XMS SEC 4 AMATH 2022 WA2

3 1 Differentiate each of the following with respect to x. (a) ( ) 3tan 2 4yx=− [1] (b) 23ln 21 xy x =  − [2] [Turn over 2022 Additional Mathematics WA 2

4 2 Find each of the following indefinite integrals. (a) ( ) 5 3 4 dxx+ [1] (b) ( )cos 2 5 dxx− [1] (c) 21e2 de x x x − +  [2] Question removed as it is not part of 2024 WA2 scope

5 3 Given that a curve has the equation ( ) 22 3 e xyx=+ , find the equation of the normal to the curve at the point where the curve cuts the y-axis. [4]

6 4 Water is poured into a bucket at a rate of 25π cm3/s. The volume, V cm3, of water in the bucket when the depth of water is x cm is given by ( ) 2π 15 3 xxV −= . Find the rate of increase in the depth of water when x = 5. [4] [Turn over

7 5 (a) Find ( ) 2d 2cos sin 2d xxx − . [1] (b) Hence, find ( )sin cos dx x x . [3] Question removed as it is not part of 2024 WA 2 scope

8 6 A piece of wire, p cm long, is bent to form the shape as shown in the diagram. ACDE is a rectangle with AE = y cm and ABC is a right-angled triangle with AB = 13x cm and BC = 5x cm. (a) Express p in terms of x and y. [2] (b) Given that the area enclosed by the shape is 96 cm2, show that 1625px x=+ . [3] A B C D E y y 13x 5x [Turn over

9 (c) Hence, find the value of x for which p has a minimum value. [4] End of Paper

10 Answers: 1a. ( ) ( ) 2212 tan 2 4 sec 2 4xx− − − 1b. 22 21xx− − 2a. ( ) 6 34 18 x c+ + 2b. ( )1 sin 2 52 xc−+ 2c. 1e 2exx c−−−+ 2d. 2 12 3ln2 x x x c x+ − − + 3. 1 38yx=− + 4. 1 cm/s 5a. 4sin 2cos 2xx−− 5b. 21 cos2 xc−+ 6a. 30 2p x y=+ 6c. 4 5