Revision Problems for Examination 1 in Algebra 1

Transcription

1 Centre for Mathematical Sciences Mathematics, Faculty of Science Revision Problems for Examination 1 in Algebra 1 Arithmetics 1 Determine a greatest common divisor to the integers a) 5431 and 1345, b) 4941 and 463 a) Reduce the fraction as far as possible b) How can you tell that the equation lacks integer solutions (x, y)? 3478x+491y = Decide whether the following Diophantine equations have solutions When this is the case find all solutions a) 15x+7y = 1 b) 5x+9y = 1 4 Solve the Diophantine equation 5 Solve the Diophantine equation 6 Solve the Diophantine equation 7 Solve the Diophantine equation 18x+3y = x 135y = 10 33x 187y = 34 98x+133y = 47 Specify the solutions (x,y) such that x > 0 and y > 0 8 Determine all integers x and y such that 17x +496y = Determine all pairs (x, y) of integers such that 19x +1y =

2 10 Find all integer solutions of the system { x+17y 11z = 8 x+4y +z = Find the smallest positive integers c such that the Diophantine equation 59x+407y = 7 93c has a solution (x,y) Determine all solutions (x,y) for this value of c 1 Find all solutions of the congruences a) x 3 mod 5, b) 3x 1 mod 7, c) 6x 9 mod 15, d) 6x 10 mod Determine which of the following congruences have solutions: a) x 1 mod 3, b) x mod 7, c) x 3 mod Prove that is divisible by 3 15 Prove that 7 (19 n + n+3 ) if n is an odd positive integer Relations 16 Determine which of the following relations are reflexive, symmetric, and/or transitive: a) The usual divisibility relation on Z given by x y iff x divides y b) Let X be a set The relation defined on subsets of X c) Let X = {1,,3} The relation R on X given by R = {(1,1),(1,),(,),(1,3),(3,3)} d) The relation Q on R given by xqy x y Q e) The relation Q on R given by xq y x y R\Q f) The relation S on R given by xsy (x y) < 0 g) The relation S on C given by zs w (z w) < 0 h) The relation L on C given by zlw z w 17 Let P = C[x] denote the set of all polynomials p(x) = a 0 + a 1 x + + a n x n with complex coefficients, with the unknown x Define a relation on P by p q p = q, where p denotes the derivative of p Prove that is an equivalence relation on P Find the equivalence class containing the polynomial p(x) = x 18 Let A = Z + Z + Define a relation on A by (a,b) (c,d) a b = c d a) Show that is an equivalence relation b) Find the equivalence class containing (9, ) c) Find an equivalence class containing exactly two elements d) Find an equivalence class containing exactly four elements

3 Induction and combinatorics 19 Show that 1+( )+( + )+ +( n 1 + (n 1) ) = ( n + n 1 ) for n =,3, 0 Show that n 3 n = (n 1) 3n for every positive integer n 1 Show that n +n = n n for every positive integer n Show that 3 Show that n k=1 n k=1 k(k +) = 3 1 n+1 1 n+, n = 1,,3, 1 k(k +1)(k +) = n(n+3) 4(n+1)(n+), n = 1,,3, 4 Show that ( ) Prove that ( ) + ( ) ( ) 3 + ( ) ( ) ( ) n 1 = 3 ( n+1 ( ) n 4 ) > n3 6 for n = 4,5,6, for n = 1,,3, 6 Show that n ( ) ( ) m+k 1 m+n k = m k n 1 k=1 for all positive integers m and n 7 How many integers between and are there which, when represented in the decimal system, contain exactly four 4s? 8 Consider the sequence (a n ) n=0 defined recursively by a 0 = 0, a 1 = 1, and a n+ = 1+ 1 (a n +a n+1 ), n = 0,1, Prove by induction that a n = 9 ( ( 1 1 ) n ) + n, n N 3 9 How many arrangements are there of all the letters in PEPPARKAKOR? 30 How many arrangements are there of 3 letters chosen from the letters in ALGE- BRA? 3

