3D Trigonometry

www.Q8maths.com

10

0580/21/M/J/13© UCLES 2013

ForExaminer′sUse

23F

D

E B

A

C

6 cm

5 cm

12 cm

NOT TOSCALE

The diagram shows a triangular prism of length 12 cm. Triangle ABC is a cross section of the prism. Angle BAC = 90°, AC = 6 cm and AB = 5 cm.

Calculate the angle between the line CE and the base ABED.

Answer ............................................... [4]_____________________________________________________________________________________

24 A = 13

24

e o B = 41

32

e o

Find

(a) AB,

Answer(a) AB = [2]

(b) B –1, the inverse of B.

Answer(b) B –1 = [2]

_____________________________________________________________________________________

8

0580/43/M/J/13© UCLES 2013

ForExaminer′s

Use

4I

GE

F

JH

40 cm

22 cm

7 cm

NOT TOSCALE

EFGHIJ is a solid metal prism of length 40 cm. The cross section EFG is a right-angled triangle. EF = 7 cm and EG = 22 cm.

(a) Calculate the volume of the prism.

Answer(a) ........................................ cm3 [2]

(b) Calculate the length FJ.

Answer(b) FJ = ......................................... cm [4]

7

0580/41/O/N/15© UCLES 2015 [Turnover

(iii) Calculate the area of triangle ABC.

Answer(a)(iii) ........................................... m2 [2]

(b)

X

Y

12 cm

22 cm

45 cm

NOT TOSCALE

A cuboid has length 45 cm, width 22 cm and height 12 cm.

Calculate the length of the straight line XY.

Answer(b) XY = .......................................... cm [4]__________________________________________________________________________________________

7

0580/23/M/J/14© UCLES 2014 [Turn over

16H G

C

BA

E F

D3 cm

4 cm

12 cm

NOT TOSCALE

ABCDEFGH is a cuboid. AB = 4 cm, BC = 3 cm and AG = 12 cm.

Calculate the angle that AG makes with the base ABCD.

Answer ................................................ [4]__________________________________________________________________________________________

12

0580/23/M/J/16© UCLES 2016

23E

H G

A

7 cm

NOT TOSCALE

5 cm

3 cm

F

B

CD

The diagram shows a cuboid. HD = 3 cm, EH = 5 cm and EF = 7 cm.

Calculate

(a) the length CE,

CE = ............................................ cm [4]

(b) the angle between CE and the base CDHG.

.................................................. [3]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.

To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series.

Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

12

0580/21/O/N/16© UCLES 2016

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.

To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series.

Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

24S R

C

BA

P Q

D

8 cm

8 cm

8 cmNOT TOSCALE

The diagram shows a cube of side length 8 cm.

(a) Calculate the length of the diagonal BS.

BS = ......................................... cm [3]

(b) Calculate angle SBD.

Angle SBD = ................................................ [2]

11

© UCLES 2011 0580/23/M/J/11 [Turn over

For

Examiner's

Use

21

4 cm

6 cm

6 cm

DC

BA

P

M

NOT TOSCALE

The diagram shows a pyramid with a square base ABCD of side 6 cm.

The height of the pyramid, PM, is 4 cm, where M is the centre of the base.

Calculate the total surface area of the pyramid.

Answer cm2

[5]

Question 22 is printed on the next page.

9

0580/41/M/J/15© UCLES 2015 [Turnover

(b) The diagram shows a pyramid with a horizontal rectangular base.

4.8 m

3 m

yNOT TOSCALE

4 m

The rectangular base has length 4.8 m and width 3 m and the height of the pyramid is 4 m.

Calculate

(i) y, the length of a sloping edge of the pyramid,

Answer(b)(i) y = ............................................. m [4]

(ii) the angle between a sloping edge and the rectangular base of the pyramid.

Answer(b)(ii) ................................................ [2]

11

0580/22/M/J/14© UCLES 2014 [Turn over

21P

D C

BA

M

6 cm

4 cm

4 cm

NOT TOSCALE

The diagram shows a pyramid on a square base ABCD with diagonals, AC and BD, of length 8 cm. AC and BD meet at M and the vertex, P, of the pyramid is vertically above M. The sloping edges of the pyramid are of length 6 cm.

Calculate

(a) the perpendicular height, PM, of the pyramid,

Answer(a) PM = .......................................... cm [3]

(b) the angle between a sloping edge and the base of the pyramid.

Answer(b) ................................................ [3]__________________________________________________________________________________________

Question 22 is printed on the next page.

10

0580/23/M/J/15© UCLES 2015

18P

C

BA

D

M

8 cm

20 cm

20 cm

NOT TOSCALE

The diagram shows a solid pyramid on a square horizontal base ABCD. The diagonals AC and BD intersect at M. P is vertically above M. AB = 20 cm and PM = 8 cm.

Calculate the total surface area of the pyramid.

Answer ......................................... cm2 [5]