SPM Mathematics 2016, Paper 2 (Questions 7 & 8)


Question 7 (5 marks):
(a) State whether the following statements are true statement or false statement.
( i ) {  }⊂{ S,E,T } ( ii ) { 1 }⊂{ 1,2,3 }={ 1,2,3 }

(b) Diagram 7 shows the first three patterns of a sequence of patterns.

Diagram 7

It is given that the diameter of each circle is 20 cm.
(i) Make a general conclusion by induction for the area of the unshaded region.

(ii) Hence, calculate the area of the unshaded region for the 5th pattern.


Solution:
(a)(i) True

(a)(ii) False

(b)(i)
Area of unshaded region (first)
= (20 × 20) – π(10)2
= 400 – 100π
= 100 (4 – π)

Area of unshaded region (second)
= 4 × 100 (4 – π)
= 400 (4 – π)

Area of unshaded region (third)
= 9 × 100 (4 – π)
= 900 (4 – π)
100 (4 – π), 400 (4 – π), 900 (4 – π), …
102 (4 – π), 202 (4 – π), 302 (4 – π), …

General conclusion = n2 (4 – π)
n = 10, 20, 30, …


(b)(ii)
Area of unshaded region (5th)
= 502 (4 – π)
= 2500 (4 – π)



Question 8 (5 marks):
Table 8 shows the information of books purchased by Maslinda.
 
Table 8

Maslinda purchased x history books and y Science books. The total number of books purchased is 5. The total price of the books purchased is RM17.

(a) Write two linear equations in terms of x and y to represent the above information.

(b) Hence, by using matrix method, calculate the value of x and of y.


Solution:
(a)
x + y = 5
4x + 3y = 17

(b)
( 1    1 4    3 )( x y )=(  5 17 )             ( x y )= 1 1( 3 )−4( 1 ) (   3    −1 −4       1 )(  5 17 )             ( x y )=−1( 3( 5 )+( −1 )( 17 ) −4( 5 )+( 1 )( 17 ) )             ( x y )=−1( 15−17 −20+17 )             ( x y )=−1( −2 −3 )             ( x y )=( 2 3 ) x=2 and y=3


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