Title Copy of T980- MATHEMATICS N6 QP AUG 2015.pdf ✅

(16030186) -2- T980(E)(A6)T Copyright reserved Please turn over DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE MATHEMATICS N6 TIME: 3 HOURS MARKS: 100 INSTRUCTIONS AND INFORMATION 1. 2. 3. 4. 5. 6. 7. 8. Answer ALL the questions. Read ALL the questions carefully. Number the answers according to the numbering system used in this question paper. Questions may be answered in any order, but subsections of questions must be kept together. Show ALL the intermediate steps. ALL the formulae used must be written down. Questions must be answered in BLUE or BLACK ink. Write neatly and legibly.
(16030186) -3- T980(E)(A6)T Copyright reserved Please turn over QUESTION 1 1.1 If z = e y x .cos  prove that x y z y x z        2 2 (3) 1.2 If x  4sin and 3 y  cos , calculate dx dy at the point where 6    (3) [6] QUESTION 2 Determine  y dx if: 2.1 y = 2 cot4 x (4) 2.2 y = 2 12  6x  3x (4) 2.3 y = e x x sin 3 (5) 2.4 y = x x sin 4 cos 4 2 3 (5) [18] QUESTION 3 Use partial fractions to calculate the following integrals: 3.1      ( 5)( 3) 5 4 2 2 x x x x dx (6) 3.2       3 2 4 2 1 2 3 2 x x x x x dx (6) [12] QUESTION 4 4.1 Calculate the particular solution of:  2 2 1 1 6 2     x x y dx dy at (3;1) (5) 4.2 Calculate the particular solution of: 2 2 dx d y dx dy  6 + 5 y = x e 5 5 , if x  0 when y  3 and  0 dx dy when x  0 . (7) [12]
(16030186) -4- T980(E)(A6)T Copyright reserved Please turn over QUESTION 5 5.1 5.1.1 Calculate the points of intersection of the two curves y = 4 2  x  and y = 4 4 2  x  . Make a neat sketch of the curves and show the area, in the first quadrant, bounded by the curves and the x-axis. Show the representative strip/element that you will use to calculate the volume generated if the area, in the first quadrant, bounded by the curves rotates about the y -axis. (3) 5.1.2 Calculate the volume generated if the area, as described in QUESTION 5.1.1, bounded by the two curves y = 4 2  x  and y = 4 4 2  x  rotates about the y -axis. (4) 5.2 5.2.1 Sketch the graph of 2 y  9  x and show the representative strip/element that you will use to calculate the volume generated when the area bounded by the graph and the x -axis rotates about the y -axis. (2) 5.2.2 Calculate the volume described in QUESTION 5.2.1. (3) 5.2.3 Calculate the distance of the centre of gravity from the x-axis of the solid generated when the area described in QUESTION 5.2.1 rotates about the y -axis. (5) 5.3 5.3.1 Make a neat sketch of the graph 1 4 2 2  y  x and show the representative strip/element that you will use to calculate the volume of the solid generated when the area, in the first quadrant, bounded by the graph, the y -axis, the x -axis and the line y = 2 is rotated about the y -axis. (2) 5.3.2 Calculate the volume of the solid generated as described in QUESTION 5.3.1. (3) 5.3.3 Calculate the moment of inertia of the solid generated when the area described in QUESTION 5.3.1 rotates about the y -axis. (5) 5.3.4 Express the answer in QUESTION 5.3.3 in terms of the mass. (1) 5.4 5.4.1 A vertical sluice gate, in the form of a parabola, is installed in a dam wall. The sluice gate is 4 m high and 8 m wide at the top. The top of the sluice gate lies in the water level. Sketch the vertical sluice gate and show the representative strip/element that you will use to calculate the area moment of the sluice gate about the water level. (2) 5.4.2 Calculate the relation between the two variables x and y that you will need to calculate the area moment of the sluice gate about the water level by means of integration. (2)