question 1

The height h of water, in metres, in a habour is modelled by the function \(\large h(t)=5.5\sin(0.5(t-1.5))+12\) where t is time after midday in hours.

a) Find the initial height of the water.

b) At what time is it when the water reaches this height again?

c) Find the maximum height of the water.

d) How much time is there in between the first and second time that the water at 16 metres?

Give heights to 3 significant figures and times to the nearest minute

Hint

Full Solution

 

 

Question 2

The following diagram shows a Ferris wheel.

The height, h metres of a seat above ground after t minutes is given by \(\large h(t)=a\ \cos(bt)+c\) , where a, b and c \(\large \in \mathbf{R}\)

The following graph shows the height of the seat.

Find the values of a, b and c

Hint

Full Solution

 

Question 3

Consider a function f, such that \(\large f(x) = 5.6\cos(\frac{\pi}{a}(x-1))+b\) , \(\large 0\le x\le 15\), \(\large a,b\in \mathbf{R}\)

The function f has a local maximum at the point (1 , 8.8) and a local minimum at the point (10 , -2.4)

a) Find the period of the function

b) Hence, find the value of a.

c) Find the value of b.

Hint

Full Solution

Question 4

Consider a function f, such that \(\large f(x) = a\sin(\frac{\pi}{15}(x+2))+b\) , \(\large a,b\in \mathbf{R}\)

The function f has passes through the points (10.5 , 5.5) and (15.5 , 2.5)

Find the value of a and the value of b

 

Hint

 

Full Solution

Question 5

The following diagram shows a ball attached to the end of a spring.

The height, h, in mtres of the ball above the ground t seconds after being released can be modelled by the function

\(\large h(t)=a\cos(\frac{\pi}{b}t)+c\) , \(\large a,b, c\in \mathbf{R}\)

The ball is release from an initial height of 4 metres.

After \(\large \frac{4}{3}\) seconds, the ball is at a height of 1.6 metres.

It takes the ball 4 seconds to return to its initial height.

Find the values of a, b and c.

Hint

Full Solution

Pythagorean Identities

Question 1

a) Show that the equation \(\large 2 \sin^2x=3 \cos x\) may be written in the form

\(\large 2 \cos^2x+3 \cos x-2=0\)

b) Hence, solve \(\large 2 \sin^2x=3 \cos x\) , for \(\large 0\le x\le2\pi\)

Hint

Full Solution

 

Question 2

 

Given that \(x=\frac{2}{cos\theta}\) and \(y=3tan\theta\)

show that \(\frac{x^2}{4}-\frac{y^2}{9}=1\)

Hint

Full Solution

Question 3

The following diagram shows triangle ABC with AB = 4 and AC = 5

DIAGRAM NOT TO SCALE

a) Given that \(\large \sin \hat A=\frac{3}{4}\), find the value of \(\large \cos \hat A\)

b) Hence, show that the length of \(\large BC=\sqrt{41-10\sqrt{7}}\)

Hint

Full Solution

 

Question 4

Prove that \(\large \frac{\sin ^3\theta}{\tan \theta}+\cos^3\theta\equiv\cos\theta\)

Hint

Full Solution

 

Double Angle Formula

Question 1

 

Let f(x) = (cos2x - sin2x)²

a) Show that f(x) can be expressed as 1 - sin4x

b) Let f(x) = 1 - sin4x. Sketch the graph of f for \(0\le x\le \pi \)

Hint

Full Solution

Question 2

 

Solve \(cos2θ=sinθ\) for \(0\le \theta \le 2\pi \)

Hint

Full Solution

Question 3

 

a) Show that \(cos2\theta-3cos\theta+2\equiv 2{ cos }^{ 2 }\theta -3cos\theta +1\)

b) Hence, solve \(cos2\theta-3cos\theta+2=0\) for \(0\le \theta \le 2\pi \)

Hint

Full Solution

 

Question 4

 

Let \(cos\theta=\frac{2}{3}\), where \(0\le \theta \le \frac { \pi }{ 2 } \)

Find the value of

a) \(sin\theta\)

b) \(sin2\theta\)

c) \(sin4\theta\)

Hint

Full Solution

Solving Trigonometric Equations

Question 1

Let f(x)= cosx and g(x) = \(\frac{2x^2}{1-x}\)

a) Show that g∘f(x) = 1 can be written as 2cos²x + cosx - 1 = 0

b) Hence solve g∘f(x)=1 for \(-\pi\le x\le \pi\)

Hint

Full Solution

Question 2

Solve \(\log _{ 3 }{ sinx-\log _{ 3 }{ cosx=0.5 } } \) for \(0\le x\le 2\pi\)

Hint

Full Solution

Question 3

1 + cosx + cos²x + cos3x + ... = \(2 + \sqrt2\)

Find x given that \(-\frac {\pi}{2}\le x\le \frac {\pi}{2}\)

Hint

Full Solution

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