4 31 In how many ways can we distribute four identical oranges into three (different) boxes? 3 A domino tile can be represented by the symbol [x y], where x and y belong to the set {,1,,3,4,5,6} The possibility that x = y is accepted Find the total number of domino tiles 33 A poker hand consists of five cards drawn from an ordinary deck of 5 cards (no jokers) (a) How many possible poker hands are there? (b) How many poker hands are there which contain only spades? (c) What is the number of poker hands containing 3 aces and kings? (d) How many poker hands contain a full house? (A full house consists of one three-of-a-kind and one pair) Complex numbers 34 a) Describe geometrically the set of complex numbers z that satisfy the inequality z +i z 1 b) Find the modulus (ie the absolute value) of the complex number ( 3+i) 13 (1 i) 7 c) Find the argument of the complex number 35 Determine the real part of ( 3+i) 100 ( 3+i) 13 (1 i) 7 36 Describe geometrically the set M of all complex numbers z such that z and Re z Determine the possible values of argz when z M 37 Draw the set M of complex numbers z such that Find the largest value of z for z M z +i 1 38 Draw the set M of complex numbers z such that Find {argz; z M} z + and Re z = 3 39 Draw the set M of complex numbers z such that z i < and Im z = 1 Describe all values assumed by argz for z M 4

5 40 Draw the set M of complex numbers z such that z +1 i 1 What values can the argument argz assume when z M? 41 Draw the set M of complex numbers z such that Find the values of z when z M z 1+i 1 4 Draw the set M of complex numbers z such that z 1 1 and z 1 i 1 Find all values which Im z assumes for z M 43 Determine all possible values of arg z, where the complex number z satisfies the relations z = and Im z = 1 Also determine the value of tan π 1 44 Write the complex number 1+i 3 1+i in the form a+ib Determine its argument in degrees Use the result to show that 45 Let θ be an arbitrary real number tan15 = 3 a) Show that cos5θ = 16cos 5 θ 0cos 3 θ +5cosθ b) Use the preceding result to determine cos π Let z, w be two complex numbers which satisfy w = (z i)/(z + i) Show that Im z > 0 if and only if w < 1 47 Show that z 1+z < 1 if and only if Re z > Show that z +4 z is real if z is a complex number such that z = 49 For which complex numbers z is pure imaginary? z + 1 z 50 Assume Re z 4 z = 0, where z is a complex number 0 Show that z = 51 Show that the set M = {z; z z (1 i)z (1+i) z + = 0} is a circle in the complex plane Find the centre and radius of the circle 5 Show that the set of points in the complex plane, whose distance to i is twice the distance to i, coincides precisely with the set of points, whose distance to 3i is equal to 5

6 Polynomials and their zeros 53 Determine the solutions of the equation z (3+i)z +1+3i = 0 The solutions shall be written in the form a+ib 54 Solve the equation (+i)z +(8 11i)z 5 5i = 0 55 Determine the complex number A so that the equation z 4z +A = 0 has the root 1+i Also determine the other root to the equation 56 Solve the equation 57 For which complex numbers z and w are z 3 = 1 i 1+i z +(4 4i)z 16i = 0 and w 8 +(4 4i)w 4 16i = 0? The answers shall be given in polar form 58 For which complex numbers z is z 4 8iz 5 = 0? 59 Determine a greatest common divisor to the polynomials x 4 4x+3 and x 3 3x +3x 1 60 Determine a greatest common divisor to the polynomials x 4 9x 4x+1 and x 3 +6x +1x+8 61 Determine a greatest common divisor to the polynomials x 5 +x 4 +x 3 +x +x+1 and x 3 +x +x+1 6 Determine a greatest common divisor to the polynomials x 5 +x 4 +x 3 +3x +x+1 and x 3 +x +x+1 63 Let P(x) be a polynomial that on division with (x 1) gives remainder 5 and on division with(x+1) gives remainder 3 Which remainder does P(x) give on division with (x 1)? 64 The polynomial P(x) gives remainder 1 on division with (x 1), remainder on division by (x ) and remainder 3 on division by (x 3) What is the remainder when P(x) is divided by (x 1)(x )(x 3)? 65 The remainder on division of the polynomial p(x) with z 3 +z +z+1 is z z+1 If p(1) =, determine the remainder when p(z) is divided by z 4 1 6

