Department of Mathematical Sciences Tutorial Problems for MATH103, Foundation Module II Autumn Semester 2004

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1 Department of Mathematical Sciences Tutorial Problems for MATH103, Foundation Module II Autumn Semester 2004 Each week problems will be set from this list; you must study these problems before the following Tuesday s tutorial. Any difficulties will be discussed during the tutorial, and final versions of your solutions are to be delivered to your tutor by 2p.m. on the Thursday, two days after the Tuesday tutorial. They will be marked and returned to you at the following Tuesday tutorial. Solutions will be given to you on the same day, at the 9 a.m. lecture. There will be one class test during the semester, probably during Week 6. You will be given plenty of warning of this. No tutorial problems are to be handed in during that week, but revision problems will be suggested which should help you prepare for the test. The examination for this module takes place in January The syllabus differs from that in years before 2000/1 in that complex numbers were moved from MATH101 to this module, and as a consequence the amount of linear algebra in this module was reduced. A mock examination paper is included at the end of these problems and also the January 2001 and January 2002 papers. Solutions to these and other January examinations will be distributed later. Papers from earlier years are not so suitable for revision. By solving the homework problems each week you will be well prepared for the examination. If you lose this problem collection then you can obtain another copy from the website: Either use VITAL or follow links from via Information for Current Students, then Some course notes and example sheets, then MATH103. Also the solutions will be posted on this website week by week, on the same day that a printed copy of the solutions is given out in class. Not all questions will be set for homework. The remaining questions (apart possibly from a small number at the end, the material for which may not be reached in the time available) are suitable for extra practice. A few questions have been marked (H) meaning that though they are on the material of this module they are a little more challenging than the other questions. Problems on algebraic manipulation and trig functions Note. The first few problems are of a revision nature and similar problems may not have been tackled in class. 1. Cancellation Which of the expressions ab + ac, ad ab + bc, b ab + ac bc can have something cancelled from numerator ( = top) and denominator (= bottom) of the fraction? 1

2 Factorise x 2 2x 35 and 25 x 2. Simplify the following expression as far as you can: (x + 5)(x 2 2x 35) + (x 7)(25 x 2 ) x 2, 2x 35 where x 2 2x 35 is assumed to be 0 to avoid a zero denominator. Which values of x does this exclude? 2. (i) Assume that x and y are real numbers and x 0, y 0. Express 1 x + 1 y Assume in addition that x + y 0. Show that the equation 1 x + 1 y = 1 x + y as a single fraction. (1) is equivalent to the equation (ii) By completing the square show that (2) is equivalent to x 2 + xy + y 2 = 0. (2) ( x y ) y2 = 0. Why is the only real solution to this x = y = 0? What can you deduce about real solutions to (1)? 3. Let x > 0. Show that x + 1 x 2. (Hint: multiply up by x and rearrange.) Now also let a > 0. Replace x by ax (which will also be > 0) in the inequality just proved. By choosing a suitable value for a prove that 3x + 1 x Phantom solutions of equations This exercise illustrates how one must be careful if an equation is squared before solving it. Note that a, for a > 0, always means the square root of a which is > 0. For example, 9 means 3, and 9 is 3. (i) Consider the equation x 1 = 4x 2 (where x is assumed to be real). Square the equation and rearrange and factorise the result to obtain x = 1 or x = 1 3. Is either of these a solution to the original equation? (ii) Do the same for the equation x + 1 = 4x 2. (iii) Do the same for the equation 2x 3 = 2x 2 7x + 7. [If you know about the graph of y = 4x 2 = 2 x you might like to draw this graph and the graphs of y = x 1, y = x + 1 to explain (i) and (ii).] The moral of this question is: if you square an equation in order to solve it, then check the resulting solutions in the original equation to see if they fit. 5. For this question and the next you need to remember the vital formulae tan θ = sin θ cos θ, sin2 θ + cos 2 θ = 1, sin(2θ) = 2sin θ cos θ. cos(2θ) = cos 2 θ sin 2 θ = 2cos 2 θ 1 = 1 2sin 2 θ. You should always remember these formulae. Express (tan θ + (tan θ) 1 ) 1 in the form asin(nθ) for constants a and n. 6. (a) Complete the square on the expression x 2 3x. Use this (rather than the quadratic formula) to solve the equation x 2 3x + 1 = 0. (b) Find all solutions of the equation 2 x 2 1 x 1 = x. 2

