North Maharashtra University 2007 B.Sc Mathematics FY 1 - Question Paper
FYBSC (Mathematics) Paper 1
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Paver 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
Adjoint and Inverse of Matrix, Rank of a Matrix and Eigen Values and Eigen Vectors
|
" 1 |
0 |
2 | ||
|
1) |
If A = |
-1 |
2 |
1 |
|
3 |
1 |
0 |
|
2) If A = If A = If A = If A = |
|
find adj A find A-1 and B = find p(A) |
find minor and cofactor of an, a23 and a32
3)
-1 1
2 0
4)
find p(AB)
5)
Find rank of A =
2 6 1 3 3 9
6)
7) Find the characteristic equation and eigen values of A =
9 - 7 3 -1
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-1 if it exist.
10) Define inverse of a matrix and state the necessary and sufficient condition for existence of a matrix.
11) Compute E12(3) , E 2(3) of order 3
|
12) If A = 13) If A = a) 0 1 2 2 4 14) If A = |
then p (A) is 1 - 2 3 4 1 2 -1 3 find (AB)-1 b) 1 c) 2 then which of the following is true ? |
|
a) adjA is nonsingular c) adjA is symmetric |
b) adjA has a zero row d) adjA is not symmetric |
then which of the following is true ?
-1 -2
-3 4
15) If A =
a) A2 = A
c) A2 is non-singular If A =
16)
|
' 1 2 " |
" 4 3" | |
|
and B = | ||
|
3 4 |
2 1 |
b) A2 is identity matrix d) A2 is singular
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
If A is a square matrix, then A-1 exists iff
17)
a) A > 0 b) |A|< 0
|
3 2 1 4 18) If A = |
d) |A| * 0 then A(adj A) is |
|
a) |
|
|
c) |
|
If A is a square matrix of order n then |KA| is
a) K|A b) (K )nA|
19)
c) Kn|A d) None of these
Let I be identity matrix of order n then a) adj A = I b) adj A = 0
20)
21)
c) adj A = n I d) None of these
Let A be a matrix of order m x n then Ia| exists iff
a) m > n c) m = n
b) m < n d) m n
|
then det. B is equal to |
|
a) 4 |
b) -6 |
C) - / |
d) -28 | |||
|
2 x 0 |
and A-1 = |
" 1 |
0 " | |||
|
If A = |
x x |
-1 |
2 |
then x = --- | ||
|
a) -1/2 |
b) -1/2 |
c) 1 |
d) 2 |
22)
and neN then An is
2 2 2 2
24) If A =
2n 2n 2n 2n
22n -1 22n -1 22n -1 22n -1
a)
c)
d)
|
2n |
2n " | |
|
b) | ||
|
2n |
2n |
22n +1 22n +1 22n +1 22n +1
then |adjA| is
-1 -3
4 2
25) If A =
a) 10 b) 1000 c) 100 d) 110
26) If a square matrix A of order n has inverses B and C then
a) BC b) B = Cn c) B = C d) None of these
|
If A is symmetric matrix then a) adjA is non-singular matrix c) adjA does not exist If AB =
28) |
b) adjA is symmetric matrix d) None of these then |
b) (AB)-1 = A-1 B-1
a) (AB)-1 = AB
c) (AB)-1 = B-1 A-1
d) None of these
29) If | A * 0 and B, C are matrices such that AB = AC then
|
a) B * C b) B * A c) B = C d) C * A | ||||||||||||||||||||
| ||||||||||||||||||||
31) If matrix A is equivalent to matrix B then
a) p(A) * p(B) b) p(A) > p(B)
c) p(A) = p(B) d) None of these
32) If A = [1 - 4 0 ] then p(A) is
a) 0 b) 1 c) 3
d) None of these
1 9 2 0
0 - 3 4 1
33) If A =
then p(A) is
1 9 2 0
a) 0 b) 1 c) 2
d) 3
If A is a matrix of order m x n then a) p(A) < min{m,n} b) p(A) < min{m,n}
34)
c) p(A) > max{m,n} d) None of these
-2 7
2 3
c) 5, -4
d) None of these
The eigen values of A = a) -5, -4 b) 5, 4
then A satisfies
a) A2 + 3A + 17I = 0 b) A2 - 3A - 17I = 0
36) IfA =
c) A2 - 3A + 17I = 0 d) A2 + 3A - 17I = 0
37) If A is a matrix and X is some scalar such that A - XI is singular then
a) X is eigen value of A b) X is not an eigen value of A
c) X = 0 d) None of these
|
0 1 1 -1 0 1 1 1 0 38) IfA = a) p(A) = 0 - 2 1 -3 2 39) IfA = a) A = A-1 |
then A-1 exists if b) p(A) = 3 c) p(A) = 1 d) None of these then which of the following is incorrect ? b) A2 = I c) A2 = 0 d) None of these |
1) If A is a square matrix of order n then prove that (adjA) = adj A and verify it for A =
2) For the following matrix, verify that (adjA) = adj A
2 2 1 3
0 1 2
A =
1 2 3 3 1 1
3) If A =
then show that adj A = A
4) If A =
show that A(adj A) is null matrix.
4 -3 -3
1 0 1
4 4 3
-3 1 0 "
2 -2 1
-1 -1 1
5) Show that the adjoint of a symmetric matrix is symmetric and verify it for A =
2 1 1 3
4 -3 -3 1 0 1 4 4 3
6) Verify that (adjA)A = A I for the matrix A =
cos 0 - sin 0 sin 0 cos 0
7) Verify that A(adjA) = (adjA)A = |A| I for the matrix A =
1 -2 3
2 3 -1 -3 1 2
8) Verify that A(adjA) = AI for the matrix A =
-1 -2 -1 2 1 0 -3 1 -1
9) Find the inverse of A =
10) Find the inverse of A =
11) Show that the matrix A = Hence find A-1
12) IfA =
-3 -1 1
4 -1"
-3 2
satisfies the equation A - 6A + 5 I = 0.
