MAT1503 Assignment 5 (QUALITY ANSWERS) 2024

This document contains workings, explanations and solutions to the MAT1503 Assignment 5 (QUALITY ANSWERS) 2024. For assistance whats-app us on 0.6.8..8.1.2..0.9.3.4... Question 1: 12 Marks (1.1) Let U and V be the planes given by: (2) U : λx + 5y − 2λz − 3 = 0, V : −λx + y + 2z + 1 = 0. Determine for which value(s) of λ the planes U and V are: (a) orthogonal, (2) (b) Parallel. (2) (1.2) Find an equation for the plane that passes through the origin (0, 0, 0) and is parallel to the (3) plane −x + 3y − 2z = 6. (1.3) Find the distance between the point (−1,−2, 0) and the plane 3x − y + 4z = −2. (3) Question 2: 11 Marks (2.1) Find the angle between the two vectors ⃗v = ⟨−1, 1, 0,−1⟩ ⃗v = ⟨1,−1, 3,−2⟩. Determine (3) whether both vectors are perpendicular, parallel or neither. (2.2) Find the direction cosines and the direction angles for the vector ⃗r = ⟨0,−1,−2, 3 (3) 4 ⟩. (2.3) HMW:Additional Exercises. Let ⃗r (t) = ⟨t,−1t , t2 − 2⟩. Evaluate the derivative of ⃗r (t)|t=1 . Calculate the derivative of V(t) · ⃗r (t) whenever V(1) = ⟨−1, 1,−3⟩ and V′(1) = ⟨1,−2, 5⟩. (2.4) HMW:Additional Exercises. Assume that a wagon is pulled horizontally by an exercising force of 5 lb on the handle at an angle of 45 with the horizontal. (a) Illustrate the problem using a rough sketch. (b) Determine the amount of work done in moving the wagon 30 lb. 36 MAT1503/101/0/2024 (2.5) HMW:Additional Exercises. Let the vector ⃗v = ⟨3500, 4250⟩ gives the number of units of two models of solar lamps fabricated by electronics company. Assume that the vector ⃗a = ⟨1008.00, 699, 99⟩ gives the prices (in Rand-ZA) of the two models of solar lamps, respectively. (a) Calculate the dot product of the two vectors ⃗a and ⃗v. (b) Explain the meaning of the resulting answer you obtain in the question above. (c) Let assume that the price of original price of the solar lamps has decreased by 10%. Identify the vector operation used for this case. (2.6) HMW: The force exerted on a rope pulling a toy wagon is 30 N. The rope is 30 above the horizontal. (a) Illustrate the problem by means of a sketch (b) Determine the force that pulls the wagon over the ground. (2.7) Show that there are infinitely many vectors in R3 with Euclidean norm 1 whose Euclidean (4) inner product with −1, 3,−5 is zero. (2.8) Determine all values of k so that ⃗u = −3, 2k,−k is orthogonal to ⃗v = 2, 5 (1) 2 ,−k . Question 3: 8 Marks (3.1) (a) Find x and y such that 4xi + (1 + i)y = 2x + 2yi. (1) (b) Let z1 = 12 + 5i and z2 = (3−2i)(2 + λi). Find λ without resorting to division such that (1) z2 = z1. (3.2) Let z = 1 − 3i and z′ = 1 + 2i. Determine the complex numbers (a) z2 − zz′ (2) (b) 1 (2) 2 (z − z)2 (c) 1 (2) 2 [z − z] + [(1 + z′)]2. Question 4: 23 Marks (4.1) Determine the complex numbers i−2667 and i−348. (2) (4.2) Let z1 = −i (6) −1+i , z2 = 1+i 1−i and z3 = 1 10 h 2(i − 1)i + (−i + √ 3)3 + (1 − i)(1 − i) i . Express z1z2 z3 , z1z2 z3 , and z1 z3z2 in both polar and standard forms. 37 (4.3) HMW: Additional Exercises: Express z1 = −i, z2 = −1−i √ 3, and z3 = − √ 3 + i in polar form and use your results to find z4 3 z2 1 z−1 2 . Find the roots of the polynomials below. (a) P(z) = z2 + a for a 0 (b) P(z) = z3 − z2 + z − 1. (4.4) (a) Find the roots of z3 − 1 (4) (b) HMW: Find the roots of z3 + 1. (c) Find in standard forms, the cube roots of −8 + 8i (3) (d) Let w = 1 + i. Solve for the complex number z from the equation z4 = w3. (4) (4.5) Find the value(s) for λ so that α = i is a root of P(z) = z2 + λz − 6. (4) Question 5: 4 Marks Find the roots of the equation: (5.1) z4 + 16 = 0, z4 − 16 = 0 and z3 − 64 = 0 (4) (5.2) Additional Exercises for practice are given below. Find the roots of (a) z8 − 16i = 0 and z8 − 16 = 0 (b) z8 + 16i = 0 and z8 + 16 = 0. Question 6: 3 Marks Determine for which value (s) of λ the real part of z = 1+λi 1−λi equals zero. Question 7: 16 Marks Use De Moivre’s Theorem to (7.1) Determine the 6th roots of w = −729i (3) (7.2) express cos(4θ) and sin(3θ) in terms of powers of cos θ and sin θ (6) 38 MAT1503/101/0/2024 (7.3) expand cos4 θ in terms of multiple powers of z based on θ (4) (7.4) express cos4θ sin3 (3) θ in terms of multiple angles. Question 8: 5 Marks (8.1) Let z = z1 (2) z2 where z1 = tan θ + i and z2 = z1. Find an expression for zn with n ∈ N. (8.2) Let z = cos θ − i(1 + sin θ). Determine (3)  2z + i −1 + iz  Question 9: 5 Marks Given that z = cos θ + i sin θ and u − iv = (1 + z)(1 + z2). Prove that v = u tan  3θ 2  r = 42 cos2  θ 2  , where r is the modulus of the complex number u ± iv. Question 10: 13 Marks Let z = cos θ + i sin θ. (10.1) Use de Moivre’s theorem to find expressions for zn and 1 (2) zn for all n ∈ N. (10.2) Determine the expressions for cos(nθ) and sin(nθ). (2) (10.3) Determine expressions for cosn θ and sinn (2) θ. (10.4) Use your answer from (10.3) to express cos4 θ and sin3 (4) θ in terms of multiple angles. (10.5) Eliminate θ from the equations (3) 4x = cos(3θ) + 3 cos θ 4y = 3 sin θ − sin(3θ).