3 (c) Find all the values of θ in the range 0 < θ < 2π which satisfy the equation cos(2θ) = 6cos θ 3. [See one of the formulae in the previous question.] Problems on Complex Numbers 7. Let z = 1 + i 2. Work out z 2 and show that z 2 + 2z = 3. What does the quadratic formula give for the solutions of z 2 + 2z + 3 = 0? 8. Let z = i 3 2. Show that z2 = 1 2 i 3 2 [it so happens that for this z, z 2 is the complex conjugate z of z], and that z 3 = 1. Without further calculation, write down z 4, z 5, z 6 and 1 z. [Note that 1,z and z 2 are three complex numbers, each having cube equal to 1.] 9. Let z = i 3 2. Show that z2 = i 3 2 and that z3 = 1. Without further calculation, write down z 4, z 5, z 6 and 1 z. Write down three different complex numbers, each having cube equal to 1. [One of these three is 1 itself: remember the phrase complex numbers includes in particular all real numbers.] 10. Find the values of (1 + i + i 2 + i 3 + i 4 ) 100 and i Write down four different complex numbers, each having fourth power equal to Let z = 1+i 2. Show that z 2 = i and deduce that z 4 = 1. Find four different complex numbers, each having fourth power equal to 1. [Hint. (z) 4 = z 4. What this says is that, if z 4 = 1, then z 4 will be 1, which is 1 since 1 is real.] 13. Find nonzero complex numbers z and w such that z 2 + zw + w 2 = 0. [Compare Exercise 2 above.] 14. Let z = 4 + 3i and w = 3 i. Find 2z 5w, (z + w) 2 and iz/w, each in the standard form a + ib with a and b real. Write down z and w. Verify that 2z 5w = 2z 5w, (z + w) 2 = (z + w) 2, iz/w = iz/w. Find also z, w and zw and verify that zw = z w. 15. Write down the binomial expansion of (x + iy) 4, and hence write down the imaginary part, Im((x + iy) 4 ). Use this result to verify that Im((cos θ + isin θ) 4 ) = sin(4θ) (you ll need some of the boxed formulae from Question 5 above). 16. Let z = 3 i. Find the real and imaginary parts of z z Find the real and imaginary parts of (1 2i) 2 (5 + i) (3i 2)(3 + 4i). 18. Let α = 2π/5. Mark w = cos α+isin α on a diagram of the complex plane and also mark w 2,w 3,w 4 and w 5. Let z = w 2. On a separate diagram mark z,z 2,z 3,z 4,z Given that w = 1 + i 3 and z = 3 i, find wz and draw a diagram of the complex plane showing the points w, z and wz. Write down the arguments of these three complex numbers and verify from your answers that arg(wz) = arg(w) + arg(z). (Remember that this means that if you take the principal values of the three arguments, each lying between π and π, the two sides are either equal or differ by 2π.) 3

4 20. Find the arguments of the following complex numbers. Indicate their positions on a diagram [you should always do this anyway when finding the argument]. 4, 3i, 5i, 1 + 3i, 3 i, 2 3i. (For the last one use a calculator, for the others don t!) 21. In each of the following cases, write z in polar form r(cos θ + isin θ) and in exponential form re iθ. Then use de Moivre s theorem to evaluate z n. Express your answers in the form x + iy. (a) z = 1 + i, n = 12; (b) z = 1 i 2, n = 6; (c) z = 1 i 3, n = (Exponential rules) Simplify without a calculator: a b a c = a b+c, (a b ) c = a bc, (ab) c = a c b c (a) , (b) , (c) Simplify as much as you can: ( 3 a 6 b 3 4 ) 1/2 1 a 6 b 2 9 a4 b 6, where a and b are taken as > Find all the solutions of the following equations, marking them on a separate diagram for each part. Express your answers in the form x + iy. (a) z 2 = 1 + i 3, (b) z 3 = i, (c) z = 0, (d) z = 8i 3. (Recall the procedure: get everything except the z term on the right hand side, so for (c) this gives z 3 = 27 = a,say. Write the r.h.s. as a e iα and z as re iθ, and equate z n = r n e niθ and a for the appropriate n. It helps to remember that the solutions of z n = a are always equally spaced around a circle centred at the origin, separated by angles 2π/n.) 25. Complete the square on the expression z 2 2iz and hence show that the equation z 2 2iz (1+2i) = 0 reduces to (z i) 2 = 2i, that is z i = ± 2i. To solve this, write w 2 = 2i and solve for w: this will give the two values of 2i. Hence write down the solutions of the equation in z. 26. (H) Let z = cos θ+isin θ. Verify that 1/z = cos θ isin θ, z+(1/z) = 2cos θ and z 1/z = 2isin θ. Use de Moivre s theorem to show that Deduce that z n + 1 z n = 2cos(nθ), zn 1 z n = 2isin(nθ). 16sin 5 θ = cos 5θ + 5cos 3θ + 10cos θ, 16sin 5 θ = sin 5θ 5sin 3θ + 10sin θ. 27. (H) Recall that any point on the line through the complex numbers w and z can be written w + λ(z w) for a real number λ, and that the middle point on the segment joining w and z is 1 2 (w + z). Let a,b,c be three complex numbers forming a triangle in the plane. Consider the medians of the triangle, joining each vertex to the midpoint of the opposite side. Show that the point 1 3 (a + b + c) lies on all three of these medians (which are therefore concurrent in this point). [Hint. For the median joining a to the midpoint 1 2 (b + c) of the opposite side you want to find a real number λ such that a + λ( b+c 2 a) = 1 3 (a + b + c).] 4