|
1 2" |
" -1 3 " | |
|
, B = | ||
|
0 3 |
7 2 |
show that adj(AB) = adjB adjA
show that A(adjA) = (adjA)A = AI
3 -3 4 2 -3 4 0 -1 1
13) If A =
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
17) What is the reciprocal of the following matrix ? | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
cos a - sin a 0 sin a cos a 0
A =
|
-1 |
2 |
-1 |
|
-1 |
1 |
2 |
|
2 |
-1 |
1 |
0 0 1
|
18) IfA = |
|
verify that (AB) -1 = B-1 A-1 |
19) Using adjoint method find the inverse of the matrix A =
20) If A is a non-singular matrix of order n then prove that adj (adjA) = A|n-2 A
21) For a non-singular square matrix A of order n , prove that
| adj (adj A) |= |A ( -1)
22) For a non-singular square matrix A of order n , prove that
adj {adj (adjA)} = |A
3-3 4
23) If A =
2 -3 4
0 -1 1
24) Find the rank of the matrix A =
2 3 2
3 2 3
1 4 1
2 -1 0 1 3 4
3 2 4 -1
25) Find the rank of the matrix A =
26) Compute the elementary matrix [E2(-3)]- . E31 (2) . E 21 (1/2) of order 3
27) Compute the matrix E 2 (1/3) . E31 . [E2(-4)]-1 for E-matrices of order 3
|
is of is of |
31) Reduce the matrix A to the normal form. Hence determine its rank,
|
" 1 |
2 |
3 |
2 | |
|
where A = |
1 |
2 |
3 |
4 |
|
2 |
6 |
7 |
5 |
32) Reduce the matrix A to the normal form. Hence determine its rank,
3 2 5 7 12 112 3 5
where A =
3 3 6 9 15
33) Find non-singular matrices P and Q such that PAQ is in normal form,
3 1 1 1 1 1 1 -1 -1
where A =
34) Find non-singular matrices P and Q such that PAQ is in normal form,
1 1 2
where A =
1 2 3
0 -1 -1
Find non-singular matrices P and Q such that PAQ is in normal form,
35)
1 3 3 " where A = 1 4 3 Also find p(A)
1 3 4
x
|
" x-1 |
1 |
2 | |
|
36) Show that the matrix A = |
0 |
x |
4 |
|
-3 |
2 |
x | |
|
has rank 3 when x * 2 and |
x * |
, find |
30)
where A =
37) Find a non-singular matrix P such that PA =
G
0
for the matrix
|
Hence find p(A). |
|
" -1 |
-2 |
-1 " |
" 1 |
1 |
-1 ' | ||
|
38) Given A = |
6 |
12 |
6 |
, B = |
2 |
-3 |
4 |
|
5 |
10 |
5 |
3 |
-2 |
3 |
verify that p(AB) < min {p(A), p(B)}
Find all values of 0 in [-n/2, n/2] such that the matrix
39)
1
0
0
0 0 cos 0 sin 0 sin 0 cos 0
is of rank 2.
A =
40) Express the following non-singular matrix A as a product of E - matrices,
1 3 3
where A =
1 4 3
1 3 4
41) Express the following non-singular matrix A as a product of E - matrices,
7 0 3 0 1 0
where A =
2 0 -1
42) Express the following non-singular matrix A as a product of E - matrices,
13 3 3
where A =
1 -5
3 2
1 2 -3 4
4 1 1
4 0 1
43) State Cayley- Hamilton Theorem. Verify it for A =
44) State Cayley- Hamilton Theorem. Verify it for A =
1 2 0 -3 -2 1 1 3 -1
3 2 -1
1 3 0
2 -1 2
45) Verify Cayley- Hamilton Theorem for A =
46) Find the characteristics equation of A =
47) Find eigen values of A =
48) If A is a non-zero eigen value of a non-singular matrix A, show that 1/A is an eigen value of A-1
1 -6 -4
0 4 2 0 -6 -3
49) If A 0 is an eigen value of a non-singular matrix A, show that |A| /A is an eigen value of adj A.
50) Let k be a non-zero scalar and A be a non-zero square matrix, show that if A is an eigen value of A then Ak is an eigen value of kA.
51) Let A be a square matrix. Show that 0 is an eigen value of A iff A is singular.
|
" 0 |
a |
b " |
" 0 |
b |
a | |||
|
52) |
Show that A = |
a |
0 |
c |
, B = |
b |
0 |
c |
|
b |
c |
0 |
a |
c |
0 |
have the same characteristic equation.
|
4 |
-1 |
|
2 |
1 |
|
1 |
3 |
|
2 |
2 |
53) Find eigen values and corresponding eigen vectors of A =
54) Find eigen values and corresponding eigen vectors of A =
1 1 3 1 3 -3 -2 -4 -4
55) Find characteristic equation of A =
Also find A-1 by using Cayley Hamilton theorem.
56) Verify Cayley Hamilton theorem for A and hence find A--2 7
where A =
3 4
1) If A, B are matrices such that product AB is defined then prove that (AB)' = b'a'
2) If A = [ aij ] is a square matrix of order n then show that A(adjA) = (adjA)A = |A| I
3) Show that a square matrix A is invertible if and only if |A| 0
4) If A, B are non-singular matrices of order n then prove that AB is non-singular and (AB)-1 = B-1 A-1
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 (An)-1 = (A-1)n , VneN
7) If A is a non-singular matrix and k 0 then prove that (kA)-1 = k A-1
8) If A is a non-singular matrix then prove that (adj A)-1 = adj A-1 =
9) State and prove the necessary and sufficient condition for a square matrix A to have an inverse.
10) If A is a non-singular matrix then show that AB = AC implies B = C Is the result true when A is singular ? Justify.
11) When does the inverse of a matrix exist ? Prove that the inverse of a matrix, if it exists, is unique.
12) If a non-singular matrix A is symmetric prove that A-1 is also symmetric.
13) Prove that inverse of an elementary matrix is an elementary matrix of the same type.
14) If A is a m x n matrix of rank r, prove that their exist non-singular matrices P
Ir 0
0 0
and Q such that PAQ =
15) Prove that every non-singular matrix can be expressed as a product of finite number of elementary matrices.
16) If A is an mxn matrix of rank r, then show that there exists a non-singular
G"
where G is rxn matrix of rank r and 0 is null
matrix P such that PA =
0
matrix of order (m-r)xn.
17) Prove that the rank of the product of two matrices can not exceed the rank of either matrix.
18) If A is an mxn matrix of rank r then show that there exists a non-singular matrix Q such that AQ = [ H 0 ] Where H is mxr matrix of rank r and 0 is null matrix of order mx(n-r).
System of Linear Equations and Theory of Equations Marks - 02
1) Examine for non-trivial solutions
x + y + z = 0 4x + y = 0 2x + 2y + 3z = 0
2) Define i) Consistent and inconsistent system ii) Equivalent system
3) Define homogeneous, non-homogeneous system of equations.
4) The equation x4+4x3-2x2-12x+9 = 0 has two pairs of equal roots, find them.