7 66 The polynomial p(z) gives the remainder z + 3 when divided by z 1 and the remainder z 1 when divided by z + 1 Determine the remainder when p(z) is divided by z Determine the remainder when z 400 +z is divided by a) z 1, b) z For which positive integers n is the polynomial x 3n x n +x n 1 divisible by the polynomial x 3 x +x 1? 69 The polynomials f(z) = z z and g(z) = z 5 z 4 +z 4z have a common zero Solve the equation g(z) = 0 70 The two equations z 4 +z 3 +5z +4z +4 = 0 and z 4 +z 3 +3z +z + = 0 have at least on common root Solve both equations completely 71 Show that z = 1 is a root to the equation Solve the equation completely 7 Show that i is a root to the equation z 3 +(1 4i)z (13 6i)z +11 i = 0 z 3 +( 3+i)z +(3 i)z 1+3i = 0 Determine the other roots and write them in the form a+bi 73 Verify that the equation z 4 (3+i)z 3 +(+3i)z (3+i)z +1+3i = 0 has the roots ±i Determine the other roots and write them in the form a+bi 74 Show that z = i is a double root to the equation z 4 (+4i)z 3 +( 5+14i)z +(+i)z 10i Then solve the equation completely 75 The number i is a zero of the polynomial Solve the equation p(z) = 0 76 The equation p(z) = z 4 (6+i)z 3 +(11+6i)z (6+11i)z +6i z 4 z 14z 10 = 0 has the root z = 1+i Solve the equation completely 77 The equation z 4 8z 3 +51z 98z +170 = 0 has the root z = 3+5i Solve the equation completely 7

8 78 Prove that the equation z 4 6z 3 +15z 18z +10 = 0 has the root z = 1+i Then solve the equation completely 79 The equation z 5 7z 4 +9z 3 57z +60z 6 = 0 has the solutions z = 1 and z = 1+i Determine the other solutions 80 The equation z 3 (5 3i)z 15iz +16+1i = 0 has a real root Solve the equation completely 81 The equation z 3 +(3+5i)z +15iz 1+16i = 0 has a pure imaginary root z = iy, where y is real Solve the equation completely 8 Solve the equation z 4 z 3 8iz +8i = 0 83 Show that if z 4 +3z 3 z +3z+1 = 0 and w = z+ 1 z, then w +3w 4 = 0 Use this to solve the first equation 84 Solve z 4 3z 3 z 3z +1 = 0 by first multiplying the equation with z and then introducing the new unknown w = z +z 1 85 Show that if z 4 6z 3 +6z +6z +1 = 0 and w = z z 1 then w 6w +8 = 0 Use this to solve the first equation 86 Solve the equation z 10 +z 8 +z 6 +z 4 +z + = 0 87 Show that if p(z) = z 4 +3z = (z az +b)(z +az +c), then a 6 +8a 9 = 0 Then solve the equation p(z) = 0 88 Show that if p(z) = z 4 +4z +3z +4 = (z az +b)(z +az +c), then a 6 +8a 4 9 = 0 Then solve the equation p(z) = 0 89 Show that if t = ( 5+) 1/3 ( 5 ) 1/3, then t 3 = 4 3t Use this to determine the value of t 90 Show that if z = ( 1+i 3 ) ( ) 1/3 + ( 1 i 3) (10 108) 1/3, then z 3 = 0 6z Use this to determine the value of z 8