5 Problems on Linear Algebra 28. The vector u = (3, 4,2) is represented by a segment starting at (4, 1,5). Where is the endpoint of the segment? The same vector is represented by a segment ending at (7,2,3). Where is the beginning point of the segment? What are the components of the vector v represented by the segment joining ( 1,2,4) to (3,0, 7)? Let A, B, C be three points with coordinates (3, 1, 2), (4, 5, 3), (1, 7, 4) respectively. State with a reason whether each of the following is true or false. (a) the position vector a of A is (3, 1,2), (b) v = (1, 4, 1) is a scalar multiple of AC, (c) v = (1, 4, 1) is a scalar multiple of AB, (d) AB + BC + CA= (0,0,0), (e) The point (2, 7, 2) lies on the straight line through A and B. 29. Let u = ( 3,1,2), v = (4,0, 8) and w = (6, 1, 4). Find the vectors (i) v w (ii) 6u + 2v (iii) v + u (iv) 5(v 4u) (v) 3(v 8w) (vi) (2u 7w) (8v + u). 30. Let P be the point (2,3, 2) and Q the point (7, 4,1). (a) Find the midpoint of the line segment connecting P and Q. (b) Find the point on the line segment connecting P and Q that is 3 4 of the way from P to Q. 31. A median of a triangle joins a vertex to the mid-point of the opposite side. Show that the three medians of a triangle whose vertices A,B,C have position vectors a, b, c all pass through the point (a + b + c) [this is called the centroid of the triangle]. 1 3 Compare Question 27 above, but NOTE that the present question allows the three vertices to be in R 3 as well as in R 2. One of the great advantages of vector notation is that calculations are often the same in R 2 or R 3, or indeed in any dimensional space. The EXCEPTION to this is the vector (cross) product, which is only defined in R Let A,B,C,D be four points in space, with position vectors a,b,c,d respectively. (We shall assume the four points do not lie in a plane. They are called the vertices of a tetrahedron.) Show that the point 1 4 (a + b + c + d) lies on the segment joining A to the centroid of triangle BCD. Deduce that all four lines joining a vertex of the tetrahedron to the centroid of the opposite triangle pass through one point. 33. Let u = (2, 2,3), v = (1, 3,4). Find (i) u (ii) v (iii) u + v. Verify that u + v < u + v. 34. Let P = (1,1, 4), Q = (3, 2,2) be two points in R 3. Find P Q and calculate the distance between P and Q. 5

6 35. In each of the following, find u v and calculate the cosine of the angle θ between u and v. (i) u = (2,3), v = (5, 7) (ii) u = (1, 5,4), v = (3,3,3) (iii) u = i + 2j + 3k, v = 3i 2j k. 36. In each of the following, find the distance between the point P and the given line. (i) 4x + 3y + 4 = 0, P = ( 3,1), (ii) y = 4x + 2, P = (2, 5), (iii) 3x + y = 5, P = (1,8). 37. (a) Suppose v 1,v 2,v 3 are three vectors in R 3 which are mutually orthogonal (v 1 v 2 = v 2 v 3 = v 3 v 1 = 0), and of unit length. Show that v 1,v 2,v 3 are linearly independent. [So start with λ 1 v 1 + λ 2 v 2 + λ 3 v 3 = 0. Try taking the dot product of this equation with v 1.] Since there are three vectors in R 3 they automatically span R 3. The three vectors are said to form an orthonormal basis of R 3. Show that, for any vector a in R 3 we have a = (a v 1 )v 1 + (a v 2 )v 2 + (a v 3 )v 3. Thus it is very easy to express any vector a as a linear combination of orthonormal basis vectors. (b) Show that the vectors ( ) 1 1 v 1 = 2,0,, 2 v 2 = (0,1,0), ( 1 v 3 = 2,0, 1 ) 2 form an orthonormal basis of R 3. (c) Express the vector (1,1,1) as a linear combination of the basis vectors in (b). 38. Let v = (1, 2, 1), w = (4,2,3). Find the vector product v w, and find a unit vector parallel to v w. Using the formula v w = v w sin θ where θ is the angle between v and w, find sin θ. Can you tell from this whether θ is acute or obtuse? Find also cosθ from the formula v w = v w cos θ. Verify from the two values you have found that sin 2 θ + cos 2 θ = 1 and find θ to 2 decimal places in degrees and in radians using a calculator. 39. Let u = 2i + j + 5k, v = i + j + k. Find (a) a vector w which is orthogonal to both u and v (b) the area of the parallelogram determined by u and v. 40. Let A = (1,0,1), B = (0,2,3) and C = (2,1,1). Find (a) a vector w which is orthogonal to the plane of the triangle ABC, (b) the area of the triangle ABC, (c) the length of the altitude of the triangle from vertex C to side AB. 41. Find the volume of the parallelepiped which has one vertex at the point P = (0, 1, 1) and the three vertices adjacent to this one at Find also A = (3,1,0), B = (1,2,2), C = (1,0,1). (a) the area of the face of the parallelepiped determined by PA and PB (b) the cosine of the angle between the normal to this face and PC. 6

