Paper I ( ALGEBRA AND TRIGNOMETRY )

Transcription

1 Paper I ( ALGEBRA AND TRIGNOMETRY ) Dr. J. N. Chaudhari Prof. P. N. Tayade Prof. Miss. R. N. Mahajan Prof. P. N. Bhirud Prof. J. D. Patil M. J. College, Jalgaon Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Nutan Maratha College, Jalgaon

2 Unit Adjoint and Inverse of Matrix, Rank of a Matrix and Eigen Values and Eigen Vectors Marks ) If A = find minor and cofactor of a, a and a, ) - If A = find adj A, ) - If A = find A - 7, - ) If A = and B = -, find ρ(ab) 5) If A = 5 7 find ρ(a), 6) Find rank of A = 6 9 7) 9-7 Find the characteristic equation and eigen values of A = - 8) Define characteristic equation of a matrix A and state Cayley-Hamilton Theorem. 9) Define adjoint of a matrix A and give the formula for A - if it exist. ) Define inverse of a matrix and state the necessary and sufficient condition for existence of a matrix. ) Compute E (), E () of order

3 - ) If A = and B = 5-6, find (AB) - ) If A = - then ρ (A) is --- a) b) c) d) ) If A = then which of the following is true? a) adja is nonsingular b) adja has a zero row c) adja is symmetric d) adja is not symmetric 5) - - If A = then which of the following is true? - a) A = A b) A is identity matrix c) A is non-singular d) A is singular 6) If A = and B = Statement I : AB singular Statement II : adj(ab) = adjb adja then which of the following is true a) Statement I is true b) Statement II is true c) Both Statements are true d) both statements are false 7) If A is a square matrix, then A - exists iff a) A > b) A < c) A = d) A 8) If A = a) c) then A(adj A) is b) - - d) -

4 9) If A is a square matrix of order n then KA is a) K A b) ( ) n K A c) K n A d) None of these ) Let I be identity matrix of order n then a) adj A = I b) adj A = c) adj A = n I d) None of these ) Let A be a matrix of order m x n then A exists iff a) m > n b) m < n c) m = n d) m n ) If AB = 5 and A = then det. B is equal to a) b) -6 c) - ¼ d) -8 x ) If A = x x and A - = - then x = --- a) -/ b) -/ c) d) ) If A = and n N then A n is ---- a) n n n n n n b) n n c) n - n - n - n - d) n + n + n + n ) If A = then adja is a) b) c) d) 6) If a square matrix A of order n has inverses B and C then a) B C b) B = C n c) B = C d) None of these

5 7) If A is symmetric matrix then a) adja is non-singular matrix b) adja is symmetric matrix c) adja does not exist d) None of these 8) If AB = - and A = then a) (AB) - = AB b) (AB) - = A - B - c) (AB) - = B - A - d) None of these 9) If A and B, C are matrices such that AB = AC then a) B C b) B A c) B = C d) C A ) If A = then - a) A = I b) A = c) A = A d) None of these ) If matrix A is equivalent to matrix B then a) ρ(a) ρ(b) b) ρ(a) > ρ(b) c) ρ(a) = ρ(b) d) None of these ) If A = [ ] then ρ(a) is a) b) c) d) None of these ) If A = 9 9 then ρ(a) is a) b) c) d) ) If A is a matrix of order m x n then a) ρ(a) min{m,n} b) ρ(a) min{m,n} c) ρ(a) max{m,n} d) None of these - 7 5) The eigen values of A = are a) -5, - b) 5, c) 5, - d) None of these

6 6) - 5 If A = then A satisfies a) A + A + 7I = b) A - A - 7I = c) A - A + 7I = d) A + A - 7I = 7) If A is a matrix and λ is some scalar such that A λi is singular then a) λ is eigen value of A b) λ is not an eigen value of A c) λ = d) None of these 8) If A = then A - exists if a) ρ(a) = b) ρ(a) = c) ρ(a) = d) None of these - 9) If A = - then which of the following is incorrect? a) A = A - b) A = I c) A = d) None of these

7 Marks : ) If A is a square matrix of order n then prove that (adja) ' = adj A ' and verify it for A = ) For the following matrix, verify that (adja) ' = adj A ' A = ) If A = - - then show that adj A = A ) If A = show that A(adj A) is null matrix., 5) Show that the adjoint of a symmetric matrix is symmetric and verify it for A = 6) Verify that (adja)a = A I for the matrix A = - - cos θ - sin θ 7) Verify that A(adjA) = (adja)a = A I for the matrix A = sin θ cos θ 8) Verify that A(adjA) = A I for the matrix A = ) Find the inverse of A =

8 ) Find the inverse of A = ) Show that the matrix A = - Hence find A - satisfies the equation A 6A + 5 I =. - ) If A =, B = show that adj(ab) = adjb adja 7, ) If A = show that A(adjA) = (adja)a = A I, ) If A = then show that adja = A ' 5) If A = show that A = A, but A - does not exist. -, 6) If A = - show that A = A -, 7) What is the reciprocal of the following matrix? A = cosα sin α -sin α cosα - 8) If A =, B = verify that (AB) - = B - A -, 9) Using adjoint method find the inverse of the matrix A = ) If A is a non-singular matrix of order n then prove that adj (adja) = A n- A

9 ) For a non-singular square matrix A of order n, prove that (adj A) adj = ( ) n A ) For a non-singular square matrix A of order n, prove that adj {adj (adja)} = A n + n A - ) If A = - - -, show that A = A - ) Find the rank of the matrix A = 5) Find the rank of the matrix A = - 6) Compute the elementary matrix [E (-)] -. E (). E ' (/) of order 7) Compute the matrix E ' (/). E. [E (-)] - for E-matrices of order 8) Determine the values of x so that the matrix x x x x x x is of i) rank ii) rank iii) rank 9) Determine the values of x so that the matrix x x x x x x is of i) rank ii) rank iii) rank ) Reduce the matrix A to the normal form. Hence determine its rank, where A = 8 6 5