5) Change the signs of the roots of the equation x7 + 5x5 - x3 + x2 + 7x + 3 = 0
6) Transform the equation x7 - 7x6 - 3x4 + 4x2 - 3x - 2 = 0 into another whose roots shall be equal in magnitude but opposite in sign to those of this equation.
7) Change of the equation 3x4 - 4x3 + 4x2 - 2x + 1 = 0 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 x4 + x2 + x + 1 = 0 , sum of roots taken one, two, three and four at time is respectively.
B) 0, 1, -1, 1
A) 1, 1, 1, 1
D) -1, 1, -1, 1
C) 1, 0, -1, 1
10) For the equation x4 + x3 + x2 + x + 1 = 0 , sum of roots taken one , two, three and four at a time is respectively.
B) -1, 1, -1, 1
A) 1, 1, 1, 1
D) -1, -1, -1, 1
C) 1, -1, 1, -1
If sum and product of roots of a quadratic equation are 1 and -1 respectively the required quadratic equation is
A) x2 + x + 1 = 0 B) x2 - x + 1 = 0
C) x2 + x - 1 = 0 D) -x2 + x + 1 = 0
The quadratic equation having roots a and P is
12)
|
A) x2 - (a + P) x + aP = 0 13) C) x2 + (a + P) x - aP = 0 The equation having roots 2, 2, -1 is A) x3 + x2 + x + 4 = 0 |
B) x2 + (a + P) x + aP = 0 D) - x2 + (a + P) x + aP = 0 B) x3 + 3x2 + 4 = 0 |
D) x3 - 3x2 + x - 4 = 0
C) x3 - 3x2 + 4 = 0
14) The equation having roots 1, 1, 1 is
A) x3 + 3x2 + 3x + 1 = 0
B) x3 - 3x2 + 3x - 1 = 0
D) x3 + 3x2 - 3x + 1 = 0
C) x3 + 3x2 - x - 1 = 0
15) Roots of equation x3 - 3x2 + 4 = 0 are 2, 2, -1, so the roots of equation x3 - 6x2 + 32 = 0 are
A) 4, 2, -1 B) 4, -4, -1
C) 4, 4, -2 D) 4, -4, -2
16) Roots of equation x2 + 2x + 1 = 0 are -1, -1 so the roots of equation x3 + 6x + 9 = 0 are
A) -3, 3 B) 3, 3
C) -3, -3 D) 3, -3
17) Roots of equation x2-2x+4=0 are 2, 2 so the roots of equation 4x2-2x+1=0 are A) 2, -2 B) 2, 2
C) 1/2, 1/2 D) -1/2, 1/2
18) Roots of equation x2 - 5x + 6 = 0 are 2, 3 so the roots of equation 6x2 - 5x + 1 = 0 are
A) 2, -3 B) 2, 3
C) 1/2, 1/3 D) -1/2, 1/3
19) Find the equation whose roots are the roots of x2 - 4x + 4 = 0 each diminished by 1.
A) x2 - 4x + 4 = 0 B) x2 - 2x + 1 = 0
C) x2 + 2x + 1 = 0 D) x2 - 2x - 1 = 0
20) Find the equation whose roots are the roots of x3 - 6x2 + 12x - 8 = 0 each diminished by 1.
A) x3 - 3x2 + 3x - 1 = 0 B) x3 + 3x2 + 3x + 1 = 0
C) x3 - 3x2 - 3x - 1 = 0 D) x3 - 3x2 - 3x + 1 = 0
21) To remove the second term from equation x4 - 8x3 + x2 - x - 3 = 0 the roots diminished by
A) 3 B) 2 C) 1 D) -2
22) To remove the second term from equation x4 - 4x3 - 18x2 - 3x + 2 = 0 the roots diminished by
A) 1 B) -1
C) 2 D) -2
Examine for consistency the following system of equations
x + z = 2
-2x + y + 3z = 3
-3x + 2y + 7z = 4
Solve the following system of equations
x + y + z = 6 2x + y + 3z = 13 5x + 2y + z = 12
, find A- . Hence solve the following system of linear
|
y 3 - x 2 |
- 2z = |
-10 | |
|
" 2 |
1 |
-1 | |
|
If A = |
1 |
-2 |
3 |
|
-1 |
3 |
-4 | |
|
equations |
2x |
+ y | |
= 1 x - 2y + 3z = 9 -x + 3y - 4z = -12
Test the following equations for consistency and if consistent solve them 2x - y - 5z + 4w = 1 x + 3y + z - 5w = 18 3x - 2y - 8z + 7w = -1 Solve the following system of equations x1 + 3x2 + 4x3 - 6x4 = 0 x2 +6 x3 = 0
2x1 + 2x2 + 2x3 - 3x4 = 0 x1 + x2 - 4x3 - 4x4 = 0
Examine for non-trivial solutions the following homogeneous system of linear equations
x + y + 3z = 0
x - y + z = 0
-x + 2y = 0
7. Solve the system of equations
x + 3y + 3z = 14 x + 4y + 3z = 16 x + 3y + 4z = 17
by i) method of inversion ii) method of reduction.
8. Examine the following systems of equation for consistency x - 2y + z - u = 1
x + y - 2z + 3u = -2 4x + y - 5z + 8u = -5 5x - 7y + 2z - u = 3
9. Test the following equations for consistency and solve them x + 2y + z = 2
3x + y - 2z = 1 4x - 3y - z = 3 x + 2y + z = 2
10. Solve the following equations 4u + 2v + w + 3t = 0
2u + v + t = 0
6u + 3v + 4w + 7t = 0
11. Solve the equation x3 - 3x2 - 6x + 8 = 0 if the roots are in A.P.
12. Solve the equation x3 - 9x2 + 14x + 24 = 0 if two of its roots are in the ratio 3:2.
13. Solve the equation 3x3 - 26x2 + 52x - 24 = 0 if the roots are in G.P.
14. Solve the equation x4 + 2x3 - 21x2 - 22x + 40 = 0 whose roots are in A.P.
15. If a, p and y are roots of the equation x3 - 5x2 - 2x + 24 = 0 find the value of i) 2 a2p ii) 2 a2 iii) 2 a3 iv) 2 a2p2
3 12 2
16. Remove the fractional coefficients from the equation x - + |x - 1 = 0
3 5 2 7 1
17. Remove the fractional coefficients from the equation x -4x -tx+! = 0
2 18 108
3 3 2 3
18. Transform the equation 5x - jx - + 1 = 0 to another with integral coefficients and unity for the coefficient of the first term.