7 42. Find an equation for the plane through the three points P = (1,1,0), Q = (1, 1,2), R = (2, 2,1). Find also the distance between this plane and the point T = (1,2,3). 43. A triangle in the plane has vertices (1,2),(3,5) and ( 5,6). Find the cosines of its three angles. 44. Let u, v and w be the same vectors as in Question 29. Find scalars a, b and c such that au + bv + cw = (2,0,4). 45. Which of the following vectors in R 3 are linear combinations of the vectors u = (1,1, 1) and v = (0,1,2)? (i) (3,2, 5), (ii) (2,1,4), (iii) (0,0,0). In each case where the given vector is a linear combination of the vectors u = (1,1, 1) and v = (0,1,2), express it in the form λ 1 u + λ 2 v. 46. In each of the following, say (with reasons) whether the given vectors in R 3 are linearly dependent or independent. (i) (1,2,1), (2,4,2); (ii) (1,2,1), (2, 3,3); (iii) (1,2,1), (2, 3,3), (1,0,0); (iv) (1,2,1), (2, 3,3), (1,0,0), (2,7, 1). In each case where the given vectors v 1,..., v r are linearly dependent, find a non-trivial solution of the vector equation λ 1 v λ r v r = 0 (that is, find values of λ 1,...,λ r not all zero satisfying the vector equation). 47. Find three linearly independent vectors in R 3, two of which belong to the plane x 2y 2z = In each of the following, say (with reasons) whether or not the given vectors span R 3. (i) (1,0,1), (0,1,1); (ii) (1,0,1), (0,1,1), (1,1,0); (iii) (1,0,1), (0,1,1), (1, 1,0); (iv) (1,0,1), (0,1,1), (1,1,0), (2,1,3). 49. In each of the following, say (with reasons) whether the given vectors in R 3 are linearly dependent or independent, and whether or not they span R 3 : (i) (3,1,1), (2,1,2); (ii) (3,1,1), (2,1,2), (1,1,4); (iii) (3,1,1), (2,1,2), (1,1,4), (0,2,3). 50. Let v 1 = (1,1,2), v 2 = (0, 1,5), v 3 = (3,4,1). Show that these vectors are linearly dependent. Choose two of them and then find another vector so that your three vectors are linearly independent. [Hint: for the extra vector you can always choose one of these: (1,0,0), (0,1,0) or (0,0,1).] 7

8 51. As in the previous question, let v 1 = (1,2,3), v 2 = (2,0,0),v 3 = (5,4,6). Show that these are linearly dependent. Choose two of them. Can you take these two and (1,0,0) to make three LI vectors? What about the chosen two and (0,1,0)? 52. Reduce the following matrices to row echelon form: (i), (ii), (iii), (iv) Find all possible products of two of the following matrices: ( ) A =, B = , C = Find also any of the products A 2,B 2,C 2 which exist. 54. Calculate A 2 and A 3 where A = Verify the associative law (AB)C = A(BC) for the matrices of Question 53. [In other words, show that working out AB and multiplying on the right by C gives the same answer as working out BC and multiplying on the left by A.] ( ) ( ) 1 1 a b 56. Let A =. Find all 2 2 matrices B = such that AB = BA. 0 1 c d ( ) ( ) ( ) ( ) Find a matrix A such that A =. Is it true that A =? Find a 2 2 matrix C (with real entries) such that C 2 = I (H) Find (i) a 2 2 matrix A such that A 0 but A 2 = 0; (ii) 2 2 matrices B and C such that B 2 = C 2 = ( ) and BC = CB. 60. In the following, A and B are square matrices of the same size. Multiply out the matrix expressions: (i) (A B)(A + B) [Hint: the answer is not A 2 B 2!] (ii) (A + 2B)(2A + 3B) (iii) 2(A + 2B) + 4(A B). 61. Remember the basic rules for transposes: Rule 1: (A + B) = A + B ; Rule 2: (AB) = B A ; Rule 3: ((A) ) = A 8

