SPM Additional Mathematics 2018, Paper 1 (Question 23 – 25)


Question 23 (4 marks):
Diagram 10 shows the position of three campsites A, B and C at a part of a riverbank drawn on a Cartesian plane, such that A and B lie on the same straight riverbank.

Diagram 10

Sam wants to cross the river from campsite C to the opposite riverbank where the campsites A and B are located.
Find the shortest distance, in m, that he can take to cross the river. Give your answer correct to four decimal places.


Solution:

Let the shortest distance from campsite C to opposite riverbank is CD. Area of ABC = 1 2 | 2     7     4      3     5   3   2   3 | = 1 2 | ( 2 )( 5 )+( 7 )( 3 )+( 4 )( 3 ) ( 3 )( 7 )( 5 )( 4 ) ( 3 )( 2 )| = 1 2 | 66| =33  units 2 Distance of AB = ( 27 ) 2 + ( 35 ) 2 = 85  m Area of ABC=33 1 2 ( AB )( CD )=33 1 2 ( 85 )( CD )=33 CD= 33( 2 ) 85 CD=7.1587 ( 4 d.p. ) Thus, the shortest distance is 7.1587 m.


Question 24 (4 marks):
A voluntary body organizes a first aid course 4 times per month, every Saturday from March until September.
[Assume there are four Saturdays in every month]

Salmah intends to join the course but she might need to spare a Saturday per month to accompany her mother to the hospital. The probability that Salmah will attend the course each Saturday is 0.8. Salmah will be given a certificate of monthly attendance if she can attend the course at least 3 times a month.

(a)
Find the probability that Salmah will be given the certificate of monthly attendance.

(b)
Salmah will qualify to sit for the first aid test if she obtains more than 5 certificates of monthly attendance.
Find the probability that Salmah qualifies to take the first aid test.


Solution:
(a)

P( X=r )= C n r p r q nr p=0.8, q=0.2, n=4, r=3, 4 P( X3 ) =P( X=3 )+P( X=4 ) = C 4 3 ( 0.8 ) 3 ( 0.2 ) 1 + C 4 4 ( 0.8 ) 4 ( 0.2 ) 0 =0.4096+0.4096 =0.8192

(b)

P( X=r )= C n r p r q nr p=0.8192, q=0.1808, n=7, r=6, 7 P( X>5 ) =P( X=6 )+P( X=7 ) = C 7 6 ( 0.8192 ) 6 ( 0.1808 ) 1 + C 7 7 ( 0.8192 ) 7 ( 0.1808 ) 0 =0.3825+0.2476 =0.6301

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