19. Remove the fractional coefficients from the equation
x4 + x2 + 13 x + 77 = 0 x 10x 25x 1000 0
20. Find the equation whose roots are reciprocals of the roots of x4 - 5x3 + 7x2 + 3x-7 = 0
21. Find the equation whose roots are the roots of x4 - 5x3 + 7x2 - 17x + 11 = 0
each diminished by 4.
Find the equatio diminished by 5.
22. Find the equation whose roots are those of 3x3 - 2x2 + x - 9 = 0 each
23. Remove the second term from equation x4 - 8x3 + x2 - x + 3 = 0
24. Remove the third term of equation x4 - 4x3 - 18x2 - 3x + 2 = 0, hence obtain the transformed equation in case h =3.
25. Transform the equation x4 + 8x3 + x - 5 = 0 into one in which the second term is vanishing.
26. Solve the equation x4+16x3+83x2+152x+84 = 0 by removing the second term.
27. Solve the equation x3 + 6x2 + 9x + 4 = 0 by Cardens method.
28. Solve the equation x3 - 15x2 - 33x + 847 = 0 by Cardens method.
29. Solve the equation z3 - 6z2 - 9 = 0 by Cardens method.
30. Solve the equation x3 - 21x - 344 = 0 by Cardens method.
31. Solve x3 - 15x - 126 = 0 by Cardens method
32. Solve 27x3 - 54x2 + 198x - 73 = 0 by Cardens method
33. Solve x3 + 3x2 - 27x + 104 = 0 by Cardens method
34. Solve x3 - 3x2 + 12x + 16 = 0 by Cardens method
35. Solve x4 - 5x2 - 6x - 5 = 0 by Descartes method.
36. Solve the biquadratic x4 + 12x - 5 = 0 by Descartes method.
37. Solve x4 - 8x2 - 24x + 7 = 0 by Descartes method.
1. For what values of a , the equations
x + y + z = 1 2x + 3y + z = a
4x + 9y - z = a2 have a solution and solve then completely in each case.
2. Investigate for what values of X and the following system of equations
x + 3y + 2z = 2 2x + 7y - 3z = -11 x + y + Xz = have
i) No solution ii) A unique solution iii) Infinite number of
solutions.
3. Show that the system of equations
ax + by + cz = 0 bx + cy + az = 0
cx + ay + bz = 0 has a non-trivial solution iff a + b + c = 0 or a = b = c
4. Find the value of X for which the following system have a non-trivial solution x + 2y + 3z = 0
2x +3 y + 4z = 0 3x + 4y + Xz = 0
5. Discuss the solutions of system of equations ( 5 - X ) x + 4y = 0
x + ( 2 - X ) y = 0 for all values of X.
6. Obtain the relation between the roots and coefficients of general polynomial equation aoxn + a1xn-1 + a2xn-2 +-----+ an-1 x + an = 0
7. Solve the equation x3 - 5x2 - 16x + 80 = 0 if the sum of two of its roots being equal to zero.
8. Solve the equation x3 - 3x2 + 4 = 0 if the two of its roots are equal.
9. Solve the equation x3-5x2 -2x+24 = 0 if the product of two of the roots is 12.
10. Solve the equation x3 - 7x2 + 36 = 0 if one root is double of another.
11. Find the condition that the roots of the equation x3 -px2+qx-r = 0 are in A.P.
12. Find the condition that the cubic equation x3 + px2 + qx + r = 0 should have two roots a and P connected by the relation aP + 1 = 0
13. If a, P and y are roots of the cubic equation x3 + px2 + qx + r = 0 find the value of i) 2 a2P ii) 2 a2 iii) 2 a3 iv) 2 a2P2
14. If a, P and y are roots of the cubic equation x3 + px2 + qx + r = 0 find the value of (P+y) (Y+a)(a+P)
15. If a, P and y are roots of the cubic equation x3 - px2 + qx - r = 0 find the value of -L_+_L + _L
p2y2 Y2a2 a2P2
16. If a, P, y and 5 are roots of biquadratic equation x4 + px3 + qx2 + rx + s = 0, find the value of the following symmetric functions
i) 2 a2P ii) 2 a2 iii) 2 a3
17. If a, P, y and 5 are roots of biquadratic equation x4 + px3 + qx2 + rx + s = 0, find the value of the following symmetric functions
i) 2 a2PY ii) 2 a2P2 iii) 2 a4
18. Remove the fractional coefficients from the equation
4 5 3,52 13 x -5x + l2x - 900 = 0
19. Find the equation whose roots are the reciprocals of the roots of
x4 - 3x3 + 7x2 + 5x - 2 = 0
20. Transform an equation aoxn + a1xn-1 + a2xn-2 +.....+ an-1 x + an = 0 into another
whose roots are the roots of given equation diminished by given quantity h.
21. If a, P, y are the roots of 8x3 - 4x2 + 6x - 1 = 0 find the equation whose roots are a + 1/2 , P + V2 , y + 1/2
22. Solve the equation x4 + 20x3 + 143x2 + 430x + 462 = 0 by removing its second term.
23. Reduce the cubic 2x3 - 3x2 + 6x - 1 = 0 to the form Z3 + 3HZ + G = 0
24. Explain Cardens method of solving equation aox3 +3a1x2+3a2x+a3 = 0
Relations, Congruence Classes and Groups
1) Let A = { 1, 2, 3,4, 5, 6, 7, 8, 9,10 }
Ai = { 1, 2, 3, 4 }, A2 = { 5, 6, 7 }, A3 = { 4, 5, 7, 9 }, A4 = { 4, 8, 10 }, A5 = { 8, 9, 10 } , A6 = { 1, 2, 3, 6, 8, 10 } Which of the following is the partition of A.
A) { A1, A2, A5 } B) { A1, A3, A5 }
C) { A2, A3, A6 } D) { A2, A3, A6 }
2) 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
3) For nN, 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
4) 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 a2 = b2 (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 = { 1, -1 } is a group w.r.t. usual
A) Addition B) Subtraction
C) Multiplication D) None of these
7) In the group ( Z6, +6 ), o(5) is
A) 2 B) 5 C) 6 D) 1
8) Linear congruence 207x = 6(mod 18) 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
10) The solution of the linear congruence 5x = 2(mod 7) is
A) x = 2 B) x = 4 C) x = 6 D) x = 3
11) 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
12) Define an equivalence relation and show that > on set of naturals is not an equivalence relation.
13) Define a partition of a set and find any two partitions of A = { a, b, c, d }
14) Define equivalence class of an element. Find equivalence classe of 2 if R ={ (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1), (2, 3), (3, 3), (4, 4),(3, 2), (5, 5)} is an equivalence relation on A = {1, 2, 3, 4, 5}
15) Define residue classes of integers modulo n. Find the residue class of 2 for the relation congruence modulo 5.
