5th Tutorial: Complex Numbers

5th Tutorial: Complex Numbers


2. Work out the following equations.


(a) X1 = 2 + j; X2 = 2 - j (d) X4 = 1


X = 1, -1 for real roots; X = j, -j for imaginary roots


3. Determine the real and imaginary components of the complex numbers below.


(b)


-3 for the real part; -3 for the imaginary part -7j (e) = 6 + 2j - 3j - j2 = 6 - j - j2 = 6 - j - j2 = 6 - j - (-1) = 7 - j


7 for the real part; -j for the imaginary part


4. Reduce the following complicated numbers to their simplest form.


(e) = 12 + 21 j\s(f) = 25


5. Condense: (h) = j I = 1-3j


6. By equating real and imaginary parts, solve the following equations for x and y :


(a)


X + yj = 9-7j


If x =0; y = -7-9j


If y =0; x = 9-7j


10. For the following complex numbers


(i) represent each on an Argand diagram


(ii) give the modulus and argument of each


(j) (m)


Tutorial 6: Differentiation 1


1. Using the rules of differentiation find the derivative of each of the following functions.


(c) =


(g) = =


(h) = =


(m) 3x3 + 2x2 + x + 1 = 9x2 + 4x + 1


2. Find the derivative of each of the following functions of x by first simplifying the expression and then using an appropriate rule of differentiation.


(c) = = =


(d) =


5. Find when


= 12q2 + = 12q2 +


8. Differentiate the following using the function of a function rule:-


(c)


(e)


(h) ln(3x2 – 5x + 8)


(l) 2sin(x2 + 1) + 3cos(x2 + 1)


(c) =


(e) = =


(f)


(l) 2cos(x2+1)(2x) + -3sin(x2+1)(2x) = 4xcos(x2 +1)- 6xsin(x2 +1)


10. Using the product rule with the specified values of u and v, differentiate the following:-


(a) y = (x + 1) (2x + 3); u = x + 1, v = 2x + 3


y’ = (x2+1)(6x2) + (2x) (2x + 3) = (6x4+6x2) + (4x4+6x)


11. Differentiate the following functions using the quotient rule:-


(a)


y’ = =


Tutorial 7: Differentiation 2


1. Find the maximum and minimum points of the following functions of x. For each one state whether it is a local maximum or a local minimum.


4. Find and classify the stationary point(s) of the following function


5. Find for the following implicit functions:


(a) (c)


7. Find the gradient of the curve at the point .


9. Find and in terms of the parameter t for each of the following:


(a) x = 2 + t; y = 1 + t2 (d) x = cos3(t), y =sin3(t)


Tutorial 8: Integration 1


1. Write down the following indefinite integrals:


(g) = +C


(j) = x + x2 + x3 + x4


2. Find the indefinite integrals of the following functions:-


(e) = (f) = -cos5x + sin4x +C


3. Evaluate the following definite integrals:-


= 41/6


= 6.68


4. In each of the following cases carry out algebraic simplification on the integrand before attempting integration and then evaluate the integral:-


= + C


7 By first sketching the area required, find the area enclosed by the following:


(i) y = x2, the x-axis and the lines x = 2 and x = 5 ;


(iv) y = ex, the x-axis and the lines x = 2 and x = 3 ;


Tutorial 9: Integration 2


1. Find the following indefinite integrals by the method of partial fractions. If the fraction is not proper then you will need to do a long division first.


(b) =


(c) . =


2. Evaluate the following definite integral:


. = 3.6027


5. Find the general solutions of the following differential equations:


(c) . = = = y = ex(x-1) + C


(d) = 3x2(4 + x3) = = = y =


Tutorial 10: Integration 3


1. Use the substitution indicated to evaluate the following definite integrals (but please note that the substitutions are obvious enough that you should be able to spot them yourself!) .


(a) (u = 2x + 3) =


(d) (u = sin x). = 1/3


2. Integrate each of the following indefinite integrals, using appropriate substitutions:


(a) = 2x4 + 2x3 + + x + C


(d) = log(sin(x)+1) + C


4. Using integration by parts, find each of the following indefinite integrals:


(b)


Let u = (2x+1), du = 2dx, dv = cosx, v = sinx


= 2xsinx + sinx – (-2cosx + C)


= 2xsinx + sinx + 2 cosx + C


5. Integrate each of the following definite integrals:


(b) = 0.6363


6. Use the method of integration by parts to show that


.


Let u = e3x, du = 3e3x dx, dv = sin2x, v = -cos2x/2


=


Tutorial 11: Differential Equations


1. Using the technique of variable separation solve the following differential equations.


(c) = y dy = x2 dx = = = = y =


(e) = = = = ln(e-y) =ln x4 = y = -ln(x4)


(i) = = = = lny = ln(1+x) = e^lny = e^ln(1+x) = y = x+1


2. Find the particular solution of the following differential equations for the given conditions.


(b)


4. For the differential equation


(a) find the particular solution


(b) given that y = –2 when x = 0, find the particular solution.


(c) find the value of y when x = 2.


6. A population is modelled by the differential equation = 4t2 + 100. If the initial population is 2000, solve the differential equation to find the population after 2 years, 10 years and 100 years.


