Cambridge International Mathematics

8 Multi-Topic Questions (Chapter 33)

51

A(¡ 3 ,4),B(5,¡2) and C(2,¡6) are three vertices of a

parallelogram ABCD.

52

Aristotle Jones wants to sail his yacht from P to Q.

In order to reach T from P, he has to sail along PS

(vectora) and then along ST (vectorb). This is called

a “tack”.

a Write the vector

¡!

PT in terms ofaandb.

If a=

μ 1

4

1

¶

and b=

μ 3

4

¡^14

¶

, find

i the components of

¡!

PT ii the length of

¡!

PT.

c The distance from P to Q is 15 km.

i How many tacks does he need to take to reach Q?

ii Write the vector

¡!

PQ in terms ofaandb.

d Find the bearing of Q from P.

53

Note: 1 hectare= 10 000m^2

On a still day, a helicopter hovers at a height of 200 m and

sprays the ground with fertiliser. The shaded part of the

diagram shows the circular area sprayed.

a If the “angle of spray” is 32 o, calculate the sprayed area

in square metres. Give your answer correct to three

significant figures.

b The farmer wants to spray a circular area of 3 hectares

from the same height. What “angle of spray” should he

use?

O

y

-4-2 246 x

2

4

-2

-4

A

B

C

-6

32°

200 m

North

a

b

Q

P

T

S

a

b Find the coordinates of the vertex D.

c Calculate the lengths of the line segments AB, BC and

AC.

d Use your answers incto show that the parallelogram

ABCD is a rectangle.

e Calculate the area of ABCD.

f The equation of the line through A and B is y=¡^34 x+^74.

i What is the gradient of this line?

ii Write down the coordinates of the point at which this line cuts they-axis.

b

³

x

y

́

is a vector with componentsxkm east and

ykm north.

Write down the vector

¡!

BA in the form

³

p

q

́

.

Adapted from June 1990, Paper 4

Adapted from June 1988, Paper 4

Adapted from June 1988, Paper 4

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