10 ) Reduce the matrix A to the normal form. Hence determine its rank, where A = ) Reduce the matrix A to the normal form. Hence determine its rank, where A = ) Find non-singular matrices P and Q such that PAQ is in normal form, where A = - - ) Find non-singular matrices P and Q such that PAQ is in normal form, where A = - - 5) Find non-singular matrices P and Q such that PAQ is in normal form, where A = Also find ρ(a) 6) Show that the matrix A = x - x x - has rank when x and x ±, find its rank when x =. 7) Find a non-singular matrix P such that PA = G for the matrix A = -. Hence find ρ(a).

11 8) Given A = , B = , verify that ρ(ab) min {ρ(a), ρ(b)} 9) Find all values of θ in [-π/, π/] such that the matrix A = cos θ sin θ sin θ cos θ is of rank. ) Express the following non-singular matrix A as a product of E matrices, where A = ) Express the following non-singular matrix A as a product of E matrices, where A = 7 - ) Express the following non-singular matrix A as a product of E matrices, where A = ) -5 State Cayley- Hamilton Theorem. Verify it for A = ) State Cayley- Hamilton Theorem. Verify it for A = - 5) Verify Cayley- Hamilton Theorem for A = ) Find the characteristics equation of A = - -

12 7) Find eigen values of A = ) If λ is a non-zero eigen value of a non-singular matrix A, show that /λ is an eigen value of A - 9) If λ is an eigen value of a non-singular matrix A, show that A /λ is an eigen value of adj A. 5) Let k be a non-zero scalar and A be a non-zero square matrix, show that if λ is an eigen value of A then λk is an eigen value of ka. 5) Let A be a square matrix. Show that is an eigen value of A iff A is singular. 5) Show that A = a b a c b c, B = b a b c a c have the same characteristic equation. 5) - Find eigen values and corresponding eigen vectors of A = 5) Find eigen values and corresponding eigen vectors of A = 55) Find characteristic equation of A = Also find A - by using Cayley Hamilton theorem. 56) Verify Cayley Hamilton theorem for A and hence find A - where A = - 7

13 Marks / 6 ) If A, B are matrices such that product AB is defined then prove that ' (AB) = B ' A ' ) If A = [ a ij ] is a square matrix of order n then show that A(adjA) = (adja)a = A I ) Show that a square matrix A is invertible if and only if A ) If A, B are non-singular matrices of order n then prove that AB is non-singular and (AB) - = B - A - 5) If A, B are non-singular matrices of same order then prove that adj(ab) = ( adjb ) ( adja ) 6) If A is a non-singular matrix then prove that (A n ) - = (A - ) n, n N 7) If A is a non-singular matrix and k then prove that (ka) - = A - k 8) If A is a non-singular matrix then prove that (adj A) - = adj A - = A A 9) State and prove the necessary and sufficient condition for a square matrix A to have an inverse. ) If A is a non-singular matrix then show that AB = AC implies B = C Is the result true when A is singular? Justify. ) When does the inverse of a matrix exist? Prove that the inverse of a matrix, if it exists, is unique. ) If a non-singular matrix A is symmetric prove that A - is also symmetric. ) Prove that inverse of an elementary matrix is an elementary matrix of the same type. ) If A is a m x n matrix of rank r, prove that their exist non-singular matrices P and Q such that PAQ = Ir 5) Prove that every non-singular matrix can be expressed as a product of finite number of elementary matrices.

14 6) If A is an mxn matrix of rank r, then show that there exists a non-singular matrix P such that PA = matrix of order (m-r)xn. G where G is rxn matrix of rank r and is null, 7) Prove that the rank of the product of two matrices can not exceed the rank of either matrix. 8) If A is an mxn matrix of rank r then show that there exists a non-singular matrix Q such that AQ = [ H ] Where H is mxr matrix of rank r and is null matrix of order mx(n-r).

15 Marks Unit - System of Linear Equations and Theory of Equations ) Examine for non-trivial solutions x + y + z = x + y = x + y + z = ) Define i) Consistent and inconsistent system ii) Equivalent system ) Define homogeneous, non-homogeneous system of equations. ) The equation x +x x x+9 = has two pairs of equal roots, find them. 5) Change the signs of the roots of the equation x 7 + 5x 5 x + x + 7x + = 6) Transform the equation x 7 7x 6 x + x x = into another whose roots shall be equal in magnitude but opposite in sign to those of this equation. 7) Change of the equation x x + x x + = into another the coefficient of whose highest term will be unity. 8) A system AX = B, of m linear equations in n unknowns, is consistent iff A) ranka rank [A, B] B) ranka = rank [A, B] C) ranka rank [A, B] D) ranka rank [A, B] 9) For the equation x + x + x + =, sum of roots taken one, two, three and four at time is respectively. A),,, B),, -, C),, -, D) -,, -, ) For the equation x + x + x + x + =, sum of roots taken one, two, three and four at a time is respectively. A),,, B) -,, -, C), -,, - D) -, -, -,