9 (a) A square matrix A is called symmetric if A = A. Give an example of a 3 3 symmetric matrix and one that is not symmetric. Let B be any m n matrix. Show that BB is symmetric. [Hint. This is easy if you let A = BB and work out A using the above rules.] Make it clear which rule or rules you are using. Show also that when B is square, B + B is symmetric. [If B isn t square, B and B aren t the same size so can t be added; that s why we have to assume B is square here.] Again make it clear which rule or rules you are using. (b) A square matrix A is called skew-symmetric if A = A. Given an example of a 3 3 matrix which is skew-symmetric and one that is not skew-symmetric. Let B be any square matix. Show that B B is skew-symmetric. Again make it clear which rule or rules you are using. 62. Find all solutions of each of the following systems of equations. Draw graphs to illustrate your answers. (i) x + y= 1 (ii) x + y = 1 (iii) x + y = 1 x + 2y= 1 2x + 2y = 2 2x + 2y = Solve the equations x + y 2z = 1 2x 3y + z = 2 3x y + z = Find the values of p,q,r which make the curve y = p + qx + rx 2 pass through the three points (1,4),(2,4),(3,2). 65. Find all solutions (x,y,z) (if there are any) of the equations x + y z = a x y 3z = b 2x + 3y z = c in the cases (i) a = 0,b = 0,c = 0, (ii) a = 1,b = 1,c = 2, (iii) a = 1,b = 2,c = Solve the equations 67. Find A 1 where A = ( ). 68. Find the inverse A 1 of the matrix 2x + y + 3z = 0 3x + 4y + 2z = 5 x + 2y 4z = A = [You will not get full credit unless you check either A 1 A = I 3 or AA 1 = I 3 by direct multiplication. Note that there is no need to check both of these.] [Method: Calculate the matrix of cofactors of A. Transpose this to get the adjoint adja. Calculate the determinant of A by the cofactor method (you ve already got the cofactors!). Then use A 1 = 1 det A adja.] 9

10 69. Find the inverse A 1 of the matrix A = (See comments in Question 68.) 70. Find the inverse of the matrix (See comments in Question 68.) 71. Use row operations to solve the system of equations x + 2y + z = a x y + z = b x + y = c for general a,b,c, and from your answers find the solutions in the three cases (i) a = 1, b = 3, c = 4; (ii) a = 5, b = 0, c = 0; (iii) a = 1, b = 1, c = Let A, B and P be n n matrices. Prove that (i) if P is invertible and PA = PB, then B = A, (ii) if P is invertible and AP = PB, then B = P 1 AP. 73. Let A and B be square matrices of the same size and suppose that A is invertible. Simplify (i) (A + B)A 1, (ii) A 1 (B + 3I)A (iii) A 1 (A + A 1 )A. (I=identity matrix of the same size as A and B), 74. Decide which of the following matrices are orthogonal. [The definition is in 6.5 of Anton [p.256 in 6th edn.], or in Towers, p.179, and states that the n n matrix A is orthogonal if and only if A 1 = A. This is equivalent to A A = I, where I is the n n identity matrix.] ; ; Prove that if A and B are orthogonal n n matrices then C = AB is also orthogonal. [This is much easier than it looks! You want to prove (AB) (AB) = I. What is another way of writing (AB)?] 76. (H) Let A be an m n matrix. Explain why A A is a square matrix. Suppose that x is a column vector (size n 1) such that A Ax = 0. Show that Ax = 0. Hint. Multiply both sides by x [on which side?] and then rewrite the left-hand side. Remember that if v is a column vector with v v = 0 then v = Compute the determinant of the matrix t t t 6 Find all the real numbers t for which this determinant is 0. 10

11 78. Let A = , B = Find deta and detb. Hence calculate det(a 2 ) and det(a 2 BA). Remember for the last part that A 2 = (A 1 ) 2. Also for any two square matrices P,Q of the same size, det(pq) = det(p)det(q), and that det(p 1 ) = 1/det(P) when det(p) Let 80. Let Compute adja, and hence find deta and A 1. A = A = Calculate adja, and find the matrix product A(adj A). What is deta? In this case you can t go on to find the inverse of A. Why not? 81. Let A = t t 1 t t t + 3 t + 7 Calculate the determinant of A. [It s worth doing the row operation R 3 R 1 first. Remember that row operations of this type, replacing R i by R i plus a multiple of any other row, do not affect the value of the determinant.] 82. Use determinants to re-do Questions 46(iii), 48(ii),(iii) and 49(ii). Recall that to check whether three vectors in R 3 are LI, or to check whether they span R 3, it is enough to check whether the 3 3 determinant of their coordinates is non-zero. If it is non-zero, then they are LI and they do span R 3 ; if it is zero then they are LD and they do not span R 3. Please remember that this applies only to n vectors in R n : the same number of vectors as the dimension of the space in which they lie. (After all, whoever heard of the determinant of a non-square matrix?) 83. (i) Show that the matrix A = is invertible if and only if α 4 and α 7. 1 α α [As in the lecture, this means the same as: Show that deta = 0 if and only if α = 4 or α = 7.] (ii) Find the inverse of A when α = 3. (iii) Find the condition on a,b,c for the equations to be consistent. x + 7y + 2z = a 3x + y + 6z = b x + 4y + 2z = c [Notice that this is the case where α = 7 so A is not invertible. We therefore expect that there will be no solutions (inconsistent case) or infinitely many solutions (consistent case).] (iv) Find all the solutions in the case a = 1, b = 3, c = 1. 11

