XMS SEC 4 AMATH 2023 WA2

1 1 Differentiate each of the following with respect to x. (a) ( ) 10 1 4 cosxx+ [2] (b) 45 tan x x − [2]

2 2 The function f is defined by ( ) 21f 32xx x=+ + for 1x− . Determine whether f is an increasing or a decreasing function. [3] 3 The point A lies on the curve 23 7 11y x x= − + . The line 5y x k+= is a normal to the curve at point A. Find the coordinates of A and the value of k. [4]

3 [Turn over 4 A balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is increasing at a constant rate of 500 cm3 per second. The balloon was empty at the start of the pumping. [The volume of sphere is 34 π3 r .] (i) Find the radius of the balloon after 20 seconds. [2] (ii) Find the rate of increase of the radius after 20 seconds. [3]

4 5 The equation of a curve is ( ) 4 1 sin 3yx=+ , where π0 2x . (i) Express d d y x in the form ( ) 3 cos3 1 sin 3k x x + , where k is a constant to be found. [1] (ii) Find the exact x-coordinate(s) of the point(s) at which the tangent to the curve is parallel to the x-axis. [5]

5 [Turn over 6 The diagram shows a pond OAB that is designed in a shape of a sector of a circle, where its radius is r m and angle AOB is  radians. The area of the pond is 10 m2. [The arc length of a sector is r and area of a sector is 21 2 r  .] (i) Given that the perimeter of the pond is P m, show that 202Pr r=+ . [3] Given that r and  can vary, (ii) find the value of r for which P has a stationary value, [3] O A B r

6 (iii) determine whether this value of P is a maximum or a minimum. [2] END OF PAPER

7 [Turn over Answer Key 1a ( ) ( ) 10 9 1 4 sin 40 1 4 cosx x x x− + + + 1b ( )( ) 2 2 4 tan 4 5 sec tan x x x x −− 2 ( ) ( ) ( ) ( ) 2 2 2 3f ' 2 32 For 1, 33 2 0 then 0 32 For 1, 2 2 0 f '( ) 0 Hence, f is a decreasing function. xx x x x x x x x −=+ + − −+   + − −   3 ( )2, 9 ; 47Ak = 4i 13.4 cm 4ii 0.223 cm/s 5i ( ) 3 12cos3 1 sin 3xx + 5ii ππand62 6ii r = 3.16 6iii P is maximum

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