16) Define prime residue class modulo n. Find the prime residue class modulo 7
17) Define a group and show that set of integers with respect to usual multiplication is not a group.
18) Define Abelian group and show that group
G = {Ja : ad be 0,a,b,c,d R j is not abelian.
19) Define finite and infinite group. Illustrate by an example.
20) Define order of an element and find order of each element in a group G = {1, -1, i, -i } under multiplication.
21) Find any four partitions of the set S = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19 }
22) Show that AxB * BxA if A = {2, 4, 6}, B = {7, 9, 11}
23) In the group (Z 8 , X8 ), find order of 3, 4, 5, 6
24) Let Z8 = { 1 3, 5, 7 }, find ( 3 )4, ( 3 )0, ( 3 )-4 in a group (Z , X8 )
25) In the group (Z 7 , X7 ), find ( 3 )2, ( 4 )-3, o( 3 ), o( 4 )
26) Find domain and range of a relation R = { (x, y) : x I y for x ,yB} where A = {2, 3, 7, 8}, B = {4, 6, 9, 14}
27) Solve the linear congruence 2x + 1 = 4 (mod 5)
28) Let X = {1, 2, 3} and R = {(1,1), (1, 2), (2, 1), (2, 2), (3, 3), (1, 3), (3, 1), (2, 3), (3, 2)} Is the relation R reflexive, symmetric and transitive ?
29) Prepare the multiplication table for the set of prime residue classes modulo 12.
30) Show that in a group G every element has unique inverse.
31) Show that the linear congruence 13x = 9 (mod 25) has only one solution.
32) Show that the linear congruence 4x = 11 (mod 6) has no solution.
33) If a = b (mod n) and c = d (mod n) then show that ac = bd (mod n)
34) A relation R is defined in the set Z of all integers as aRb iff 7a - 4b is divisible by 3. Prove that R is symmetric.
35) Let ~ be an equivalence relation on a set A and a, b A. Show that b [a] iff [a] = [b]
36) If in a group G, every element is its own inverse then prove that G is abelian.
37) In a group every element except identity element is of order two. Show that G is abelian.
38) If R and S are equivalence relations in set X. Prove that RnS is an equivalence relation.
39) In the set R of all real numbers, a relation ~ is defined by a~b if 2 + ab >0. Show that ~ is reflexive, symmetric and not transitive.
1. 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.
2. 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.
3. 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.
4. Show that in the set of integers x ~ y iff x2 = y2 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 e 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) (Z5, +5) and (Z5, x5) ii) (Z7, +7) and (Z7, x7)
8. Prepare the composition tables for addition and multiplication of
Z6 = {0, 1 2, 3, 4, 5}
9. Show that a Zn has a multiplicative inverse in Zn iff (a, n) = 1
10. Find the remainder when 8103 is divided by 13.
11. Show that G = { 1, -1, i, -i }, where i = >/-I , is an abelian group w.r.t. usual multiplication of complex numbers.
12. Show that the set of all 2 x 2 matrices with real numbers w.r.t multiplication of matrices is not a group.
13. Show that G = { A : A is non-singular matrix of order n over R } is a group w.r.t. usual multiplication of matrices.
14. Let Q+ denote the set of all positive rationals. For a, b e Q+ define a * b = 4
Show that ( Q+ , * ) is a group.
multiplication is a group but it is not an abelian group.
a b c d
15. Show that G =
: ad - bc 0 & a,b,c,d e R > w.r.t. matrix
16. Let Zn be the set of residue classes modulo n with a binary operation
a+nb = a + b = r where r is the remainder when a + b is divided by n Show that (Zn , +n ) is a finite abelian group.
17. Let Z'n denote the set of all prime residue classes modulo n. Show that Z'n is an abelian group of order (n) w.r.t. x n
18. Let G be a group and a, b e R be such that ab = ba. Prove that (ab) n = a n b n , n e Z
19. If in a group G every element is its own inverse then prove that G is an abelian group.
20. Let G = { (a, b) / a, b e R , a 0 } Define on G as
(a, b) (c, d) = ( ac, bc + d ). Show that ( G, ) is a non-abelian group.
21. Let f1, f2 be real valued functions defined by f1(x) = x and f2(x) = 1-x ,VxeR. Show that G = { f1, f2 }is group w.r.t. composition of mappings.
22. Let G be a group and V a, b e G, (ab)n = an bn for three consecutive integers n. Show that G is an abelian group.
23. Show that a group G is abelian iff (ab)2 = a2 b2 , V a, b e G
24. Prove that a group having 4 elements must be abelian.
25. Using Fermats Theorem, Show that 510 - 310 is divisible by 11.
26. Using Fermats Theorem find the remainder when 2105 is divided by 11.
27. Solve
i) 8x = 6 (mod 14) ii) 13x = 9 (mod 25)
28. Let * be an operation defined by a * b = a + b + 1 V a, b e Z where Z is the set of integers. Show that < Z, *> is an abelian group.
cosa -sma Lsina cosaJ
29. Let Aa =
, where a eR and G = { Aa : a eR . Prove that G is an
abelian group under multiplication of matrices.
+ + Cc U
30. Let Q be the set of all positive real numbers and define * on Q by a*b = . Show that (Q+, *) is an abelian group.
31. Find the remainder when 273 + 143 is divided by11.
32. Show that the set G = {[J 1, [J _1], [_Q1 ], ["J1 _1]} is a group w.r.t. multiplication.
33. Show that G = R - {1} is an abelian group under the binary operation a * b = a + b - ab, V a, b e G
34. If the elements a, b and ab of a finite group G are each of order 2 then show that ab = ba.
35. A relation R is defined in the set of integers Z by xRy iff 7x - 3y is divisible by 4. Show that R is an equivalence relation in Z.
36. A relation R is defined in the set of integers Z by xRy iff 3x + 4y is a multiple of 7. Show that R is an equivalence relation in Z.
37. 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 (1, 4).
38. Define congruence modulo n relation and prove that congruence modulo n is an equivalence relation in Z.
39. Show that the set of all 2x2 matrices with real numbers w.r.t. addition of matrices is a group.
1. Let ~ be an equivalence relation on set A. Prove that any two equivalence classes are either disjoint or identical.
2. 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.
