Paper I ( ALGEBRA AND TRIGNOMETRY ) Dr. J. N. Chaudhari M. J. College, Jalgaon Prof. P. N. Tayade Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Prof. Miss. R. N. Mahajan Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Prof. P. N. Bhirud Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Prof. J. D. Patil Nutan Maratha College, Jalgaon
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Paper I
( ALGEBRA AND TRIGNOMETRY )
Dr. J. N. Chaudhari M. J. College, Jalgaon Prof. P. N. Tayade Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Prof. Miss. R. N. Mahajan Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Prof. P. N. Bhirud Dr. A. G. D. Bendale Mahila Mahavidyalaya, Jalgaon Prof. J. D. Patil Nutan Maratha College, Jalgaon
Unit – 01
Adjoint and Inverse of Matrix, Rank of a Matrix and
Eigen Values and Eigen Vectors
Marks – 02
1) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
0 1 3
1 2 1 2 0 1
,
find minor and cofactor of a11, a23 and a32
2) If A = ⎥⎦
⎤⎢⎣
⎡ 2 3
1- 2
, find adj A
3) If A = ⎥⎦
⎤⎢⎣
⎡ 2 7
3 1-
, find A-1
4) If A = ⎥⎦
⎤⎢⎣
⎡ 2 3
2 1- and B = ⎥
⎦
⎤⎢⎣
⎡ 0 2
1 1-
, find ρ(AB)
5) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
3 2 1
7 5 2 1 0 1
,
find ρ(A)
6) Find rank of A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
9 33 1
6 2
7) Find the characteristic equation and eigen values of A = ⎥⎦
⎤⎢⎣
⎡ 1- 3
7- 9
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 = ⎥⎦
⎤⎢⎣
⎡ 4 3
2- 1 and B = ⎥
⎦
⎤⎢⎣
⎡ 2 3-
6 5
, find (AB)-1
13) If A = ⎥⎦
⎤⎢⎣
⎡ 3 1-
2 1 then ρ (A) is ---
a) 0 b) 1 c) 2 d) 4
14) If A = ⎥⎦
⎤⎢⎣
⎡ 4 2
2 1then 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
15) If A = ⎥⎦
⎤⎢⎣
⎡ 4 3-
2- 1-then which of the following is true ?
a) A2 = A b) A2 is identity matrix
c) A2 is non-singular d) A2 is singular
16) If A = ⎥⎦
⎤⎢⎣
⎡ 4 3
2 1 and B = ⎥
⎦
⎤⎢⎣
⎡ 1 2
3 4
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
17) If A is a square matrix, then A-1 exists iff
a) A > 0 b) A < 0
c) A = 0 d) A ≠ 0
18) If A = ⎥⎦
⎤⎢⎣
⎡ 4 1
2 3 then A(adj A) is
a) ⎥⎦
⎤⎢⎣
⎡ 0 10
10 0 b) ⎥
⎦
⎤⎢⎣
⎡ 4 2-
3 1
c) ⎥⎦
⎤⎢⎣
⎡ 10 0
0 10 d) ⎥
⎦
⎤⎢⎣
⎡ 3 1-
2- 4
19) If A is a square matrix of order n then KA is
a) K A b) ( )nK1 A
c) Kn A d) None of these
20) Let I be identity matrix of order n then
a) adj A = I b) adj A = 0
c) adj A = n I d) None of these
21) 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
22) If AB = ⎥⎦
⎤⎢⎣
⎡ 5 4
11 4 and A = ⎥
⎦
⎤⎢⎣
⎡ 2 1
2 3 then det. B is equal to
a) 4 b) -6 c) - ¼ d) -28
23) If A = ⎥⎦
⎤⎢⎣
⎡ x x 0 2x
and A-1 = ⎥⎦
⎤⎢⎣
⎡ 2 1-
0 1 then x = ---
a) -1/2 b) -1/2 c) 1 d) 2
24) If A = ⎥⎦
⎤⎢⎣
⎡ 2 2
2 2 and n∈N then An is ----
a) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
n2 n2
n2 n2 b) ⎥
⎦
⎤⎢⎣
⎡2n 2n
2n 2n
c) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
1-2n2 1-2n2
1-2n2 1-2n2 d)
⎥⎥⎦
⎤
⎢⎢⎣
⎡
++
++
12n2 12n2
12n2 12n2
25) If A = ⎥⎦
⎤⎢⎣
⎡ 2 4
3- 1- then adjA is
a) 10 b) 1000 c) 100 d) 110