16 ) If sum and product of roots of a quadratic equation are and respectively the required quadratic equation is A) x + x + = B) x x + = C) x + x = D) x + x + = ) The quadratic equation having roots α and β is A) x (α + β) x + αβ = B) x + (α + β) x + αβ = C) x + (α + β) x αβ = D) x + (α + β) x + αβ = ) The equation having roots,, - is A) x + x + x + = B) x + x + = C) x x + = D) x x + x = ) The equation having roots,, is A) x + x + x + = B) x x + x = C) x + x x = D) x + x x + = 5) Roots of equation x x + = are,, -, so the roots of equation x 6x + = are A),, - B), -, - C),, - D), -, - 6) Roots of equation x + x + = are -, - so the roots of equation x + 6x + 9 = are A) -, B), C) -, - D), - 7) Roots of equation x x+= are, so the roots of equation x x+= are A), - B), C) /, / D) -/, / 8) Roots of equation x 5x + 6 = are, so the roots of equation 6x 5x + = are A), - B), C) /, / D) -/, /

17 9) Find the equation whose roots are the roots of x x + = each diminished by. A) x x + = B) x x + = C) x + x + = D) x x = ) Find the equation whose roots are the roots of x 6x + x 8 = each diminished by. A) x x + x = B) x + x + x + = C) x x x = D) x x x + = ) To remove the second term from equation x 8x + x x = the roots diminished by A) B) C) D) - ) To remove the second term from equation x x 8x x + = the roots diminished by A) B) - C) D) -

18 Marks. Examine for consistency the following system of equations x + z = -x + y + z = -x + y + 7z =. Solve the following system of equations x + y + z = 6 x + y + z = 5x + y + z = x - y - z = -. If A = - - -, find A -. Hence solve the following system of linear - equations x + y - z = x - y + z = 9 -x + y - z = -. Test the following equations for consistency and if consistent solve them x - y - 5z + w = x + y + z 5w = 8 x - y - 8z + 7w = - 5. Solve the following system of equations x + x + x - 6x = x +6 x = x + x + x - x = x + x - x - x = 6. Examine for non-trivial solutions the following homogeneous system of linear equations x + y + z = x - y + z = -x + y = x - y + z =

19 7. Solve the system of equations x + y + z = x + y + z = 6 x + y + z = 7 by i) method of inversion ii) method of reduction. 8. Examine the following systems of equation for consistency x y + z u = x + y z + u = - x + y 5z + 8u = -5 5x 7y + z u = 9. Test the following equations for consistency and solve them x + y + z = x + y z = x y z = x + y + z =. Solve the following equations u + v + w + t = u + v + t = 6u + v + w + 7t =. Solve the equation x x 6x + 8 = if the roots are in A.P.. Solve the equation x 9x + x + = if two of its roots are in the ratio :.. Solve the equation x 6x + 5x = if the roots are in G.P.. Solve the equation x + x x x + = whose roots are in A.P. 5. If α, β and γ are roots of the equation x - 5x - x + = find the value of i) α β ii) α iii) α iv) α β 6. Remove the fractional coefficients from the equation x x + x =

20 7. Remove the fractional coefficients from the equation x 5 x 7 x+8 = 8 8. Transform the equation 5x x x + = to another with integral coefficients and unity for the coefficient of the first term. 9. Remove the fractional coefficients from the equation x + x + x + 77 = 5. Find the equation whose roots are reciprocals of the roots of x 5x + 7x + x 7 =. Find the equation whose roots are the roots of x 5x + 7x 7x + = each diminished by.. Find the equation whose roots are those of x x + x 9 = each diminished by 5.. Remove the second term from equation x 8x + x x + =. Remove the third term of equation x x 8x x + =, hence obtain the transformed equation in case h =. 5. Transform the equation x + 8x + x 5 = into one in which the second term is vanishing. 6. Solve the equation x +6x +8x +5x+8 = by removing the second term. 7. Solve the equation x + 6x + 9x + = by Carden s method. 8. Solve the equation x 5x x + 87 = by Carden s method. 9. Solve the equation z 6z 9 = by Carden s method.. Solve the equation x x = by Carden s method.. Solve x 5x 6 = by Carden s method. Solve 7x 5x + 98x 7 = by Carden s method. Solve x + x 7x + = by Carden s method. Solve x x + x + 6 = by Carden s method 5. Solve x 5x 6x 5 = by Descarte s method. 6. Solve the biquadratic x + x 5 = by Descarte s method. 7. Solve x 8x x + 7 = by Descarte s method.

21 Marks / 6. For what values of a, the equations x + y + z = x + y + z = a x + 9y - z = a have a solution and solve then completely in each case.. Investigate for what values of λ and μ the following system of equations x + y + z = x + 7y - z = - x + y + λz = μ have i) No solution ii) A unique solution iii) Infinite number of solutions.. Show that the system of equations ax + by + cz = bx + cy + az = cx + ay + bz = has a non-trivial solution iff a + b + c = or a = b = c. Find the value of λ for which the following system have a non-trivial solution x + y + z = x + y + z = x + y + λz = 5. Discuss the solutions of system of equations ( 5 - λ ) x + y = x + ( - λ ) y = for all values of λ. 6. Obtain the relation between the roots and coefficients of general polynomial equation a x n + a x n- + a x n a n- x + a n = 7. Solve the equation x 5x 6x + 8 = if the sum of two of its roots being equal to zero. 8. Solve the equation x x + = if the two of its roots are equal. 9. Solve the equation x 5x x+ = if the product of two of the roots is.