3. If a = b ( mod n ) and c = d ( mod n ) for a, b, c, d e Z and n e N then prove that
i) ( a + c ) = ( b + d ) ( mod n )
ii) ( a - c ) = ( b - d ) ( mod n )
iii) ac = bd ( mod n )
4. Write the algorithm to find solution of linear congruence, ax = b ( mod n ), for a, be Z and n e N
5. State and prove Fermats 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-1)-1 = a, V a e G
7. If G is a group then prove that
i) identity of G is unique
ii) (a-1)-1 = a, V a e G
iii) (ab)-1 = b-1a-1, V a , b e G.
8. Let G be a group and a,b,c e 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 e G. Prove that the equations i) ax = b ii) ya = b have unique solutions in G
10. Let G be a group and a e G. Prove that (an)-1 = (a-1)n , V n e N
11. Let G be a group and a e G. For m , n e N,
Prove that i) aman = am+n ii) (am)n = amn
12. 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.
13. Solve the following linear congruence equations
i) 3x = 2 ( mod 8 ) ii) 6x = 5 ( mod 9 )
14. Define a group. Show that any element a e G has a unique inverse in G. Further show that (a*b)-1 = b"1*a"1, V a , b e G.
15. If R is an equivalence relation on a set A then for any a , b e A, prove that
i) [a] = [b] or [a] n [b] =
ii) u { [a] / a e A } = A
16. If ~ is an equivalence relation on set A and A and a , b e A then show that
i) a e [a] for all a e A
ii) b e [a] iff [a] = [b]
iii) a ~ b iff [a] = [b]
17. Define residue classes of integers modulo n. Show that the number of residue classes of integers modulo n are exactly n.
Unit - 04 Subgroups and Cyclic Groups
Marks - 02
1) Define subgroup. Give example
2) Define proper and improper subgroups. Give example.
3) Define a cyclic group. Give example.
4) Define left coset and right coset..
5) State Lagranges Theorem.
6) State Fermats Theorem.
7) State Eulers Theorem.
8) Show that nZ = { nr / rZ ) is a subgroup of (Z, +), where nN.
9) Show that (5Z, +) is a subgroup of (Z, +)
10) Is group (Q+, ) a subgroup of (R, +) ? Justify.
11) Determine whether or not H = {ix : xR} under addition is a subgroup of G = group of complex numbers under addition.
12) Find all possible subgroups of G = { 1, -1, i, -i ) under multiplication.
13) Find proper subgroups of ( Z, + )
14) Write all subgroups of the multiplicative group of 6th roots of unity.
15) Find all proper subgroups of the group of non-zero reals under multiplication.
16) Give an example of a proper subgroup of a finite group.
17) Give an example of a proper finite subgroup of an infinite group.
18) Give an example of a proper infinite subgroup of an infinite group.
19) Is union of two subgroups a subgroup ? Justify.
20) Prove that cyclic group ia abelian.
21) Show by an example that abelian group need not be cyclic.
22) Let G = { 1, -1, i, -i ) be a group under multiplication and H = { 1, -1 ) be its subgroup. Find all right cosets of H in G.
23) Find the order of each proper subgroup of a group of order 15. Are they cyclic.
24) Find generators of Z6 under addition modulo 6.
25) Verify Eulers theorem by taking m = 12, a = 7.
26) Verify Lagranges theorem for Z9 under addition modulo 9.
27) Let G = Z be a additive group of integers and H = 3Z a subgroup of G then H+2 is
A) {3. 6. 9. 12, ----} B) {2, 5, -1, 8, -4, ---}
C) {1, 2, 3, 4, 5, ----} D) {1, 4, -2, 7, -5, ---}
28) Let G = { 1, -1, i, -i ) be a group under multiplication and H = { 1, -1 ) is a subgroup of G then H(-i) is
A) {-1, 1} B) {-i, i} C) {i, -1} D) {1, 1}
29) If n = 12 then 0(12) is
A) 5 B) 4 C) 7 D) 12
30) If p is prime then generators of a cyclic group of order p
A) p B) p-1 C) p2 D) p+1
31) A cyclic group having only one generator can have at the most --- element A) 1 B) 3
C) 2 D) None of these
32) In additive group Z12, ( 4 ) = --A) {4 ,8 } B) {4 ,8,0} C) {2 ,4,0} D) Z12
1) If H is a subgroup of G and x e G, show that xHx-1 = { xhx-1 / h e H } is a subgroup of G
2) Let G be a group. Show that H = Z(G) = { xe G / xa = ax, V a e G } is a subgroup of G.
3) Let G be an abelian group with identity e and H = { xeG / x2 = e }. Show that H is a subgroup of G.
4) Let G be the group of all non-zero complex numbers under multiplication. Show that H ={ a+ib e G / a2+b2 = 1 } is a subgroup of G
5) Show that H = { xe G : xb2 = b2x , V be G } is subgroup of G.
6) Write all subgroups of the multiplicative group of non-zero residue classes modulo 7.
7) Determine whether H1 = { 0,4,8} and H2 = { 0,5,10 } are subgroups of
( Z12 , +12)
8) Let G be a finite cyclic group of order n, and G = < a >. Show that G = < am > ( m, n ) = 1 , where 0 < m < n
9) Find all subgroups of (Z12 , +12 ).
10) Find all subgroups of ( ZS , X7 )
11) Find all generators of additive group Z20
12) Let G = { 1, -1, i, -i } be a group under multiplication and H = { 1, -1 } be its subgroup. Find all right coset of H in G
13) Compute the right cosets of 4Z in ( Z, + ).
14) Let Q = { 1, -1, i, -i , j , -j , k, -k } be a group under multiplication and H = { 1, -1, i, -i } be its subgroup. Find all the right and left cosets of H in G
15) Let G = (Z8 , +8 ) and H = { 0,4 }. Find all right cosets of H in G
16) Let H be a subgroup of a group G and a e G. Show that Ha = { x e G / xa-1 e H }
17) Let G = { 1,2,3,4,5,6,7,8,9,10 }. Show that G is a cyclic group under multiplication modulo 11. Find all its generators, all its subgroups and order of every element. Also verify the Lagranges theorem.
18) List all the subgroups of a cyclic group of order 12.
19) Find order of each element in ( Z7 , +7 )
20) If Z8 is a group w.r.t. addition modulo 8
i) Show that Z8 is cyclic.
ii) Find all generators of Z8
iii) Find all proper subgroups of Z8
21) Show that every proper subgroup of a group of order 35 is cyclic.
22) Show that every proper subgroup of a group of order 77 is cyclic.
23) Let G be a group of order 17. Show that for any a e G either o(a) =1 or o(a) = 17.
24) Let A, B be subgroups of a finite group G , whose orders are relatively prime. Show that A n B = { e }.
