3. (d) None of these / Geen van hierdie

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1 SECTION A (24 marks) / AFDELING A (24 punte) The questions in Section A must be completed on SIDE 2 of the optical reader form in SOFT PENCIL. First circle your answers on this paper and then transfer these to the optical reader form. Die vrae in Afdeling A moet op KANT 2 van die merkleesvorm in SAGTE POTLOOD voltooi word. Omkring eers jou antwoorde op hierdie vraestel en dra dit dan oor op die merkleesvorm. Question 1 / Vraag 1 If A a ij is a 3 3 matrix with a ij 2 ij then As A a ij n 3 3 matriks is met a ij 2 ij dan is 1. (a) A A T 1. (b) A is symmetric / A simmetries 1. (c) deta 0 1. (d) None of these / Geen van hierdie Question 2 / Vraag 2 Given that A, B and C are n n matrices. If AB AC can be simplified to B C then Gegee dat A, B en C is n n matrikse. As AB AC vereenvoudig kan word na B C dan is 2. (a) A is the zero matrix / A die nulmatriks 2. (b) deta 0 2. (c) deta 0 2. (d) None of these / Geen van hierdie Question 3 / Vraag 3 Given that ax by c and dx ey f are the equations of two non-vertical, non-horizontal lines. The lines will intersect if Gegee dat ax by c en dx ey f die vergelykings van twee nie-vertikale, nie-horisontale lyne is. Die lyne sal sny as 3. (a) ae bd 0 3. (b) ae bd 0 3. (c) a d b e c f 3. (d) None of these / Geen van hierdie 1

2 Question 4 / Vraag 4 a b c Given that / Gegee dat d e f 3 then / dan is g h i 4. (a) 6 4. (b) (c) 6 4. (d) (d) None of these / Geen van hierdie 2a 2b 2c 2d g 2e h 2f i 2g 2h 2i Question 5 / Vraag 5 A b represents a consistent system of linear equations with A in row echelon form Consider the statements: I If a kk 0 for k 1, 2,..., n then the system has a unique solution. II If a kk 0 for some k then the system has infintely many solutions. Which of the statements is / are true? A b stel n nie-strydige stelsel lineêre vergelykings voor met A in ry-trapvorm Beskou die bewerings: I As a kk 0 vir k 1, 2,..., n dan het die stelsel n unieke oplossing. II As a kk 0 vir n sekere k, dan het die stelsel oneindig veel oplossings. Watter van die bewerings is waar? 5. (a) Only I / Slegs I 5. (b) Only II / Slegs II 5. (c) Both I and II / Beide I en II 5. (d) Neither I nor II / Nie I of II nie Question 6 / Vraag 6 Given that A, B and X are all n n matrices with AXA B and deta 0. Then Gegee dat A, B en X almal n n matrikse is met AXA B en deta 0. Dan is 6. (a) X B A 2 6. (b) X A 1 BA 1 6. (c) X A 1 2 B 6. (d) None of these / Geen van hierdie 2

3 Question 7 / Vraag 7 z 4 given in polar form is i 3 z 4 gegee in poolvorm is i 3 7. (a) 4cis0 7. (b) 4cis 2 7. (c) 4cis 7. (d) 4cis 2 7. (d) None of these / Geen van hierdie Question 8 / Vraag 8 Choose the statement that is not true or choose (e). Kies die stelling wat nie waar is nie of kies (e). 8. (a) z. z z 2 8. (b) z 1. z 2 z 1. z 2 8. (c) As / If z r cisdan is / then iz r cis 2 8. (d) 1 cis n cisn 8. (e) Die bewerings is almal waar. The statements are all true. Question 9 / Vraag 9 If x 3 i is divided by x 2i then the remainder is As x 3 i gedeel word deur x 2i dan is die res 9.(a) 7i 9. (b) 9 9. (c) 9i 9. (d) None of these / Geen van hierdie 3