9 Mixed Problems 91 Determine whether the following Diophantine equations are soluble When this is the case find all solutions a) 13x +31y = b) 45x+54y = Find the coefficient of x 4 in the expansion of ( x x ) 9 93 Consider the polynomials f(x) = x 4 +3x 3 x 13x 10 and g(x) = x 3 +x 7x 15 a) Determine a greatest common divisor to f and g b) Solve the equations f(x) = 0 and g(x) = 0 94 Verify that, for every n Z +, ( ) ( ) n n = 1 ( ) n n n 1 n+1 n 95 Find the solutions of the following equations in the form a+ib, a,b R ( ) a) z +(3+i)z ++14i = 0 b) z 4 = +i 1 6+i 96 Prove the following double inequality for all integers n : 13 4 < 1 n n n < n n+1 97 Determine whether the following Diophantine equations are soluble; solve them when this is the case a) 7x+7y = 7 b) 7x 7y = Let f(x) = x 4 4x+3 a) Determine a greatest common divisor to the polynomials f and f (f is the derivative of f) b) Solve the equation f(x) = 0 99 Evaluate the sum ( ) n S n = Prove that the equation ( n 1 ) +4 ( n ) + + n ( n n z 3 (10 5i)z +(5 38i)z +65i = 0 has a real solution Solve the equation completely ), n = 0,1,, 9

10 101 Solve the equation ( ) 1 +i 6 z 6 +6iz 5 15z 4 0iz 3 +15z +6iz 1 = 1+i The solutions shall be written in the form a+ib where a,b R 10 Prove the identity n k 3 1 k 3 +1 = (n +n+1), n =,3,4, 3n(n+1) k= 103 Determine whether the following Diophantine equations are soluble; solve them when this is the case a) 111x 33y = 11 b) 111x 33y = a) Determine a greatest common divisor to the polynomials f(x) = x 4 +6x 3 +14x +16x+8 and g(x) = x 3 +5x +8x+4 b) Find all zeros of f 105 A farmer with 50 cows and 40 sheep shall single out 4 animals for slaughter He wants at least one animal of each sort In how many ways can this be done? 106 Prove that the equation z 3 +( 3+i)z +(6 3i)z 6+8i = 0 has a pure imaginary root Solve the equation completely 107 Use the identity (1+x) m (1+x) n = (1+x) m+n to prove that ( ) ( )( ) ( )( ) ( m+n m n m n m = r 0 r 1 r 1 r 108 Solve the equation z 6 +6z 5 +15z 4 +0z 3 +15z +6z +1 = i Mark the locations of the zeros in a figure )( n 0 ) 109 Prove that n k=1 k+ k!+(k +1)!+(k +)! = 1 1 (n+)!, n Z Determine the number of positive divisors of the integer

11 Answers 1 a) 3, b) 3 a) 53 74, b) 47 does not divide a) No solution b) x = 4+9n, y = 11 5n where n Z 4 x = 117 3n, y = 91+18n where n Z 5 x = n, y = n where n Z 6 x = 3+11n, y = 5+19n where n Z 7 x = n, y = n where n Z The only positive solution is (x,y) = (3,1) 8 The solutions are (8,),( 8,),(8, ),( 8, ),(4,5),( 4,5),(4, 5),( 4, 5) 9 The solutions are (9,11), ( 9,11), (9, 11), ( 9, 11), (1,8), ( 1,8), (1, 8), and ( 1, 8) 10 x = 17+98n, y = 10 13n, z = 5+7n, n Z 11 c = 14 The solutions are (x,y) = ( 5+11n, 7n), n Z 1 a) {4+5k; k Z}, b) {5+7k; k Z}, c) {k Z; k 4 k 9 k 14 mod 15}, d) No solutions 13 a) and c) have solutions 16 a) Reflexive, transitive b) Reflexive, transitive c) Reflexive, transitive d) Equivalence relation e) Symmetric f) Symmetric, Transitive g) Symmetric e) Reflexive, Symmetric 17 The equivalence class is {x+c; c C} 18 b) {(3,4),(9,),(81,1)} c) Eg {(4,1),(,)} d) Eg {(64,1),(8,),(4,3),(,6)} (a) ( 5 5 ) ( ; (b) 13 5) = 187; (c) 4; (d) a) The half plane on and above the line Re z +Im z = 0 b) 33 c) 5π