22 . Solve the equation x 7x + 6 = if one root is double of another.. Find the condition that the roots of the equation x px +qx r = are in A.P.. Find the condition that the cubic equation x + px + qx + r = should have two roots α and β connected by the relation αβ + =. If α, β and γ are roots of the cubic equation x + px + qx + r = find the value of i) α β ii) α iii) α iv) α β. If α, β and γ are roots of the cubic equation x + px + qx + r = find the value of (β+γ) (γ+α)(α+β) 5. If α, β and γ are roots of the cubic equation x - px + qx - r = find the value of β γ + γα + α β 6. If α, β, γ and δ are roots of biquadratic equation x + px + qx + rx + s =, find the value of the following symmetric functions i) α β ii) α iii) α 7. If α, β, γ and δ are roots of biquadratic equation x + px + qx + rx + s =, find the value of the following symmetric functions i) α βγ ii) α β iii) α 8. Remove the fractional coefficients from the equation x 5 x + 5 x = Find the equation whose roots are the reciprocals of the roots of x x + 7x + 5x =. Transform an equation a x n + a x n- + a x n a n- x + a n = into another whose roots are the roots of given equation diminished by given quantity h.. If α, β, γ are the roots of 8x x + 6x = find the equation whose roots are α + /, β + ½, γ + /. Solve the equation x + x + x + x + 6 = by removing its second term.. Reduce the cubic x x + 6x = to the form Z + HZ + G =. Explain Carden s method of solving equation a x +a x +a x+a =

23 Unit Relations, Congruence Classes and Groups Marks - ) Let A = {,,,, 5, 6, 7, 8, 9, } A = {,,, }, A = { 5, 6, 7 }, A = {, 5, 7, 9 }, A = {, 8, }, A 5 = { 8, 9, }, A 6 = {,,, 6, 8, } Which of the following is the partition of A. A) { A, A, A 5 } B) { A, A, A 5 } C) { A, A, A 6 } D) { A, A, A 6 } ) Let A = Z +, the set of all positive integers. Define a relation on A as arb iff a divides b then this relation is not --- A) Reflexive B) Symmetric C) Transitive D) Antisymmetric ) For n N, a, b Z and d = ( a, n ), linear congruence ax b (mod n) has a solution iff ---- A) d I b B) x I b C) n I d D) a I b ) If the Linear congruence ax b (mod n) has a solution then it has exactly ---- non-congruent modulo n solutions A) a B) b C) n D) (a, n) 5) If a b (mod p) then p I a+b or p I a-b only when p is --- A) Even B) Odd C) Prime D) Composite 6) G = {, - } is a group w.r.t. usual A) Addition B) Subtraction C) Multiplication D) None of these

24 7) In the group ( Z 6, + 6 ), o(5 ) is A) B) 5 C) 6 D) 8) Linear congruence 7x 6(mod 8) has A) No solution B) Nine solutions C) Three solutions D) One solution 9) The number of residue classes of integers modulo 7 are A) one B) five C) six D) seven ) The solution of the linear congruence 5x (mod 7) is A) x = B) x = C) x = 6 D) x = ) The set of positive integers under usual multiplication is not a group as following does not exist A) identity B) inverse C) associativity D) commutativity ) Define an equivalence relation and show that > on set of naturals is not an equivalence relation. ) Define a partition of a set and find any two partitions of A = { a, b, c, d } ) Define equivalence class of an element. Find equivalence classe of if R ={ (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ),(, ), (5, 5)} is an equivalence relation on A = {,,,, 5} 5) Define residue classes of integers modulo n. Find the residue class of for the relation congruence modulo 5. 6) Define prime residue class modulo n. Find the prime residue class modulo 7 7) Define a group and show that set of integers with respect to usual multiplication is not a group. 8) Define Abelian group and show that group G = :,,,, is not abelian. 9) Define finite and infinite group. Illustrate by an example.

25 ) Define order of an element and find order of each element in a group G = {, -, i, -i } under multiplication. ) Find any four partitions of the set S = {,, 5, 7, 9,,, 5, 7, 9 } ) Show that AxB BxA if A = {,, 6}, B = {7, 9, } ) In the group (Z 8, X 8 ), find order of,, 5, 6 ) Let Z 8 = {,, 5, 7 }, find ( ), ( ), ( ) - in a group (Z 8, X 8 ) 5) In the group (Z 7, X 7 ), find ( ), ( ) -, o( ), o( ) 6) Find domain and range of a relation R = { (x, y) : x I y for x, y B} where A = {,, 7, 8}, B = {, 6, 9, } 7) Solve the linear congruence x + (mod 5) 8) Let X = {,, } and R = {(,), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )} Is the relation R reflexive, symmetric and transitive? 9) Prepare the multiplication table for the set of prime residue classes modulo. ) Show that in a group G every element has unique inverse. ) Show that the linear congruence x 9 (mod 5) has only one solution. ) Show that the linear congruence x (mod 6) has no solution. ) If a (mod n) and c (mod n) then show that ac (mod n) ) A relation R is defined in the set Z of all integers as arb iff 7a b is divisible by. Prove that R is symmetric. 5) Let ~ be an equivalence relation on a set A and a, b A. Show that b [a] iff [a] = [b] 6) If in a group G, every element is its own inverse then prove that G is abelian. 7) In a group every element except identity element is of order two. Show that G is abelian. 8) If R and S are equivalence relations in set X. Prove that R S is an equivalence relation. 9) In the set R of all real numbers, a relation ~ is defined by a~b if + ab >. Show that ~ is reflexive, symmetric and not transitive.

