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4.10 SPM Practice (Long Questions)

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Question 11 (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



Question 12 (6 marks):
 Given A=( 4    −2 3    −1 ), B=m( −1    n −3    4 ) and I=( 1    0 0    1 ).
(a) Find the value of m and of n if AB = I.
(b) Write the following simultaneous linear equation as matrix equation:
4x – 2y = 3
3x – y = 2
Hence, by using matrix method, calculate the value of x and of y.

Solution:
(a)
If AB=I, then B= A −1 A −1 = 1 4( −1 )−( −2 )( 3 ) ( −1    2 −3    4 ) A −1 = 1 2 ( −1    2 −3    4 ) By comparison: B= A −1 m( −1    n −3    4 )= 1 2 ( −1    2 −3    4 ) m= 1 2  ; n=2


(b)
4x−2y=3 3x−y=2 ( 4    −2 3    −1 )( x y )=( 3 2 )                ( x y )= 1 2 ( −1    2 −3    4 )( 3 2 )                ( x y )= 1 2 ( ( −1 )( 3 )+( 2 )( 2 ) ( −3 )( 3 )+( 4 )( 2 ) )                ( x y )= 1 2 (   1 −1 )                ( x y )=(    1 2 − 1 2 ) x= 1 2  and y=− 1 2