SPM Additional Mathematics (Model Test Paper)


SPM Additional Mathematics (Model Test Paper)

Section B
[40 marks]
Answer any four questions from this section.

Question 8
Diagram 4 shows part of a curve x = y2 + 2. The gradient of a straight line QR is –1.
Find
(a)    the equation of PQ,                                                                                              [2 marks]
(b)   the area of shaded region,                                                                                     [4 marks]

(c)    the volume of revolution, in terms of π, when the shaded region is rotated through 360o about the y-axis.                                                                                                             [4 marks]

Answer and Solution:
(a)
y– 2 = –1 (x – 6)
y= –x + 6 + 2
y = –x + 8

(b)
y= –x + 8, at y-axis, x = 0
y= 8
Point R = (0, 8)
Area of shaded region
= Area under the curve + Area of triangle
= 0 2 x dy + 1 2 ( 82 )( 6 ) = 0 2 ( y 2 +2 ) dy+ 1 2 ( 6 )( 6 ) = [ y 3 3 +2y ] 0 2 +18 =[ 2 3 3 +2( 2 )0 ]+18 = 20 3 +18 =24 2 3 uni t 2

(c)
Volume of revolution

= Volume under the curve + Volume of cone
=π 0 2 x 2 dy + 1 3 π r 2 h =π 0 2 ( y 2 +2 ) 2 dy+ 1 3 π ( 6 ) 2 ( 6 ) =π 0 2 ( y 4 +4 y 2 +4 ) dy+72π =π [ y 5 5 + 4 y 3 3 +4y ] 0 2 +72π =π[ 2 5 5 + 4 ( 2 ) 3 3 +4( 2 )0 ]+72π =π[ 32 5 + 32 3 +8 ]+72π = 376 15 π+72π =97 1 15 π unit 3


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