26 Marks -. Let Z be the set of all integers. Define a relation R on Z by xry iff x-y is an even integer. Show that R is an equivalence relation.. Let P be the set of all people living in a Jalgaon city. Show that the relation has the same surname as on P is an equivalence relation.. Let P(X) be the collection of all subsets of X ( power set of X ). Show that the relation is a proper subset of in P(X) is not an equivalence relation.. Show that in the set of integers x y iff x = y is an equivalence relation and find the equivalence classes. 5. Let S be the set of points in the plane. For any two points x, y S, define x y if distances of x and y is same from origin. Show that is an equivalence relation. What are the equivalence classes? 6. Consider the set NxN. Define (a, b) (c, d) iff ad = bc. Show that is an equivalence relation. What are the equivalence classes? 7. Find the composition table for i) (Z 5, + 5 ) and (Z 5, x 5 ) ii) (Z 7, + 7 ) and (Z 7, x 7 ) 8. Prepare the composition tables for addition and multiplication of Z 6 = {,,,,, 5 } 9. Show that a Z n has a multiplicative inverse in Z n iff (a, n) =. Find the remainder when 8 is divided by.. Show that G = {, -, i, -i }, where i = multiplication of complex numbers., is an abelian group w.r.t. usual. Show that the set of all x matrices with real numbers w.r.t multiplication of matrices is not a group.. Show that G = { A : A is non-singular matrix of order n over R } is a group w.r.t. usual multiplication of matrices.. Let Q + denote the set of all positive rationals. For a, b Q + define a b = ab. Show that ( Q+, ) is a group.

27 a b 5. Show that G = : ad bc & a,b,c,d R w.r.t. c d multiplication is a group but it is not an abelian group. matrix 6. Let Z n be the set of residue classes modulo n with a binary operation a+ n b = a + b = r where r is the remainder when a + b is divided by n Show that (Z n, + n ) is a finite abelian group. 7. Let Z' n denote the set of all prime residue classes modulo n. Show that Z' n is an abelian group of order φ(n) w.r.t. x n 8. Let G be a group and a, b R be such that ab = ba. Prove that (ab) n = a n b n, n Z 9. If in a group G every element is its own inverse then prove that G is an abelian group.. Let G = { (a, b) / a, b R, a } Define on G as (a, b) (c, d) = ( ac, bc + d ). Show that ( G, ) is a non-abelian group.. Let f, f be real valued functions defined by f (x) = x and f (x) = x, x R. Show that G = { f, f }is group w.r.t. composition of mappings.. Let G be a group and a, b G, (ab) n = a n b n for three consecutive integers n. Show that G is an abelian group.. Show that a group G is abelian iff (ab) = a b, a, b G. Prove that a group having elements must be abelian. 5. Using Fermat s Theorem, Show that 5 is divisible by. 6. Using Fermat s Theorem find the remainder when 5 is divided by. 7. Solve i) 8x 6 (mod ) ii) x 9 (mod 5) 8. Let be an operation defined by a b = a + b + a, b Z where Z is the set of integers. Show that < Z, > is an abelian group. 9. Let A α = cosα sinα, where α R and G = { Aα : α R. Prove that G is an sinα cosα abelian group under multiplication of matrices.

28 . Let Q + be the set of all positive real numbers and define on Q + by a b =. Show that (Q +, ) is an abelian group.. Find the remainder when 7 + is divided by.. Show that the set G =,,, is a group w.r.t. multiplication.. Show that G = R {} is an abelian group under the binary operation a b = a + b ab, a, b G. If the elements a, b and ab of a finite group G are each of order then show that ab = ba. 5. A relation R is defined in the set of integers Z by xry iff 7x y is divisible by. Show that R is an equivalence relation in Z. 6. A relation R is defined in the set of integers Z by xry iff x + y is a multiple of 7. Show that R is an equivalence relation in Z. 7. Consider the set NxN, the set of ordered pairs of natural numbers. Let ~ be a relation in NxN defined by (x, y) ~ (z, u) if x + u = y + z. Prove that ~ is an equivalence relation. Determine the equivalence class of (, ). 8. Define congruence modulo n relation and prove that congruence modulo n is an equivalence relation in Z. 9. Show that the set of all x matrices with real numbers w.r.t. addition of matrices is a group.

29 Marks / 6. Let be an equivalence relation on set A. Prove that any two equivalence classes are either disjoint or identical.. Prove that every equivalence relation on a non-empty set S induces a partition on S and conversely every partition of S defines an equivalence relation on S.. If a b ( mod n ) and c d ( mod n ) for a, b, c, d Z and n N then prove that i) ( a + c ) ( b + d ) ( mod n ) ii) ( a - c ) ( b - d ) ( mod n ) iii) ac bd ( mod n ). Write the algorithm to find solution of linear congruence, ax b ( mod n ), for a, b Z and n N 5. State and prove Fermat s Theorem. 6. If G is a group then prove that i) identity of G is unique ii) Every element of G has unique inverse in G. iii) (a - ) - = a, a G 7. If G is a group then prove that i) identity of G is unique ii) (a - ) - = a, a G iii) (ab) - = b - a -, a, b G. 8. Let G be a group and a,b,c G. Prove that i) ab = ac b = c left cancellation law. ii) ba = ca b = c Right cancellation law. 9. Let G be a group and a, b G. Prove that the equations i) ax = b ii) ya = b have unique solutions in G. Let G be a group and a G. Prove that (a n ) - = (a - ) n, n N

30 . Let G be a group and a G. For m, n N, Prove that i) a m a n = a m+n ii) (a m ) n = a mn. Define an abelian group. If in a group G the order of every element ( except identity element ) is two then prove that G is an abelian group.. Solve the following linear congruence equations i) x ( mod 8 ) ii) 6x 5 ( mod 9 ). Define a group. Show that any element a G has a unique inverse in G. Further show that (a b) - = b - a -, a, b G. 5. If R is an equivalence relation on a set A then for any a, b A, prove that i) [a] = [b] or [a] [b] = φ ii) { [a] / a A } = A 6. If is an equivalence relation on set A and A and a, b A then show that i) a [a] for all a A ii) b [a] iff [a] = [b] iii) a b iff [a] = [b] 7. Define residue classes of integers modulo n. Show that the number of residue classes of integers modulo n are exactly n.