12 84. Consider the equations x + 2y 3z = a 3x y + 2z = b x 5y + 8z = c Reduce the augmented matrix to REF and hence find the condition for the equations to be consistent. In each of the following cases, either state that there are no solutions, or find all solutions: (i) a = 0, b = 0, c = 0; (ii) a = 2, b = 1, c = 3; (iii) a = 2, b = 1,c = Find all solutions of the following system of equations for each value of the real number α: x 2y + z = 1 x y + αz = 1 2x αy + 4z = Find the eigenvalues and eigenvectors of the matrix ( ) Write down the characteristic equation of each of the following matrices, and find an eigenvector corresponding to each (real) eigenvalue: ( ) ( ) (a) ; (b) (H) Let θ be a real number. Find the characteristic polynomials of the matrices ( ) ( ) cos θ sin θ cos θ sin θ A θ = and B sin θ cos θ θ =. sin θ cos θ Show that, unless θ = nπ for some integer n, then A θ has no real eigenvalues. Find the eigenvalues of B θ. 89. (i) Find the eigenvalues of the matrix A = (ii) For each eigenvalue of A, find an eigenvector of length 1. (iii) Write down an orthonormal basis of R 3 consisting of eigenvectors of A, and a diagonal matrix D and an orthogonal matrix P such that P 1 AP = D. 90. Find all solutions of the homogeneous system of equations 3x + y + z + t = 0 2x 3y + 4z = 0 5x + 9y 5z + 3t = 0. 12

13 91. (i) Find a basis of R 3 which contains the vector (1, 1,1). (ii) Find a basis of R 3 which contains the vectors (1,1,1) and (1, 1, 1). (iii) Does there exist a basis of R 3 which contains the vectors (1,1,1) and ( 1, 1, 1)? Give reasons for your answers. 92. (i) Find a basis for the span S of the vectors (2,2,4,3), ( 2,1,2,1), (3,1,2,2), and (1, 1, 2,0). (ii) Extend your basis to a basis of R 4. (iii) Decide whether (7,7,14,11) is in S. Remember that for (i) you put the vectors as the rows of a matrix, reduce to REF, and take the non-zero rows of the result. You should find three such rows. (This part could equally well be phrased Find a basis for the subspace S of R 4 spanned by the vectors... ) For (ii), you need to find a vector v which, together with the three nonzero rows of the REF, makes four linearly independent vectors, that is a new REF with four non-zero rows. Hint: It s always possible to choose one v as of the standard basis vectors (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1). 93. Let L be the line of intersection of the two planes x + y 2z = 4, 2x + 3y + z = 1. Let L be the line through the two points A = (1,0,1) and B = ( 9,7,2). (i) Find in parametric form an expression for the general point of L. (ii) Write down the vector AB and an expression for the general point of L. (iii) Determine the point in which L meets the plane x + y + z = 6. (iv) Decide whether L meets L. (v) Decide whether L meets the line of intersection M of the two planes x + y + z = 6, x + 2y 3z = Diagonalise the matrix ( ) Find the eigenvalues and eigenvectors of the matrix ( ) 2 1 A =. 1 0 Is A diagonalisable? 96. Show that the matrix A = has three distinct eigenvalues. Find corresponding eigenvectors, and hence write down a diagonal matrix D and an invertible matrix P such that P 1 AP = D. ( ) Find the eigenvalues of the matrix A =

14 98. Let B = Find a nonzero vector v = (x,y,z) satisfying (B + 2I)v = 0. Which real number λ is therefore an eigenvalue of B? Write down a corresponding unit length eigenvector. ( ) Find the eigenvalues of the matrix A = Let B = Find a nonzero vector v = (x,y,z) satisfying (B 2I)v = 0. Which real number λ is therefore an eigenvalue of B? Write down a corresponding unit length eigenvector. 14

15 A Mock January Examination Paper Here is a typical examination January examination paper. Solutions will be distributed towards the end of term. Section A carries 55 marks and three questions from Section B carry 45 marks. This total out of 100 is multiplied by 8 10 and the Key Skills mark for the module (out of 10) and the homework mark (out of 10) added in to give your final mark out of 100. Section A 1. Let z = 1 + 2i. Find the real and imaginary parts of z + 3 z. 2. Let z = 2 3 6i. Express z in the form re iθ. (As usual, r > 0 and θ is real.) Indicate the position of z on an Argand diagram. Use de Moivre s theorem to find the real and imaginary parts of z 3. [6 marks] 3. Verify that (2 + 8i) 2 = i. By means of the quadratic formula, or completing the square, solve the quadratic equation z 2 + 2iz i = Let A,B,C be three points in the plane, with position vectors a,b,c respectively. Write down the position vector of the midpoint M of AB. Show that the point with position vector 1 3 (a + b + c) lies on the line MC. 5. Let A = ( 1,2,0), B = (3,1,4) and C = ( 2,2,3). (i) Find the vectors AB, (ii) Find the area of the triangle ABC. AC and BC. Find the cosine of the angle BAC. [3 marks] [2 marks] (iii) Find the volume of the parallelepiped with one vertex at (0,0,0) and three adjacent vertices at A,B and C. [2 marks] 6. Find an equation for the plane which has normal vector n = 3i j + 4k and which passes through the point (1,1,3). Find the distance of this plane from the point (2,4,0). 7. Find the values of p,q,r such that the curve y = p+qx+rx 2 passes through the points (1,1), (2,5) and ( 1,5). 8. For each set of vectors (i) and (ii) decide, giving reasons, whether the vectors are linearly independent and whether they span R 3. (i) ( 1, 3,4), (2,6, 8); (ii) ( 1, 3,4), (1,2,0), (1,0,8). 9. Find the determinants of the matrices A = 4 1 2, B = Let Write down the determinants of A 2 B 1 and B 2I, where I is the 3 3 identity matrix. [6 marks] A = For each of the values λ = 0,1 and 2, find a nonzero vector v = (x,y,z) satisfying (A λi)v = 0. Deduce that λ = 0,1, 2 are the eigenvalues of A, and write down corresponding unit length eigenvectors. [9 marks] Section B 15