4 Question 10 / Vraag 10 The solution(s) of z 2 i is /are Die oplossing(s) van z 2 i is 10. (a) (b) 1 2 i 2 or / of 1 2 i (d) None of these / Geen van hierdie 10. (c) 1 i Question 11 / Vraag 11 The equation of the hyperbola with vertices3, 0 and 3, 0 and with foci 5, 0 and 5, 0 is Die vergelyking van die hiperbool met toppunte 3, 0 en 3, 0 en met fokuspunte 5, 0 en 5, 0 is 11. (a) 16x 2 9y (b) 16y 2 9x (c) x2 16 y (d) None of these / Geen van hierdie Question 12 / Vraag 12 The parabola y 2 4x has directrix Die parabool y 2 4x het riglyn 12. (a) y (b) x (c) y (d) x (e) None of these / Geen van hierdie 4

5 SECTION B (26 marks) / AFDELING B (26 punte) Question 13 / Vraag 13 [4] Find the equation of the plane that contains the point 2, 1, 3 and is parallel to both the vectors 1, 0, 1 and 2, 1, 0. Bepaal die vergelyking van die vlak wat die punt 2, 1, 3 bevat en ewewydig is aan beide die vektore 1, 0, 1 en 2, 1, 0. Question 14 / Vraag 14 [3] 1 0 Given / Gegee A. For what value(s) of x is A 1 A T (ie A is orthogonal)? x 1 Vir watter waarde(s) van x is A 1 A T (dws A is ortogonaal)? 5

6 Question 15 / Vraag 15 [4] Given / Gegee x 1 2x 2 x 3 x 4 0 x 1 2x 2 x 4 4 x 1 2x 2 2x 3 4x Write the system in the form A b / Skryf die stelsel in die vorma b Use Gaussian elimination to reduce A to row echelon form. Show all operations. Gebruik Gauss-eliminasie om A in ry-trapvorm te skryf. Toon alles bewerkings Find the solution (if any) of the system of equations and give the solution in vector form.. Bepaal die oplossing (indien enige) van die stelsel vergelykings en gee die oplossing in vektor-vorm. 6

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8 Question 16 / Vraag 16 [6] 16.1 The complex number z i in polar form is (mark your choice): Die komplekse getal z i in poolvorm is (merk jou keuse): z 4cis 3 z 2cis 3 z 4cis 6 z 2cis Use your choice in 16.1 to show that z 6 is a real number. (No need to simplify powers). Gebruik jou keuse in 16.1 om aan te toon dat z 6 n reële getal is. (Nie nodig om magte te vereenvoudig nie) De Moivre s Theorem for n-th roots. If z r cis, then the n-th roots of z are given by w k r 1 n De Moivre se stelling vir nde magswortels. Indien z r cis, dan word die nde magswortels van z gegee deur w k r 1 n k 0, 1,..., n 1. Find all values of z 1 2 (answers in a ib form). Bepaal alle waardes van z 1 2 (antwoorde in a ib vorm). cis 2k n, k 0, 1,..., n 1. cis 2k n, 8

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10 Question 17/ Vraag 17 [4] One of the zeros of the function fx x 3 9x 2 25x 25 is 2 i. Where does the function have a real root? Een van die nulpunte van die funksie fx x 3 9x 2 25x 25 is 2 i. Waar het die funksie n reële wortel? 10

11 Question 18 / Vraag 18 [4] Consider the conic sections / Beskou die keëlsneë 9x 1 2 4y 2 36 x 2 9y Sketch the graphs of the two conic sections on the same set of axes. Indicate all foci, asymptotes and vertices of the conic sections on the sketch. Skets die grafieke van die twee keëlsneë op dieselfde assestelsel. Toon alle fokuspunte, asimptote en toppunte van die keëlsneë op die skets Consider / Beskou 9x 2 2 4y 2 36 x 2 9y 2 9 How many points of intersection do the two curves have? Hoeveel snypunte sal die twee krommes hê? 11

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13 Question 19 / Vraag 19 [2] Use Cramer s rule to solve for x and y in terms of x and y if Gebruik Cramer se reël om x and y op te los in terme van x en y as cosx siny x sinx cosy y 13

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