31 Unit Subgroups and Cyclic Groups Marks ) Define subgroup. Give example ) Define proper and improper subgroups. Give example. ) Define a cyclic group. Give example. ) Define left coset and right coset.. 5) State Lagrange s Theorem. 6) State Fermat s Theorem. 7) State Euler s Theorem. 8) Show that nz = { nr / r Z ) is a subgroup of (Z, +), where n N. 9) Show that (5Z, +) is a subgroup of (Z, +) ) Is group (Q+, ) a subgroup of (R, +)? Justify. ) Determine whether or not H = {ix : x R} under addition is a subgroup of G = group of complex numbers under addition. ) Find all possible subgroups of G = {, -, i, -i ) under multiplication. ) Find proper subgroups of ( Z, + ) ) Write all subgroups of the multiplicative group of 6 th roots of unity. 5) Find all proper subgroups of the group of non-zero reals under multiplication. 6) Give an example of a proper subgroup of a finite group. 7) Give an example of a proper finite subgroup of an infinite group. 8) Give an example of a proper infinite subgroup of an infinite group. 9) Is union of two subgroups a subgroup? Justify. ) Prove that cyclic group ia abelian. ) Show by an example that abelian group need not be cyclic. ) Let G = {, -, i, -i ) be a group under multiplication and H = {, - ) be its subgroup. Find all right cosets of H in G. ) Find the order of each proper subgroup of a group of order 5. Are they cyclic. ) Find generators of Z 6 under addition modulo 6. 5) Verify Euler s theorem by taking m =, a = 7. 6) Verify Lagrange s theorem for Z 9 under addition modulo 9.

32 7) Let G = Z be a additive group of integers and H = Z a subgroup of G then H+ is A) { , ----} B) {, 5, -, 8, -, ---} C) {,,,, 5, ----} D) {,, -, 7, -5, ---} 8) Let G = {, -, i, -i ) be a group under multiplication and H = {, - ) is a subgroup of G then H(-i) is A) {-, } B) {-i, i} C) {i, -} D) {, } 9) If n = then () is A) 5 B) C) 7 D) ) If p is prime then generators of a cyclic group of order p --- A) p B) p- C) p D) p+ ) A cyclic group having only one generator can have at the most --- element A) B) C) D) None of these ) In additive group Z, ( ) = --- A) {, 8 } B) {, 8, } C) {,, } D) Z.

33 Marks ) If H is a subgroup of G and x G, show that xhx - = { xhx - / h H } is a subgroup of G ) Let G be a group. Show that H = Z(G) = { x G / xa = ax, a G } is a subgroup of G. ) Let G be an abelian group with identity e and H = { x G / x = e }. Show that H is a subgroup of G. ) Let G be the group of all non-zero complex numbers under multiplication. Show that H ={ a+ib G / a +b = } is a subgroup of G 5) Show that H = { x G : xb = b x, b G } is subgroup of G. 6) Write all subgroups of the multiplicative group of non-zero residue classes modulo 7. 7) Determine whether H = {,, 8 } and H = {, 5, } are subgroups of ( Z, + ) 8) Let G be a finite cyclic group of order n, and G = < a >. Show that G = < a m > ( m, n ) =, where < m < n 9) Find all subgroups of (Z, + ). ) Find all subgroups of ( Z 7, X 7 ) ) Find all generators of additive group Z ) Let G = {, -, i, -i } be a group under multiplication and H = {, - } be it s subgroup. Find all right coset of H in G ) Compute the right cosets of Z in ( Z, + ). ) Let Q = {, -, i, -i, j, -j, k, -k } be a group under multiplication and H = {, -, i, -i } be its subgroup. Find all the right and left cosets of H in G 5) Let G (Z 8, + 8 ) and H = {, }. Find all right cosets of H in G 6) Let H be a subgroup of a group G and a G. Show that Ha = { x G / xa - H } 7) Let G = {,,,,5,6,7,8,9, }. Show that G is a cyclic group under multiplication modulo. Find all its generators, all its subgroups and order of every element. Also verify the Lagrange s theorem.

34 8) List all the subgroups of a cyclic group of order. 9) Find order of each element in ( Z 7, + 7 ) ) If Z 8 is a group w.r.t. addition modulo 8 i) Show that Z 8 is cyclic. ii) Find all generators of Z 8 iii) Find all proper subgroups of Z 8 ) Show that every proper subgroup of a group of order 5 is cyclic. ) Show that every proper subgroup of a group of order 77 is cyclic. ) Let G be a group of order 7. Show that for any a G either o(a) = or o(a) = 7. ) Let A, B be subgroups of a finite group G, whose orders are relatively prime. Show that A B = { e }. 5) Find the order of each element in the group G = {, w, w }, where w is complex cube root of unity, under usual multiplication. 6) Find all subgroups of group of order. How many of them are proper? 7) Find the remainder obtained when 5 is divided by. 8) Find the remainder obtained when 9 is divided by 7. 9) Using Fermat s theorem, find the remainder when i) 9 87 is divided by. ii) 5 + is divided by ) Find the remainder obtained when 5 7 is divided by 8.