25) Find the order of each element in the group G = { 1, w, w2 }, where w is complex cube root of unity, under usual multiplication.
26) Find all subgroups of group of order 41. How many of them are proper ?
27) Find the remainder obtained when 354 is divided by 11.
28) Find the remainder obtained when 3319 is divided by 7.
29) Using Fermats theorem, find the remainder when
i) 987 is divided by 13 .
ii) 541 + 4112 is divided by 13
30) Find the remainder obtained when 1527 is divided by 8.
1) A non-empty subset H of a group G is a subgroup of G iff
a, b e H >.ab-1e H.
2) A non-empty subset H of a group G is a subgroup of G iff
i) a, b e H >.ab-1e H ii) a e H a-1e H .
3) Prove that Intersection of two subgroups of a group is a subgroup
4) Let H, K be subgroups of a group G. Prove that H U K is a subgroup of G, iff either HcK or K c 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-1 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 e H Ha = H ii) a e H aH = H
9) Let H be a subgroup of a group G. Prove that Ha = Hb ab 1 e H
aH = bH b-1a e H , V a,b e G .
10) 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.
11) If H is a subgroup of a finite group G. Then prove that 0(H) / 0(G)
12) Prove that every group of prime order is cyclic and hence abelian
13) Order of every element a of a finite group G is a divisor of order of a group
i.e. 0(a) / 0(G)
14) If a is an element of a finite group G then a o(G) = e
15) If an integer a is relatively prime to a natural number n then prove that a(n) = 1 (mod n) , being the Eulers function .
16) Prove that If P is a prime number and a is an integer such that Pfa, then a p-1 = 1(mod p)
De-moivers Theorem, Elementary Functions.
1) State De-Moivers Theorem for integral indices.
2) List n - nth roots of unity.
3) Write 3- distinct cube roots of unity.
4) Find the sum of all n- nth roots of unity.
5) Simplify ( cos 30 + i sin 30 )8 . ( cos 40 - i sin 40 ) -2
6) Simplify ( +\ ) ( 1+V i ), using De-Moivers Theorem.
i ( 1-V3 i )
7) Find 4- fourth roots of unity.
8) Solve the equation x2 - i = 0, using De-Moivers Theorem.
5 + n i
9) Separate into real and imaginary parts of e 2
10) Separate into real and imaginary parts of e ( 5 + 3 i )2
11) Define sin z and cos z, zeC.
12) Define sinh z and cosh z, zeC.
13) Prove that cos2z + sin2z = 1, using definitions of cos z and sin z.
14) Prove that tan z = 2 tan2
1 - tan2z
15) Prove that sin iz = i sinh z
16) Prove that sinh (iz) = i sin z
17) Prove that cos (iz) = cosh z
18) Prove that cosh (iz) = cos z
19) Prove that tanh (iz) = i tan z
20) Prove that tan (iz) = i tanh z
The four fourth roots of unity are If z = V3 - i, then z12 =
e -ni = , and e 4ni =
22)
23)
24)
25)
26)
27)
28) 29)
Period of sin z is ----
Period of cos z is ----
Period of sinh z is ----
Period of cosh z is ----
( 1+ i )10
( cos0 + isin0 )7 has seven distinct values. T I I F I I
( cos0 + isin0 )34 has 4 distinct values. T I I F I I
Re (e z) = e Re (z) T I I F r
30) e z = e
z
T
F
31) Match
i)
ii)
iii)
iv)
v)
1
-1
e z
- sinh z
_2_
e iz - e -iz
b) sinh2 z - cosh2 z
c) i sin (iz)
d) sec z .cos z
a) sinh2 z + cosh2 z
Choose the correct answer ( 32 to 40 )
32) Consider
a) The sum of the n, nth roots of unity is always 1
b) The product of any two roots of unity is a root of unity.
33) A value of log i is
A) ni B) ni / 2 C) 0 D) - ni / 2
34) 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
35) 2n is period of
A) cos z B) tan z C) e z D) cot z
36) 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
37) sinh2 z - cosh2 z is equal to
A) cosh 2z B) 1 C) -1 D) sinh 2z
38) If w is an imaginary 9th root of unity, then w + w2 +.....+ w8 is equal to
A) 9 B) 0 C) 1 D) -1
39) A square root of 2i is
A) 1 - i B) 1 + i C) V2 D) V2 i
40) ( cos n/4 + i sin n/4 )-2 is
A) i B) - i C) 1 D) -1
1. Simplify using De-Moivers Theorem, the expression
( cos 29 - i sin 29 )7 ( cos 39 + i sin 39 )-5 ( cos 49 + i sin 49 )12 ( cos 59 - i sin 59 ) -6
2. Simplify
( cos 9 + i sin 9 )8/7 ( cos 9 - i sin 9 ) 12/7 ( cos 9 + i sin 9 )12/7 ( cos 49 - i sin 49 ) 5/4
= cos [ ( n/2 - 0 ) n ] + i sin [ ( n/2 - 0 ) n ]
1 + sin 9 + i cos 9 ) nn 1 + sin 9 - i cos 9 )
3. Prove that
4. If a and P are roots of x2 - 2x + 2 = 0 and n is a positive integer, then prove that
n+2
an + pn = 2 2 cos (nn/4)
5. Evaluate ( 1 + i V3 )10 + ( 1 - i -J3 )10
6. Prove that ( 1 + i )- 10 = 2 -11( -1 + i >/3 )
7. Prove that ( -1 + i )7 = -8( 1 + i )
8. Prove that ( 1 + i V3 )8 + ( 1 - i V3" )8 = -256
9. If x =cos a + i sin a , y =cos P + i sin P prove that
X-y = i an (
10. Find ( 3 + 4i ) * + ( 3 - 4i ) 1/2
11. Find all values of ( 1 - i *J3> )1/4
12. Find all values of ( 1 + i )1/5 Show that their continued product is 1 + i.
(1 -J3 x3/4
13. Find the continued product of the four values of I 2 + i 2
14. If w is a complex cube root of unity, prove that ( 1 - w ) 6 = -27
Using De-Moivers Theorem, solve the following equations ( 15 to 25 )
