Nelson Mandela University Mathematics for Engineering Technologists 1a PDF Past Paper - MATV101 - March 25, 2024

Mathematics for Engineering Technologists 1a MATV101 Semester Test 1: 25 March 2024 Total: 20 Marks Time: 50 minutes Special Instructions: Answer ALL questions on the multiple-choice answer shee...

Mathematics for Engineering Technologists 1a MATV101 Semester Test 1: 25 March 2024 Total: 20 Marks Time: 50 minutes Special Instructions: Answer ALL questions on the multiple-choice answer sheet provided. Do all your workings on the pages provided, clearly number each question’s workings. 1. Which of the following statements is TRUE? A) The equation x 2 = −3 has no solutions. B) The equation x 2 = −3 has REAL solutions. C) The equation x 2 = −3 has COMPLEX solutions. D) The equation x 2 = −3 has REAL and COMPLEX solutions. E) None of the above. 2. Let z =−4 + 7i, w =3 − 6i. Determine 4 z − 6 w. A) 2 − 8i B) −34 + 64i C) −34 − 8i D) −22 + 43i E) None of the above. v 3. Let v =− 8 3i, w =−11 + 5i. Determine. w A) −103 − 7i 103 7 B) − + i 73 73 103 7 C) − − i 146 146 1 1 D) − + i 2 2 E) None of the above.  ab 4. Let a =−3 − 8i, b =14 − i. Determine. b + 3a 99 359 A) + i 26 130 B) −50 − 109i C) −12 − 32i 731 995 D) − i 106 106 E) None of the above. u − 3w 5. Let z =10 − 2i, w =−4 + 3i, u =x + yi. Determine u given =w. z A) −46 + 47i 10 2 B) − i 3 3 10 2 C) − + i 3 3 D) 22 − 29i E) None of the above. 6. Let u ∈  be the additive identity. Then: A) Re ( u ) = 1. B) For z ∈ , z + u = 1. C) u = 1. D) For z ∈ , z + ( − z ) =u. E) None of the above.  4π   −2π  7. The statement 2cis   = 2cis   is:  3   3  4π −2π True, since the angles and have the same terminal side A) 3 3 on the Argand Plane. B) True, since both numbers have a modulus of 2. 4π −2π C) False, since ≠. 3 3 D) False, since in rectangular form the numbers are different. E) None of the above.   5π  8. The principle argument of 4cis   is:  3  π A) 3 π B) − 3 2π C) 3 5π D) 3 E) None of the above. 9. Given z = =− 3 − i, then z 3 =...  125π  A) 8cis  −   216   125π  B) 6cis  −   216   5π  C) 6cis  −   2   5π  D) 8cis  −   2  E) None of the above. 10. Given w4 =−2 + 2 3i, then w3 =...  5π  A) 2cis    3  1  20π  B) 4 4 cis    3   19π  C) 2cis    12  1  41π  D) 2 2 cis    12  E) None of the above. 11. p = [ P+PT =... i] (A) cannot be calculated 1 2.T (B) 2y y2 ] [ 2+X (C) ] [2 ! X 2y 2+ : (D) l2!.1: 2y rl (E) none of the other options given 12. A has size 5 x 5; B is a 5 by 2 matrix The third row of A is [6 1 0 1 OJ The third row of Bis [1 1] The first column of A is [1 0 The first column of B is [o 1 AB= (ci;j) p xq Then c3; 1 =... (A) 1 (B) 2 (C) 6 (D) 7 (E) none of the other options given  3 1 0 A is an example of a.... matrix (A) lower triangular (B) upper triangular (C) diagonal (D) symmetric (E) none of the other options given 3 1 6 det(C) is calculated using co-factor expansion using column two. So we get det(C) = 3A1;2 + lA2;2 + 6A3;2 Then we have A1;2 =... (A) I I (J3) (-1)(1+2) I I (C) (-l)(lx2) I I (D) -3(0) (E) none of the other options given  1 0 2 [ 1 15 0 1 4 ]. 0 0 6 4 =... (A) not4 able to be calculated 6 12 (B) [2 6p] (C) 8 rn ] 16 (D) [3 2 8p] (E) none of the other options given 16. 6x + 3y - z = 11 4y + z = 10 z=4 \,Ve apply Cramer's rule to determine the solution for variable x X =... 11 3 -1 1 (A) - 10 4 1 24 4 0 1 6 3 -1 1 (B) 24-0 4 1 0 0 1 11 3 -1 (C) 24---;-- 10 4 1 4 0 1 6 3 -1 (D) 24 7 0 4 1 0 0 1 (E) none of the other options given 17. A quick way to find x- 1 for a 2 x 2 matrix is X= [a then x- 1 = (det(X))- 1 [ d - bJ C -c a 2 If G= [ then c-1 =... p (B) 6 [ ! P (C) 6(6 - 0) (D) ¼(O - 6) (E) none of the other options given Then the following matrices will have a mutiplicative inverse (A) X·' y & z (B) X & y (C) X & z (D) y & z (E) none of the other options given 19. Given the following linear system x - 3y + llz = 7 2x + 4y = 8 3x + 9y + z = 7 Vve start using the Gaussian elimination technique to the augmented matrix and get... hint : look at column 1 [ -3 11 7 (A) 10 -22 -6 18 -32 -14l [ -3 _t] 11 (B) -2 -22 18 -32 [ -3 11 (C) 10 -22 0 -32 l4l [ ] -3 11 (D) 1 11 6 12 (E) none of the other options given 20. The following is an augmented matrix representation of a linear system [ ] 0 0 0 0 1 0 0 0 1 The solution set is... (A) {(x1;x2;x3) : (2;2; O)} (B) {(x1;x2;x3) : (1; -1;0)} (C) {(x1;x2;x3;x4): (2;2;2;0)} (D) {(x1;x2;x3;x4): (2;0;2;0)} (E) none of the other options given ANSWERS: 1. C 2. B 3. C 4. E 5. A 6. D 7. A 8. B 9. D 10. A