26) If a square matrix A of order n has inverses B and C then
a) B≠C b) B = Cn c) B = C d) None of these
27) 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
28) If AB = ⎥⎦
⎤⎢⎣
⎡ 1 2-
3 1 and A = ⎥
⎦
⎤⎢⎣
⎡ 2- 4
7- 3 then
a) (AB)-1 = AB b) (AB)-1 = A-1 B-1
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
30) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
3- 2- 1
4 3 1 4- 2- 2
then
a) A2 = I b) A2 = 0 c) A2 = A d) None of these
31) 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
32) If A = [ ] 0 4 1 − then ρ(A) is
a) 0 b) 1 c) 3 d) None of these
33) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
0 2 9 11 4 3 00 2 9 1
then ρ(A) is
a) 0 b) 1 c) 2 d) 3
34) 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
35) The eigen values of A = ⎥⎦
⎤⎢⎣
⎡ 3 2
7 2- are
a) -5, -4 b) 5, 4 c) 5, -4 d) None of these
36) If A = ⎥⎦
⎤⎢⎣
⎡ 2 3
5- 1 then A satisfies
a) A2 + 3A + 17I = 0 b) A2 - 3A - 17I = 0
c) A2 - 3A + 17I = 0 d) A2 + 3A - 17I = 0
37) 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 = 0 d) None of these
38) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
0 1 1
1 0 1 1 1 0
then A-1 exists if
a) ρ(A) = 0 b) ρ(A) = 3 c) ρ(A) = 1 d) None of these
39) If A = ⎥⎦
⎤⎢⎣
⎡ 2 3-
1 2- then which of the following is incorrect ?
a) A = A-1 b) A2 = I c) A2 = 0 d) None of these
Marks : 04
1) If A is a square matrix of order n then prove that (adjA) ' = adj A '
and verify it for A = ⎥⎦
⎤⎢⎣
⎡ 3 1
2 2
2) For the following matrix, verify that (adjA) ' = adj A '
A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 1 3
3 2 1 2 1 0
3) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3 4 4
1 0 1 3- 3- 4
then show that adj A = A
4) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 1- 1-
1 2- 2 0 1 3-
,
show that A(adj A) is null matrix.
5) Show that the adjoint of a symmetric matrix is symmetric and verify it for
A = ⎥⎦
⎤⎢⎣
⎡ 3 1
1 2
6) Verify that (adjA)A = A I for the matrix A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3 4 4
1 0 1 3- 3- 4
7) Verify that A(adjA) = (adjA)A = A I for the matrix A = ⎥⎦
⎤⎢⎣
⎡ θ cos θsin
θsin - θ cos
8) Verify that A(adjA) = A I for the matrix A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
2 1 3-
1- 3 2 3 2- 1
9) Find the inverse of A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1- 1 3-
0 1 2 1- 2- 1-
10) Find the inverse of A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 1- 3-
2 1 1- 1- 2 1
11) Show that the matrix A = ⎥⎦
⎤⎢⎣
⎡ 2 3-
1- 4 satisfies the equation A2 – 6A + 5 I = 0.
Hence find A-1
12) If A = ⎥⎦
⎤⎢⎣
⎡ 3 0
2 1, B = ⎥
⎦
⎤⎢⎣
⎡ 2 7
3 1-
, show that adj(AB) = adjB adjA
13) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 1- 0
4 3- 2 4 3- 3
,
show that A(adjA) = (adjA)A = A I
14) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 2- 2
2- 1 2 2- 2- 1-
then show that adjA = 3A '
15) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3- 2- 1
4 3 1- 4- 2- 2-
,
show that A2 = A, but A-1 does not exist.
16) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
0 1 0
1- 0 0 0 0 1
,
show that A3 = A-1
17) What is the reciprocal of the following matrix ?
A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 0 0 0 cos sin 0 sin - cos
αααα
18) If A = ⎥⎦
⎤⎢⎣
⎡ 1 1
1 3, B = ⎥
⎦
⎤⎢⎣
⎡ 1 2
1- 3
, verify that (AB) -1 = B-1 A-1
19) Using adjoint method find the inverse of the matrix A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 1- 2
2 1 1- 1- 2 1-
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
A) (adj adj = ( )21nA −
22) For a non-singular square matrix A of order n , prove that
adj {adj (adjA)} = 332A +− nn A-1
23) If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 1- 0
4 3- 2 4 3- 3
,
show that A3 = A-1
24) Find the rank of the matrix A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 4 1
3 2 3 2 3 2
25) Find the rank of the matrix A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
4 2 3
4 3 1 0 1- 2
26) Compute the elementary matrix [E2(-3)]-1. 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
28) Determine the values of x so that the matrix ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
x2 x
x x2 2 x x
is of
i) rank 3 ii) rank 2 iii) rank 1
29) Determine the values of x so that the matrix ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
x1 x
x x1 1 x x
is of
i) rank 3 ii) rank 2 iii) rank 1
30) Reduce the matrix A to the normal form. Hence determine its rank,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
10 8 6
5 4 3 3 2 1
31) Reduce the matrix A to the normal form. Hence determine its rank,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
5 7 6 24 3 2 1
2 3 2 1
32) Reduce the matrix A to the normal form. Hence determine its rank,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
159 6 3353 2 11
12 7 5 2 3
33) Find non-singular matrices P and Q such that PAQ is in normal form,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1- 1- 1
1 1 1 1 1 3
34) Find non-singular matrices P and Q such that PAQ is in normal form,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1- 1- 0
3 2 1 2 1 1
35) Find non-singular matrices P and Q such that PAQ is in normal form,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
4 3 1
3 4 1 3 3 1
Also find ρ(A)
36) Show that the matrix A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
x2 3-
4 x0 2 1 1- x
has rank 3 when x ≠ 2 and x ≠ ± 2 , find its rank when x = 2.
37) Find a non-singular matrix P such that PA = ⎥⎦
⎤⎢⎣
⎡0 G for the matrix
A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
4 2 3
4 3 1 0 1- 2
.
Hence find ρ(A).
38) Given A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
5 10 5
6 12 6 1- 2- 1-
, B = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3 2- 3
4 3- 2 1- 1 1
,
verify that ρ(AB) ≤ min {ρ(A), ρ(B)}
39) Find all values of θ in [-π/2, π/2] such that the matrix
A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
θ cos θsin 0
θsin θ cos 0 0 0 1
is of rank 2.
40) Express the following non-singular matrix A as a product of E – matrices,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
4 3 1
3 4 1 3 3 1
41) Express the following non-singular matrix A as a product of E – matrices,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1- 0 2
0 1 0 3 0 7
42) Express the following non-singular matrix A as a product of E – matrices,
where A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1 0 4
1 1 4 3 3 13
43) State Cayley- Hamilton Theorem. Verify it for A = ⎥⎦
⎤⎢⎣
⎡ 2 3
5- 1
44) State Cayley- Hamilton Theorem. Verify it for A = ⎥⎦
⎤⎢⎣
⎡ 4 3-
2 1
45) Verify Cayley- Hamilton Theorem for A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1- 3 1
1 2- 3- 0 2 1
46) Find the characteristics equation of A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
2 1- 2
0 3 1 1- 2 3
47) Find eigen values of A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3- 6- 0
2 4 0 4- 6- 1
48) If λ is a non-zero eigen value of a non-singular matrix A, show that 1/λ is an
eigen value of A-1
49) If λ ≠ 0 is an eigen value of a non-singular matrix A, show that A /λ 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 λ is
an eigen value of A then λk 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.