15. x4 - x3 + x2 - x + 1 = 0
16. x4 + x3 + x2 + x + 1 = 0
17. x8 - x4 + 1 = 0
18. x9 - x5 + x4 - 1 = 0
19. x10 + 11x5 + 10 = 0
20. 16x4 - 8x3 + 4x2 - 2x + 1 = 0
21. x3 + x2 + x + 1 = 0
22. x6 - 1 = 0
23. x4 + 1 = 0
24. z7 - z4 + z3 - 1 = 0
25. z12 - z6 + 1 = 0
26. Express cos50 in terms of cosines of multiple of angle 0.
27. Express cos60 in terms of cosines of multiple of angle 0..
28. Express sin50 in terms of sines of multiple of angle 0.
29. Prove that cos8 0 = 1/128 [ cos 80 + 8 cos 60 + 28 cos 40 + 56 cos 20 + 35 ]
30. Prove that cos7 0 = 1/64 [ cos 70 + 7 cos 50 + 21 cos 30 + 35 cos 0 ]
31. Prove that sin 70 = 7cos60 sin0 - 35 cos40 sin30 + 21 cos20 sin50 - sin70
32. Prove that cos 50 = cos50 - 10 cos30 sin20 + 5 cos0 sin40
33. Prove that sin 50 = 5 cos40 sin0 - 10 cos20 sin30 + sin50
34. If sin a + sin p + sin y = cos a + cos P + cos y prove that
a) sin 3a + sin 3p + sin 3y = 3 sin ( a + p + y )
b) cos 3a + cos 3p + cos 3y = 3 cos ( a + p + y )
35. Express 70 in powers of sin 0 only.
36. Prove that sin 60 = 32 cos50 - 24cos30 + 6 cos0
sin 0
37. Prove that sin 60 = 32 sin50 - 32 sin30 + 6 sin0
cos 0
38. Using definitions of cos z and sin z, prove that sin2z + cos2z = 1
39. If z1 and z2 are complex numbers, show that cos ( z1 + z2 ) = cos z1 cos z2 - sin z1 sin z2
40. Prove that cosh ( z1 + z2 ) = cosh z1 cosh z2 + sinh z1 sinh z2
41. Prove that a) 2cosh2z - 1 = cosh 2z
b) 2sinh2z + 1 = cosh 2z
42. Find the general values of a) Log (-i ) b) Log ( -5 ) Separate into real and imaginary parts of ( 43 to 55 )
|
43. |
log ( 4 + 3i ) |
|
44. |
log ( 3 + 4i ) |
|
45. |
sin ( x + iy ) |
|
46. |
cos ( x + iy ) |
|
47. |
tan ( x + iy ) |
|
48. |
sec ( x + iy ) |
|
49. |
cosec ( x + iy ) |
|
50. |
cosh ( x + iy ) |
|
51. |
coth ( x + iy ) |
|
52. |
sech ( x + iy ) |
|
53. |
cosech ( x + iy ) |
|
54. |
tanh ( x + iy ) |
|
55. |
cot ( x + iy ) |
Prove the following ( 56 to 60 )
56. sinh 2z = 2 sinh z cosh z
57. sinh 2z = 2 tanh.z
1 - tanh2 z
58. cosh 2z = 1 + tanh2 z
1 - tanh2 z
59. tanh 2z = 2 tanh2
1 + tanh2 z
60. cosh 3z = 4 cosh2 z - 3 cosh z
61. If cos ( x + iy ) = cos a + i sin a show that cos 2x + i cosh 2y = 2
62. If sin ( x + iy ) = tan a + i sec a show that cos 2x cosh 2y = 3
63. If sin (a + ip ) = x + iy , prove that
2
2
,2
y
y-
= 1 and
= 1
+ -
cosh2 P sinh2 P If x + iy = cosh (u + iv ) , show that
sin2 a cos2 a
64.
y2 =1
2
2
y
x
x
= 1 and
+ -
cosh2 u sinh2 u
22 cos v sin v
If x + iy = cosh ( u + iv ) , show that x2 sech2u + y2 cosech2u = 1
65.
66.
67.
68.
If x + iy = cosh ( u + iv ) , show that ( 1 + x)2 + y 2 = ( cosh v + cos u ) If x + iy = cos ( u + iv ) , show that ( 1 - x)2 + y 2 = ( cosh v - cos u )2
sin ( x - a ) sin ( x + a )
If cos ( x + iy ) = r( cos a + i sin a ), show that 2y = log
If u = log [ tan (4 + x) ]. prove that tanh u = tan x
69.
sin 2x sinh 2y
If tan ( x + iy ) = A + iB then show that A =
B
70.
71. Prove that sin[ log ( i ) ] = -1
-i0
1 + ie
72. Show that sin
is purely real.
i log
-i0
1 - ie
Find the 5-5th roots of -1.
73.
( 1 + )7 ( 43 - i )11
74. Find the modulus and principal value of the argument of
Express ( i|)7 in the form a + ib, where a and b are reals. ( 1 + i )10
75.
76.
77.
10
If z = -(>/3 + i ), find z
If x2. + 1 = 2 x. cos 0 ( i = 1, 2,3 ), then prove that one of the value of xi x2 x3
+ 1 is 2 cos (01 + 02 + 03 )
x1x2x3
78. If 2 cos a = x + 1 and 2 cos P = y + 1, prove that one of the values of
xmyn + 1 is 2 cos ( ma + nP )
xmyn
79. If 2 cos 0 = x + -1 and 2 cos = y + 1, prove that
rn
= 2i sin ( m0 - n )
yn xm
80. Solve the equation x2 - i = 0, using De-moivres theorem.
1) State and prove De-Moivers Theorem for integral indices.
2) State and prove De-Moivers Theorem for rational indices.
3) State De-Moivers Theorem. Obtain the formula for n-nth roots of unity.
4) 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 7th 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 = 2 ( x 2 + y2 ) m/2n . cos [ (m/n) tan-1 (y/x) ]
10) If 2 cos 0 = x + and 2 cos = y + 1 then show that
x y + = 2 cos ( m0 - n6 ) yn xm
11) Define sin z, cos z and sinh z, cosh z. Prove that sin z and cos z are periodic functions with period 2n.
12) Define tan z. Prove that tan z is a periodic function with period n.
13) Define sinh z, and cosh z. Prove that sinh z and cosh z are periodic functions with period 2ni.
14) Obtain the relation between circular functions sinz, cosz and hyperbolic functions sinhz, coshz.
15) Define Log z , z e C Separate Log z into real and imaginary parts.
16) Prove that i log [ x - = n - 2 tan-1x
17) Prove that cos { i log ( +i )} = ~ b2
18) Prove that tan { i log ( +i )} = 22
19) Using definition prove that cosh2 z - sinh2 z = 1
20) If sin -1 (a + ip ) = u + iv, prove that sin2u and cosh2 v are the roots of the quadratic equation X2 - ( 1 + a2 + p2 ) X + a2 = 0
Reduce the matrix A to the normal form. Hence determine its rank,
1 2 3 3 4 5
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