SPM Additional Mathematics 2017, Paper 2 (Question 10 & 11)


Question 10 (10 marks):
(a) Prove that 2 tan x cos2 x = sin 2x.
(b) Hence, solve the equation 4 tan x cos2 x = 1 for 0 ≤ x ≤ 2π.
(c)(i) Sketch the graph of y = sin 2x for 0 ≤ x ≤ 2π.
(c)(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation 4π tan x cos2 x = x – 2π for 0 ≤ x ≤ 2π.
State the number of solutions.

Solution: 
(a)

2tanx cos 2 x=sin2x Left hand side =2tanx cos 2 x =2× sinx cosx × cos 2 x =2sinxcosx =sin2x = Right hand side ( Proven )


(b)
4tanx cos 2 x=1, 0x2π 2( 2tanx cos 2 x )=1 2sin2x=1 sin2x= 1 2 Basic angle= π 6 2x= π 6 ,( π π 6 ),( 2π+ π 6 ),( 3π π 6 ) 2x= π 6 , 5π 6 , 13π 6 , 17π 6 x= π 12 , 5π 12 , 13π 12 , 17π 12



(c)(i)
y = sin 2x, 0 ≤ x ≤ 2π.




(c)(ii)
4πtanx cos 2 x=x2π 2π( 2tanx cos 2 x )=x2π 2πsin2x=x2π sin2x= x 2π 2π 2π sin2x= x 2π 1 y= x 2π 1


Number of solutions = 4

Question 11 (10 marks):
Diagram 6 shows a curve y = 2x2 – 18 and the straight line PQ which is a tangent to the curve at point K.

It is given that the gradient of the straight line PQ is 4.
(a) Find the coordinates of point K
(b) Calculate the area of the shaded region.
(c) When the region bounded by the curve, the x-axis and the straight line y = h is rotated through 180o about the y-axis, the volume generated is 65π unit3.
Find the value of h.


Solution: 
(a)
y=2 x 2 18 dy dx =4x Gradient of straight line PQ=4 4x=4 x=1 When x=1,  y=2 ( 1 ) 2 18=16 Coordinates of K=( 1,16 ).


(b)
At x-axis, y=0 2 x 2 18=0 2 x 2 =18 x 2 =9 x=±3 The curve cuts the x-axis at ( 3,0 ) and ( 3,0 ). Area of shaded region = Area of triangleArea under the curve = 1 2 ( 51 )( 16 ) 1 3 ydx =32 1 3 ( 2 x 2 18 )dx =32| [ 2 x 3 3 18x ] 1 3 | =32| ( 2 ( 3 ) 3 3 18( 3 ) )( 2 ( 1 ) 3 3 18( 1 ) ) | =32| ( 1854 2 3 +18 ) | =32| 18 2 3 | =3218 2 3 =13 1 3  units 2


(c)
Volume generated=65π π h 0 x 2 dy =65π π h 0 ( y 2 +9 )dx =65π y=2 x 2 18 x 2 = y 2 +9 [ y 2 4 +9y ] h 0 =65 0( h 2 4 +9h )=65 h 2 4 9h=65 h 2 +36h+260=0 ( h+10 )( h+26 )=0 h=10   or   h=26 ( rejected ) Thus, h=10


Leave a Comment