35 Marks / 6 ) A non-empty subset H of a group G is a subgroup of G iff a, b H.ab - H. ) A non-empty subset H of a group G is a subgroup of G iff i) a, b H.ab - H ii) a H a - H. ) Prove that Intersection of two subgroups of a group is a subgroup ) Let H, K be subgroups of a group G. Prove that H U K is a subgroup of G, iff either H K or K H 5) Show that every cyclic group is abelian. Is the converse true? Justify. 6) Show that If G is a cyclic group generated by a, then a - also generated by G. 7) Show that every subgroup of a cyclic group is cyclic. 8) Let H be a subgroup of a group G. prove that i) a H Ha = H ii) a H ah = H 9) Let H be a subgroup of a group G. Prove that Ha = Hb ab H ah = bh b - a H, a,b G. ) Let H be a subgroup of a group G. Prove that i) Any two right cosets of H are either disjoint or identical. ii) Any two left cosets of H are either disjoint or identical. ) If H is a subgroup of a finite group G. Then prove that (H) / (G) ) Prove that every group of prime order is cyclic and hence abelian ) Order of every element a of a finite group G is a divisor of order of a group i.e. (a) / (G) ) If a is an element of a finite group G then a o(g) = e 5) If an integer a is relatively prime to a natural number n then prove that a φ(n) (mod n), φ being the Euler s function. 6) Prove that If P is a prime number and a is an integer such that P a, then a p- (mod p)

36 Unit - 5 De-moiver s Theorem, Elementary Functions. Marks ) State De-Moiver s Theorem for integral indices. ) List n nth roots of unity. ) Write - distinct cube roots of unity. ) Find the sum of all n- nth roots of unity. 5) Simplify ( cos θ + i sin θ ) 8. ( cos θ - i sin θ ) - 6) Simplify (+ i ) (+ i ), using De-Moiver s Theorem. i ( i ) 7) Find - fourth roots of unity. 8) Solve the equation x i =, using De-Moiver s Theorem. 9) Separate into real and imaginary parts of 5 π e + i ) Separate into real and imaginary parts of e ( 5 + i ) ) Define sin z and cos z, z C. ) Define sinh z and cosh z, z C. ) Prove that cos z + sin z =, using definitions of cos z and sin z. ) Prove that tan z = tan - tan z z 5) Prove that sin iz = i sinh z 6) Prove that sinh (iz) = i sin z 7) Prove that cos (iz) = cosh z 8) Prove that cosh (iz) = cos z 9) Prove that tanh (iz) = i tan z ) Prove that tan (iz) = i tanh z

37 ) The four fourth roots of unity are ----, ----,---- and ---- ) If z = - i, then z = ---- ) e - π i = ----, and e π i = ---- ) Period of sin z is ---- Period of cos z is ) Period of sinh z is ---- Period of cosh z is ) Express ( i ) (+ i ) in the form p + iq where p, q are reals. 7) ( cosθ + isinθ ) 7 has seven distinct values. T F 8) ( cosθ + isinθ ) / has distinct values. T F 9) Re ( e z ) = e Re (z) T F ) z e = e z T F ) Match a) sinh z + cosh z i) b) sinh z - cosh z ii) - c) i sin (iz) iii) z e d) sec z.cos z iv) - sinh z v) e iz e iz

38 Choose the correct answer ( to ) ) Consider a) The sum of the n, nth roots of unity is always b) The product of any two roots of unity is a root of unity. A) Both a) & b) are true B) Only a) is true C) Only b) is true D) Both are false ) A value of log i is A) πi B) πi / C) D) - πi / ) The real part of sin ( x + iy ) is A) sin x. cosh y B) cos x. sinh y C) sinh x. cos y D) cosh x. sin y 5) π is period of A) cos z B) tan z C) z e D) cot z 6) a) cos (iz) = cosh z b) sin (iz) = i sinh z A) Both are true B) Both are false C) Only a) is true D) Only b) is true 7) sinh z - cosh z is equal to A) cosh z B) C) - D) sinh z 8) If w is an imaginary 9 th root of unity, then w + w w 8 is equal to A) 9 B) C) D) - 9) A square root of i is A) - i B) + i C) D) i ) ( cos π/ + i sin π/ ) - is A) i B) - i C) D) -

39 Marks. Simplify using De-Moiver s Theorem, the expression ( cos θ ( cos θ i sin θ )7 ( cos θ + i sin θ ) -5 + i sin θ ) ( cos 5θ - i sin 5θ ) -6. Simplify ( cos θ ( cos θ + + i i sin θ ) 8/7 ( cos θ sin θ ) /7 ( cos θ - i sin θ ) /7 - i sin θ ) 5/. Prove that n + sin θ + i cos θ ) sin θ - i cos θ ) = cos [ ( π/ - θ ) n ] + i sin [ ( π/ - θ ) n ] +. If α and β are roots of x x + = and n is a positive integer, then prove that n+ α n + β n = cos (nπ/) 5. Evaluate ( + i ) + ( - i ) 6. Prove that ( + i ) - = - ( -+ i ) 7. Prove that ( - + i ) 7 = -8( + i ) 8. Prove that ( + i ) 8 + ( - i ) 8 = If x =cos α + i sin α, y =cos β + i sin β prove that x y x + y = i tan α β. Find ( + i ) ½ + ( - i ) /. Find all values of ( - i ) /. Find all values of ( + i ) /5 Show that their continued product is + i.. Find the continued product of the four values of + i /. If w is a complex cube root of unity, prove that ( w ) 6 = -7

