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Essential Mathematical Skills and Techniques-exercise semsister 1, Exercises of Mathematics

The document is a collection of exercises and solutions for MAS152: Essential Mathematical Skills and Techniques, focusing on first-semester topics. It covers a wide range of mathematical concepts, including complex numbers, functions of real variables, partial differentiation, vectors, and hyperbolic functions. The exercises are designed to reinforce understanding through problems on curve sketching, domain and range analysis, inverse functions, binomial expansions, and vector operations. Each section includes step-by-step solutions and hints, making it a practical resource for students to practice and master foundational mathematical skills. The document is well-structured, with clear explanations and examples, making it suitable for both self-study and classroom use.

Typology: Exercises

2024/2025

Available from 03/14/2025

charles-khama
charles-khama 🇮🇹

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3/14/25, 9:00 PM MAS152 - Semester 1 exercises MASI52 Essential Mathematical Skills & Techniques Examples 5: Complex Numbers 1 1, Express the following in the form z=a+ib (where aand b are real numbers) and write down the complex conjugate, modulus and principal argument of each of the resulting complex. numbers. (a) 3 + 4i) + 2 + 21) © Vi () @ +398 -2i) @ a+ira ~i) 2. Plot the following on the Argand diagram (@) 243i, ) 2-i, © 4-3, @ 342i 3. Given that x and y are real, write x yin iii in the form a+ib (where a and b are real). Hence find the values of x and y which satisfy the equation Tei 1-1 ° 4, Find the locus of the complex numbers z that satisfy [z+ 2i-3|=|z+ 3i Find the complex number that lies on this locus and has arg z= 7/4. 5. Find the roots of the equation P-(2 + i?+z-2 [Hint: look for an ‘obvious’ root.] Answers 1. @) 2=5 + 64, Z=5 ~6i, [z|=(2] ‘en ‘Argz = 0.8761, Argz () 2=12 + Si, Z=12 ~5i, |z|=[z)= 13, Arge = 0.3948, Argz= (©) z=-i, z=i, (Z| 1, Args (2, Argz = n/2 @ 2=-14i,7 zl=[z)= 2, Argz = 3n/4, Argz =—31/4, 3. Qx/5-y) + iG ~y-w/5);x= 5, y= 2. 4, Locus: y=-3x+ 2. 2=(1+ i/2 5. 2+i, 41 about:blank 1/46