16 11. Express the complex number a = 4 2 4i 2 in the form a e iα. Find all the solutions of the equation z 3 = a in the form z = re iθ and indicate their positions clearly on an Argand diagram. Using a calculator, express one of them in the cartesian form z = x + iy, giving the answer correct to two decimal places. Without further calculation indicate clearly on a separate diagram the solutions of the equation z 3 = a = i 2, giving a brief reason for your answer. [15 marks] 12. Let A = (i) Show that A is invertible if and only if α 1. (ii) Find the inverse of A when α = 2. (iii) Show that the system of equations α 4 1 α 3 0 x + 3y z = a 2x + 7y 3z = b x 2y = c is consistent if and only if c = b 3a. 13. Let A = (1,1,0), B = (1, 3, 4) and C = (2,1,1). (i) Find an equation for the plane passing through A,B and C. (ii) Find in parametric form the line of intersection L of the plane in (i) and the plane 5x+y+z = 4. (iii) Find the point in which the line L meets the line through A and B. 14. Vectors v 1,v 2,v 3,v 4 in R 4 are defined by v 1 = (1,1,3,2), v 2 = (2,3, 1,5), v 3 = (1,2,4,3), v 4 = (3,3,1,6). (i) Show that v 1,v 2,v 3,v 4 are linearly dependent. (ii) Let S be the span of v 1,v 2,v 3,v 4. Find linearly independent vectors with the same span S. (iii) Which, if any, of the standard basis vectors (1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1) belong to S? Give reasons. MATH103 January 2001 Examination All answers from Section A and the best three answers from Section B will be taken into account. Section A 1 Let z = 1 3i. Find the real and imaginary parts of z + 1 z 2. 2 Let z = 3 + i 3. Express z in the form re iθ. (As usual, r > 0 and θ is real.) Indicate the position of z on an Argand diagram. Use de Moivre s theorem to find the real and imaginary parts of z 6. [6 marks] 3 Verify that ( 4+5i) 2 = 9 40i. By means of the quadratic formula, or completing the square, solve the quadratic equation z 2 iz i = 0. 4 Let A,B,C be three points with position vectors a,b,c respectively. Write down the position vector of the point P which is on the segment AB, three-quarters of the distance from A to B. Write down the 16

17 position vector of the point Q which is on the segment CB, three-quarters of the distance from C to B. Find the vector PQ and verify that it is 1 4 AC. 5 Let A = (1,2,0), B = ( 1,3,5) and C = (2, 1,3). (i) Find the vectors AB, AC and AB AC. Find the area of the triangle ABC. (ii) Find the length of the altitude of the triangle ABC drawn through the vertex A. (iii) Find an equation for the plane containing the triangle ABC. [3 marks] [2 marks] [2 marks] 6 Find an equation for the plane which has normal vector n = (2, 1,3) and which passes through the point (2,0,1). Find the distance of this plane from the point (3,1, 6). 7 Find the values of p,q,r such that the curve y = p + qx + rx 2 passes through the points (1,6), ( 1,8) and (2,17). 8 For each set of vectors (a) and (b) decide, giving reasons, whether the vectors are linearly independent and whether they span R 3. (a) (1,0,1), (3,2, 1); (b) (1,0,1), (3,2, 1), (1, 2,3). 9 Find the determinants of the matrices A and B: A = 4 3 6, B = Write down the determinants of A 1 B and B + 3I, where I is the 3 3 identity matrix. 10 Let A = [6 marks] For each of the values λ = 1,2 and 3, find a nonzero vector v = (x,y,z) satisfying (A λi)v = 0. Deduce that λ = 1,2,3 are the eigenvalues of A, and write down corresponding unit length eigenvectors. [9 marks] Section B 11 Express the complex number a = 8 8i 3 in the form a e iα. Find all the solutions of the equation z 4 = a in the form z = re iθ and indicate their positions clearly on an Argand diagram. For one of the solutions, express it in the cartesian form z = x + iy. Without further calculation indicate clearly on a separate diagram the solutions of the equation z 4 = a = 8 + 8i 3, giving a brief reason for your answer. [15 marks] 12 Let A = 2 7 α α 0 (i) Show that A is invertible if and only if α 2. (ii) Find the inverse of A when α = 0. (iii) Find a condition which a,b and c must satisfy for the system of equations 2x + 7y 3z = a x + 3y z = b x 2y = c to be consistent. 17