52) Show that A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
0 c b
c 0 a b a 0
, B = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
0 c a
c 0 b a b 0
have the same characteristic equation.
53) Find eigen values and corresponding eigen vectors of A = ⎥⎦
⎤⎢⎣
⎡ 1 2
1- 4
54) Find eigen values and corresponding eigen vectors of A = ⎥⎦
⎤⎢⎣
⎡ 2 2
3 1
55) Find characteristic equation of A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
4- 4- 2-
3- 3 1 3 1 1
Also find A-1 by using Cayley Hamilton theorem.
56) Verify Cayley Hamilton theorem for A and hence find A-1
where A = ⎥⎦
⎤⎢⎣
⎡ 4 3
7 2-
Marks – 04 / 06
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 , ∀n∈N
7) If A is a non-singular matrix and k ≠ 0 then prove that (kA)-1 = k1 A-1
8) If A is a non-singular matrix then prove that (adj A)-1 = adj A-1 = AA
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
and Q such that PAQ = ⎥⎦
⎤⎢⎣
⎡ 0 0
0 rI
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
matrix P such that PA = ⎥⎦
⎤⎢⎣
⎡0G
, where G is rxn matrix of rank r and 0 is null
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 = [ ] 0 H Where H is mxr matrix of rank r and 0 is
null matrix of order mx(n-r).
Unit - 02
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.
A) 1, 1, 1, 1 B) 0, 1, -1, 1
C) 1, 0, -1, 1 D) -1, 1, -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.
A) 1, 1, 1, 1 B) -1, 1, -1, 1
C) 1, -1, 1, -1 D) -1, -1, -1, 1
11) 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
12) The quadratic equation having roots α and β is
A) x2 – (α + β) x + αβ = 0 B) x2 + (α + β) x + αβ = 0
C) x2 + (α + β) x – αβ = 0 D) – x2 + (α + β) x + αβ = 0
13) The equation having roots 2, 2, -1 is
A) x3 + x2 + x + 4 = 0 B) x3 + 3x2 + 4 = 0
C) x3 – 3x2 + 4 = 0 D) x3 – 3x2 + x – 4 = 0
14) The equation having roots 1, 1, 1 is
A) x3 + 3x2 + 3x + 1 = 0 B) x3 – 3x2 + 3x – 1 = 0
C) x3 + 3x2 – x – 1 = 0 D) x3 + 3x2 – 3x + 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
Marks – 04
1. Examine for consistency the following system of equations
x + z = 2
-2x + y + 3z = 3
-3x + 2y + 7z = 4
2. Solve the following system of equations
x + y + z = 6
2x + y + 3z = 13
5x + 2y + z = 12
2x - 3y - 2z = -10
3. If A = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
4- 3 1-
3 2- 1 1- 1 2
, find A-1. Hence solve the following system of linear
equations 2x + y - z = 1 x - 2y + 3z = 9 -x + 3y - 4z = -12
4. 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
5. 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
6. Examine for non-trivial solutions the following homogeneous system of linear
equations
x + y + 3z = 0
x - y + z = 0
-x + 2y = 0
x - y + z = 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 α, β and γ are roots of the equation x3 - 5x2 - 2x + 24 = 0 find the value of
i) ∑ α2β ii) ∑ α2 iii) ∑ α3 iv) ∑ α2β2
16. Remove the fractional coefficients from the equation x3 – 21 x2 + 3
2 x – 1 = 0
17. Remove the fractional coefficients from the equation x3 – 25 x2–18
7 x+1081 = 0
18. Transform the equation 5x3 – 23 x2 – 4
3 x + 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 + 103 x2 + 25
13 x + 100077 = 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.
22. Find the equation whose roots are those of 3x3 – 2x2 + x – 9 = 0 each
diminished by 5.
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 Carden’s method.
28. Solve the equation x3 – 15x2 – 33x + 847 = 0 by Carden’s method.
29. Solve the equation z3 – 6z2 – 9 = 0 by Carden’s method.
30. Solve the equation x3 – 21x – 344 = 0 by Carden’s method.