Tiêu đề Copy of T980 - MATHEMATICS N6 QP APRIL 2015.pdf ✅

Copyright reserved Please turn over T980(E)(M31)T APRIL EXAMINATION NATIONAL CERTIFICATE MATHEMATICS N6 (16030186) 31 March 2015 (Y-Paper) 13:00–16:00 Calculators may be used. This question paper consists of 5 pages and 1 formula sheet of 7 pages.
(16030186) -2- T980(E)(M31)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. 9. 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. Marks indicated are percentages. Write neatly and legibly.
(16030186) -3- T980(E)(M31)T Copyright reserved Please turn over QUESTION 1 1.1 If z =         x y arc 2 cot calculate the following: 1.1.1 x z   (1) 1.1.2 y z   (1) 1.2 A rectangular container with a length of 3,5 m, breadth of 1,6 m and a depth of 0,8 m is expanding along its length at 0,04 m/s and along its breadth at 0,035 m/s. The container is contracting along its depth at 0,05 m/s. Calculate the approximate change in the volume of the container. HINT: h dh V b b V l l V V             (4) [6] QUESTION 2 Determine  y dx if: 2.1 y = ec ax 3 cos 1 (4) 2.2 y = 2 37 18x  9x (5) 2.3 y = cot 2x 5 (5) 2.4 y = e x x sin2 2 (4) [18] QUESTION 3 Use partial fractions to calculate the following integrals: 3.1      (3 1) (2 3) 33 7 6 2 2 x x x x dx (6) 3.2      (2 )( 1) 3 3 2 2 x x x x dx (6) [12]
(16030186) -4- T980(E)(M31)T Copyright reserved Please turn over QUESTION 4 4.1 Solve the following differential equation: x .sin x dx  y dx  x dy 2 (5) 4.2 Calculate the particular solution of: 2 2 2 dx d y dx dy  4 + 4 y = t 20 , if e y  0 when t  0 and  0 dt dy when t  0. (7) [12] QUESTION 5 5.1 5.1.1 Calculate the points of intersection of y = 2 4  x and y = 2 4  4x . Make a neat sketch of the two curves and show the representative strip/element that you will use to calculate the volume generated when the area bounded by the two curves is rotated about the y -axis. (2) 5.1.2 Calculate the magnitude of the volume described in QUESTION 5.1.1 by means of integration. (3) 5.2 5.2.1 Sketch the graph of x y e 3  . Show the representative strip/element that you will use to calculate the area bounded by the graph, x  0 , 0 y  and x  2 . (2) 5.2.2 Calculate the area described in QUESTION 5.2.1. (3) 5.2.3 Calculate the second moment of area about the y -axis of the area described in QUESTION 5.2.1. (6) 5.2.4 Express the second moment of area, calculated in QUESTION 5.2.3, in terms of the area. (1) 5.3 5.3.1 Calculate the points of intersection of x  y  7  0 and xy  6 . Make a neat sketch of the two graphs and show the representative strip/element that you will use to calculate the volume generated when the area bounded by the two graphs is rotated about the x -axis. (3) 5.3.2 Calculate the volume described in QUESTION 5.3.1. (4) 5.3.3 Calculate the moment of inertia of the solid generated when the area described in QUESTION 5.3.1 is rotated about the x -axis. Express the answer in terms of the mass. (6)