12 36 The right half of the closed disc with centre and radius argz varies in the interval [ π 4, π 4 ] 37 Closed disc with centre i and radius 1 Largest value of z is Closed line with end points 3 i 3 and 3+i 3 5π 6 argz 7π 6 39 Open line with endpoints 3+i and 3+i π 6 < argz < 5π 6 40 Closed disc with centre 1+i and radius 1 π 41 Closed disc with centre 1 i and radius 1 π 4 1 Im z 1 argz π argz 0 43 argz = π 1 or 5π 1 tan π 1 = i 3 1 The argument is cos π 10 = The expression is purely imaginary if and only if Re z = Im z 51 Centre 1+i, radius 3 53 The solutions are +i and 1+i 54 The solutions are +i and 1+5i 55 A = 4+i The solutions are 3 i and 1+i 56 The solutions are i and ± 3 i 57 w = e i(π 8 +k π ) or w = e i(π 4 +k π ), k = 0,1,,3 z = 4i or z = 4 = 4e iπ 58 ±(+i) and ±(1+i) 59 x x+1 60 x +4x+4 61 x+1 6 x+1 63 x+4 64 x (z3 +5z 3z +5) 66 z +z a) 3 b) z ,3,4,5,7,8,9,11,1,13,15,, that is all integers not of the form 4k + 69 The solutions are 0, and 3 e i (k+1)π 3, k = 0,1, 1

13 70 The first equation has the double solutions 1 ± 7 i, and the second has the single solutions 1 ± 7 i and ±i 71 The solutions are 1, +i and 4+3i 7 z 1 = i, z = i, z 3 = 1+i 73 The other solutions are +i and 1+i 74 The other two solutions are 1+3i and 3 i 75 The solutions are 1,, 3 and i 76 The solutions are 1±i and 1± 3 77 The solutions are 3+5i, 3 5i, 1+i and 1 i 78 The solutions are 1±i and ±i 79 The solutions are 1, 1±i and ±3i 80 The solutions are 4, i and 1 i 81 The solutions are 4i, 1 i and +i 8 The solutions are 1 and e (π 6 +nπ 3 ), n = 0,1, 83 z 1, = ± 3, z 3,4 = 1±i 3 84 z 1, = ± 3, z 3,4 = 1±i 3 85 z 1, = 1±, z 3,4 = ± 5 86 The solutions are ±i and 8 e i(±3π 16 +k π ), k = 0,1,,3 87 The zeros of p are 1±i 7 and 1± 5 88 The zeros of p are 1±i 15 and 1±i 3 89 t = 1 90 z has the value 1+3i 91 a) No solutions (gcd(13,31) equals 3 which is not a divisor of 13431) b) x = 100 6n, y = 1 5n, n Z 9 ( 9 ) 3 = a) x +4x+5 (and non-zero constant multiples of this polynomial) b) The zeros of f are ±i,, and 1 The zeros of g are ±i and 3 95 a) 1 3i, 4+i b) ± 1 ±i 1 (four cases) 97 a) Lacks solutions (gcd(7,7) = 9 which does not divide 7) b) x = 100+8n, y = 1+3n, n Z 13

14 98 a) gcd(f,f ) = c(x 1), (c a non-zero constant) b) The equation has the double root x = 1 and simple roots at x = 1±i 99 S n = 3 n 100 The equation has the solutions 5, 3 i, and 3i 101 The equation has the solutions ± 3+i, ± 3 3i, 3i, and 5i 103 a) No solutions b) x = 6+11n, y = 0+37n, n Z a) For example x +4x+4 b) f has a double zero at, simple zeros at 1±i ( )( 40 ) ( )( 40 ) ( + 50 )( ) 106 i, i, and 1+i 108 The equation has the solutions z k = cos ( π 1 + π 3 k) + isin ( π 1 + π 3 k) 1, k = 0,1,,3,4,