18 13 Let L denote the line of intersection of the planes in R 3 with equations x y + 2z = 3 and 2x + y + z = 0. Let L denote the line joining the points A = ( 1,1,1) and B = ( 2,3,1). (i) Find in parametric form an expression for the general point of L. (ii) Write down the vector AB and an expression for the general point of L. (iii) Determine the point at which L meets the plane [3 marks] x y + 2z = 3. (iv) Decide whether or not L meets L. 14 Vectors v 1,v 2,v 3,v 4 in R 4 are defined by v 1 = (1, 1,3,1), v 2 = (2,1, 1,2), v 3 = (3,0,2,4), v 4 = (2, 2,6,4). (i) Show that v 1,v 2,v 3,v 4 are linearly dependent. (ii) Let S be the span of v 1,v 2,v 3,v 4. Find linearly independent vectors with the same span S. Extend these linearly independent vectors to a basis of R 4. (iii) Show that the vector (2,4, 8,5) lies in S. MATH103 January 2002 Examination All answers from Section A and the best three answers from Section B will be taken into account. Section A 1 Let z = 1 + 4i. Find the real and imaginary parts of z + 2 z. 2 Let z = 2 3 2i. Express z in the form re iθ. (As usual, r > 0 and θ is real.) Indicate the position of z on an Argand diagram. Use de Moivre s theorem to find the real and imaginary parts of z 3. [7 marks] 3 Verify that (5 + i) 2 = i. By means of the quadratic formula, or completing the square, solve the quadratic equation z 2 + (1 3i)z 8 4i = 0. 4 Let A,B,C,D be four points with position vectors a,b,c,d respectively. Write down the position vectors of the mid-point P of AB; the mid-point Q of BC; the mid-point R of CD and the mid-point S of DA. Show that the vector PQ equals the vector SR. 5 Let A = ( 1,1,2), B = (0,2,4) and C = ( 3,1,3). (i) Find the vectors AB, AC and AB AC. Write down the area of the triangle ABC. (ii) Find the angle BAC. (iii) Find an equation for the plane containing the triangle ABC. [3 marks] [3 marks] [3 marks] 6 Find the values of p,q,r such that the curve y = p + qx + rx 2 passes through the points (0,2), (1,0) and (3,2). 7 For each set of vectors (a) and (b) decide, giving reasons, whether the vectors are linearly independent and also whether they span R 3. (a) (2, 1,3), (4, 2,5), (b) (2, 1,3), (4, 2,5), ( 2, 1,4). [6 marks] 18

19 8 Find the determinants of the matrices A and B: A = 2 3 4, B = Write down the determinants of BA 1 and B + I, where I is the 3 3 identity matrix. [6 marks] ( ) Find the eigenvalues of the matrix A =. [3 marks] Let B = Find a nonzero vector v = (x,y,z) satisfying (B 3I)v = 0. Deduce that λ = 3 is an eigenvalue of B, and write down a corresponding unit length eigenvector. Section B 11 Express the complex number a = i 2 in the form a e iα. Find all the solutions of the equation z 3 = a in the form z = re iθ and indicate their positions clearly on an Argand diagram. For one of the solutions, express it in the cartesian form z = x + iy. Without further calculation indicate clearly on a separate diagram the solutions of the equation z 3 = a = 4 2 4i 2. [15 marks] 12 Let A = 2 1 α α (i) Show that A is invertible if and only if α 1 and α 7. (ii) Find the inverse of A when α = 0. (iii) Find a condition which a,b and c must satisfy for the system of equations to be consistent. 2x + y + 8z = a x + 7y + z = b x y + 3z = c 13 Let L denote the line of intersection of the planes in R 3 with equations x + y z = 5 and 2x + 3y + 4z = 7. Let L denote the line joining the points A = (1,2,3) and B = (0, 2,4). (i) Find in parametric form an expression for the general point of L. (ii) Write down the vector AB and an expression for the general point of L. (iii) Determine the point at which L meets the plane (iv) Decide whether or not L meets L. 14 Vectors v 1,v 2,v 3,v 4 in R 4 are defined by 2x y + z = 0. v 1 = (1,0, 2,3), v 2 = (3,1,0, 2), v 3 = (2,0,1,1), v 4 = (2,1, 3,0). [3 marks] (i) Show that v 1,v 2,v 3,v 4 are linearly dependent. (ii) Let S be the span of v 1,v 2,v 3,v 4. Find linearly independent vectors with the same span S. Extend these linearly independent vectors to a basis of R 4. (iii) Show that the vector (6,1, 1,2) lies in S. 19