Tutorial 12: More Complex Numbers


1. By first expressing in polar form, find the Cartesian form of the complex numbers with the following modulus and argument. In each case sketch an Argand diagram to check that your answer is reasonable.


(a) (d)


2. Give the Cartesian form and the modulus and argument of each of the following. In each case sketch an Argand diagram to check that your answer is reasonable.


(d) .


3. Given that and find the following in polar form (without converting to Cartesian form).


(a) (c) (f)


6. Use the result that if then to find the square root of .


Tutorial 13: Vectors


1. If a = 3i – 2j and b = i + 4j find a, 3a + 2b, a – b, and .


a = = 3.61


3a + 2b= 3a + 2 b = 11i + 2j = = 11.18


a-b = 3i – i – 2j – 4j = 2i -6j


= 3î – 2ĵ


5. Find the lengths of the diagonals of the parallelogram with sides given by the


vectors 3i + 2j and 4i – j.


6. Show that the points A(1, 2, 4), B(2, 1, 0) and C(3, 0, –4) lie on a straight line.


Tutorial 14: Lines, Planes and Intro to Matrices


1. Write down the vector equation of the straight line which passes through the point A (1,2, –1) in the direction of the vector 2i + j – k.


2. Find the Cartesian equations of the planes containing the points P, Q, R where


(i) P (2, 1, 3) , Q (1, 2, 1) and R (–1, -2, –2)


3. Find the common points of the following pairs of lines.


(iii) r = i + j + k + (j – 3k) and r = i + (k – j)


5. Find the intersections of the following pairs of planes


(i) r = 2i + (j – 2k) + k and r = 2i + j + k + k + (2k – j)


7. Find (a) A + B (b) A - B (d) 2A + 5B


where A = , B =


A+B =


A-B =


2A+5B =


9. Find (a) AB (b) BA (c) AT (d) AI I IA where I is the identity matrix and :


(i) A = , B =


AB = =


BA = =


AT =


10. Evaluate the following matrix products, where possible :


(a) =


(b) = cannot be evaluated


(c) = [1(2) + -2(1) + 4(-3)] = -12


14. Write down A-1 where A = and hence solve


Tutorial 15: Determinants


1. Evaluate the following determinants :


(a) = D = (4x3) – (-3x1) = 12+ 3 = 15


(c) D = (0 -2 + 8) – (0 + 2 – 16) = D = 6 – (-14) = D = 20


2. Evaluate the following in the easiest way possible :


(d)


Tutorial 16: Multiplying Vectors


1. Find a . b, a b and the area of the parallelogram with sides a and b, in each of the following cases:


(i) a = 2i + j + 3k and b = i – 4j + 2k


(v) a = 3i + 2k and b = i – 4j + 5k


5. Find a . b c when:


(i) a = i – 2j – 3k, b = 2i + j – k and c = i + 3j – 2k


(v) a = i – j – 2k, b = 4i + j + k and c = – 4i + 3j


6. Find the volume of the parallelepiped whose edges are represented by the vectors:


(i) a = 2i – 3j + 4k, b = i + 2j – k and c = 3i – j + 2k.


(v) a = 2i + j – 3k, b = 2i + 4j – k and c = 2i – 5j – k.


7. Find a unit vector perpendicular to both a = 3i + j – 2k and b = i –2j + k.


Tutorial 17: Solving Systems Of Linear Equations


1. Use Gaussian elimination to convert each of the following systems into echelon form, then use back-substitution to find the solution.


N.B. be careful to write the equations so that each variable appears in its own column, and in (c) you will find it helpful to re-order the equations.


(c)


2x2 + 5x3 = 6


x1 – 2x3 = 4


2x1 + 4x2 = –2


(e)


2x + 5y = 10


4x – 3z = –26


y + 4z = 12


2. Use Gaussian elimination to solve each of the following systems, where possible. When there is more than one solution, give the solution set in Cartesian form.


(b)


2x2 + 5x3 = 6


x1 – 2x3 = 4


2x1 + 4x2 = –2


(f)


x1 + 3x2 + 2x3 + 5x4 = 10


3x1 – 2x2 – 5x3 + 4x4 = –5


2x1 + x2 – x3 + 5x4 = 5


Tutorial 18: More Matrices


2. Find the inverse of each of the following matrices using the row operation method. You should check your answer by checking that AA-1 = I. If you have made an error, then search for it in the way described at the end of the notes.


(a)


3. Use the inverse matrix method to solve the following :


(b) x =


6. Find the inverse of each of the following matrices using the cofactors method. You should check your answer by checking that AA-1 = I.


(a)


7. (a) Write down A-1 where A = and hence solve


Tutorial 19: Series and Sequences


1. Find the sums of the following series:


(c)


S = a1 / (1-r); r = = s = 1/ (1- =


2. Find (in terms of n) for the series:


(a)


S = n/2 (a1 + an) =


5. Express as fractions the following recurring decimals:


(b)


6. The 1st, 3rd and 9th terms in an A.P. are also the first three terms in a G.P. If the first term is 2, find the common difference of the A.P.


8. Find the next two terms, and an expression of the nth term, for the following sequences:


(b)


(d)


9. Determine whether the following sequences converge. Give the limit of any that do converge. Otherwise state whether the terms tend to or , or whether the sequence oscillates.


(a)


(b)


(c)


(e)


(f)


(i)

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