40 Using De-Moiver s Theorem, solve the following equations ( 5 to 5 ) 5. x x + x x + = 6. x + x + x + x + = 7. x 8 x + = 8. x 9 x 5 + x = 9. x + x 5 + =. 6x 8x + x x + =. x + x + x + =. x 6 - =. x + =. z 7 z + z = 5. z z 6 + = 6. Express cos 5 θ in terms of cosines of multiple of angle θ. 7. Express cos 6 θ in terms of cosines of multiple of angle θ.. 8. Express sin 5 θ in terms of sines of multiple of angle θ. 9. Prove that cos 8 θ = /8 [ cos 8θ + 8 cos 6θ + 8 cos θ + 56 cos θ + 5 ]. Prove that cos 7 θ = /6 [ cos 7θ + 7 cos 5θ + cos θ + 5 cos θ ]. Prove that sin 7θ = 7cos 6 θ sinθ - 5 cos θ sin θ + cos θ sin 5 θ - sin 7 θ. Prove that cos 5θ = cos 5 θ - cos θ sin θ + 5 cosθ sin θ. Prove that sin 5θ = 5 cos θ sinθ - cos θ sin θ + sin 5 θ. If sin α + sin β + sin γ = cos α + cos β + cos γ prove that a) sin α + sin β + sin γ = sin ( α + β + γ ) b) cos α + cos β + cos γ = cos ( α + β + γ ) 5. Express sin sin 7θ θ in powers of sin θ only. 6. Prove that 7. Prove that sin sin 6θ θ sin cos 6θ θ = cos 5 θ - cos θ + 6 cosθ = sin 5 θ - sin θ + 6 sinθ

41 8. Using definitions of cos z and sin z, prove that sin z + cos z = 9. If z and z are complex numbers, show that cos ( z + z ) = cos z cos z - sin z sin z. Prove that cosh ( z + z ) = cosh z cosh z + sinh z sinh z. Prove that a) cosh z = cosh z b) sinh z + = cosh z. Find the general values of a) Log (-i ) b) Log ( -5 ) Separate into real and imaginary parts of ( to 55 ). log ( + i ). log ( + i ) 5. sin ( x + iy ) 6. cos ( x + iy ) 7. tan ( x + iy ) 8. sec ( x + iy ) 9. cosec ( x + iy ) 5. cosh ( x + iy ) 5. coth ( x + iy ) 5. sech ( x + iy ) 5. cosech ( x + iy ) 5. tanh ( x + iy ) 55. cot ( x + iy ) Prove the following ( 56 to 6 ) 56. sinh z = sinh z cosh z 57. sinh z = 58. cosh z = 59. tanh z = tanh - tanh z z + tanh z - tanh z tanh tanh z + z

42 6. cosh z = cosh z cosh z 6. If cos ( x + iy ) = cos α + i sin α show that cos x + i cosh y = 6. If sin ( x + iy ) = tan α + i sec α show that cos x cosh y = 6. If sin (α + iβ ) = x + iy, prove that x y + = and x y = cosh β sinh β sin α cos α 6. If x + iy = cosh (u + iv ), show that x y cosh + u sinh = and x y u cos v sin = v 65. If x + iy = cosh ( u + iv ), show that x sech u + y cosech u = 66. If x + iy = cosh ( u + iv ), show that ( + x) + y = ( cosh v + cos u ) 67. If x + iy = cos ( u + iv ), show that ( - x) + y = ( cosh v - cos u ) 68. If cos ( x + iy ) = r( cos α + i sin α ), show that y = log sin ( x - α ) sin ( x + α ) 69. If u = log [ tan ( π + x ) ]. prove that tanh u = tan x 7. If tan ( x + iy ) = A + ib then show that A = B sinh sin x y 7. Prove that sin[ log ( i i ) ] = - 7. Show that + iθ sin i log ie is purely real. ie iθ 7. Find the 5-5 th roots of. 7. Find the modulus and principal value of the argument of (+ i ) 7 ( i ) 75. Express ( i ) 7 (+ i ) in the form a + ib, where a and b are reals. 76. If z = -( + i ), find z 77. If x i + = x i cos θ ( i =,, ), then prove that one of the value of x x x + x x is cos (θ + θ + θ ) x

43 78. If cos α = x + x and cos β = y + y, prove that one of the values of x m y n + x m y n is cos ( mα + nβ ) 79. If cos θ = x + x and cos φ = y + y, prove that n xm y = i sin ( mθ - nφ ) yn xm 8. Solve the equation x i =, using De-moivre s theorem.

44 Marks / 6 ) State and prove De-Moiver s Theorem for integral indices. ) State and prove De-Moiver s Theorem for rational indices. ) State De-Moiver s Theorem. Obtain the formula for n-nth roots of unity. ) Find n-nth roots of unity and represent them geometrically. 5) Show that the product of any two roots of unity is the root of unity. 6) Show that the 7 th roots of unity form a series in G.P. and find their sum. 7) Show that the sum of n-nth roots of unity is zero. 8) Find n-nth roots of a complex number z = x + iy. 9) Prove that ( x + iy ) m/n + ( x - iy ) m/n = ( x + y ) m/n. cos [ (m/n) tan - (y/x) ] ) If cos θ = x + x and cos φ = y + y then show that x m yn + = cos ( mθ nφ ) yn xm ) Define sin z, cos z and sinh z, cosh z. Prove that sin z and cos z are periodic functions with period π. ) Define tan z. Prove that tan z is a periodic function with period π. ) Define sinh z, and cosh z. Prove that sinh z and cosh z are periodic functions with period πi. ) Obtain the relation between circular functions sinz, cosz and hyperbolic functions sinhz, coshz. 5) Define Log z, z C Separate Log z into real and imaginary parts. 6) Prove that i log [ i ] x = π - tan - x x + i 7) Prove that cos { log ( a i b ) } i + = a - i b 8) Prove that tan { i log ( a - i b ) } a + i b = a b a + b ab a - b 9) Using definition prove that cosh z sinh z = ) If sin - (α + iβ ) = u + iv, prove that sin u and cosh v are the roots of the quadratic equation λ - ( + α + β ) λ + α =