9387.matrices mcq
DESCRIPTION
Mcq's on matricesTRANSCRIPT
MCQ QUESTION BANK ON MATRICES Type I Rank and Normal Form
Q.1)Which of the following matrix is in normal form?
A)
1 2 5
0 1 9
0 0 5
B)
1 4 3 1
0 0 1 4
0 1 3 8
0 0 0 1
C)1 0
0 1
D) 1 0 0
1 0 0
Q.2) Echelon form of matrix 1 1 1 1 1
5 5 5 5 5
8 8 8 8 8
is
A) 1 1 1 1 1
4 4 4 4 4
7 7 7 7 7
B) 0 0 0 0 0
4 4 4 4 4
7 7 7 7 7
C)1 1 1 1 1
0 0 0 0 0
0 0 0 0 0
D)1 1 1 1 1
6 6 6 6 6
9 9 9 9 9
Q.3) Rank of a matrix is nothing but
A) number of zero rows in that matrix B) number of zero rows in its echelon form of matrix
C) number of non-zero rows in that matrix D) number of non-zero rows in its echelon form of
matrix.
Q.4) The rank of matrix 1 2 3
2 2 2
3 3 3
A
=
is equal to
A) 4 B) 3 C) 2 D) 1
Q5) If 1 2 3
3 4 5
4 5 6
A
=
and det(A)=0 then rank of a matrix A is
A) Greater than or equal to 3 B) Strictly less than 3
C) Less than or equal to 3 D) Strictly greater than 3 .
Q.6) Which of the following matrix is in normal form?
A)
1 0 0
0 1 0
0 0 0
B)
0 0 0
0 1 0
0 0 1
C)
1 1 1
0 1 0
1 1 1
D)
0 0 0
0 0 0
0 1 0
Q.7) Which of the following matrix is in the Normal form?
A) 1 1 0 0
0 1 0 0
0 0 1 0
B) 1 0 0 0
0 1 0 1
0 0 0 1
C)
0000
0100
0010
0001 D)
100
110
111
Q.8) The rank of matrix 10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
is
A) 10 B) 5 C)2 D)1.
Q.9)The rank of the matrix 1 1 1
1 1 0
1 1 1
−
is ,
A)0 b) 1 c) 2 d) 3
Q.10) For matrix A of order mxn, the rank r of matrix A is
A) min{ , }r m n≥ B) max{ , }r m n≥
C) min{ , }r m n≤ D) max{ , }r m n≤
Q.11)For non singular matrix A If PAQ is in normal form then 1
A−
is equal to
A)PQ B) QP C) P+Q D) Q-P
Q.12) A 5×7 matrix has all its entries equal to -1 , then rank of matrix is
A)7 B) 5 C) 1 D) zero
Q.13) The rank of the following matrix by determinant method
421
134
432is
A)2 B) 3 C) 1 D) 0
Q.14)If P=3 then the rank of matrix
=
3
3
3
PP
PP
PP
A .
A)1 B) 2 C) 3 D) 0
Type II) System of Linear Equations & LD/ID, Linear Transform and Orthogonal Transforms.
Q.15) Given system of linear equations x-4y+5z=-1, 2x-y+3z=1,3x+2y+z=3 has
A) unique solution B) no solution
C) infinitely many solutions D) n-r solutions
Q.16) In given system of linear equations AX=B,
if Rank (A) = rank (A/B) =Number of unknowns then the system is,
A) inconsistent & system has no solution B)Consistent & system has infinite solutions
C) Consistent & system has unique solution D) None of the above
Q.17) In given system of linear equations AX=B, A is square matrix of order n.
If Rank (A) =rank (A/B) <Number of unknowns then the system is,
A)inconsistent & system has no solution B)Consistent & system has infinite solutions
C) Consistent & system has unique solution D) None of the above
Q.18) In given system of linear equationsAX=B, if det(A) ≠ 0 then system has
A)Unique solution B)No solution C)infinite
solutions D) None of the above
Q.19)In set of vectors , if at least one vector of the set can be expressed as a linear
combination of the remaining vectors then these vectors are called
A)Linearly independent B)linearly dependent
C)Orthogonal vectors D) none of these
Q.20) If two linear transformations are y Ax= and z By= then composite transformation which
Converts vector x in to a vector z is
A) z BAx= B) 1 1z A B x− −=
C) 1 1z B A x− −= D) z ABx=
Q.21) A Linear transformation y Ax= is said to be orthogonal if A is
A) Orthonormal matrix B) Orthogonal matrix
C) Symmetric Matrix D)Singular Matrix
Q.22)Non- Homogeneous system of linear equations AX=B is consistent if and only if
A) ( ) ([ / ])A A Bρ ρ> B) ( ) ([ / ])A A Bρ ρ≠
C) ( ) ([ / ])A A Bρ ρ< D) ( ) ([ / ])A A Bρ ρ=
Q.23) A is mxn matrix and AX=B is system of linear equations then AX=B has unique solution if and only
if
A) ( ) ([ / ])A A B mρ ρ= = B) ( ) ([ / ])A A B nρ ρ= =
C) ( ) ([ / ])A A Bρ ρ= D) ( ) ([ / ])A A B mρ ρ= <
Q.24) A is mxn matrix and AX=B is system of linear equations then AX=B has infinite number of solution
if and only if
A) ( ) ([ / ])A A B mρ ρ= = B) ( ) ([ / ])A A B nρ ρ= <
C) ( ) ([ / ])A A B mρ ρ= < D) ( ) ([ / ])A A B mρ ρ= >
Q.25) Every homogeneous system of linear equations is
A) Always consistent B) May or may not be consistent
C) Never consistent D) none of these
Q.26)In a given system of equations AX=B, if )/()( BAA ρρ ≠ then the system of equations is,
A)Consistent B ) Inconsistent
C)Has a Unique solution. D)Infinite solutions.
Q.27)The system of linear equations 4x+2y=7 , 2x+y=6 has
A)A unique solution B)No solution
C) infinite no. of solution D)Exactly two distinct solutions.
Q.28)Consider following system of linear equations in three real variables x, y & z
2x-y+3z=1, 3x+2y+5z=2, -x+4y+z=3. The system has
A) No solution B) An infinitely many solutions
C) unique solution D) more than one but finite no. of solutions
Q.29) Consider following system of linear equations in three real variables x, y & z
2x-y+3z=0,3x+2y+5z=0,-x+4y+z=0. The system has solution
A) x=0, y=0,z=0 B) x=1,y=3,z=0 C) x=-9,y=5,z=1 D)x=-1,y=-2,z=6
Q.30)A is a 3×4 real matrix & AX=B is an inconsistent system of linear equation Then the highest possible
rank of A is
A)1 B) 2 C) 3 D) 4
Q.31) For what value of b the matrix 51
513
bA
b
− =
is an orthogonal?
A) 5± B) 13± C) 12± D) 16±
Q. 32) The determinant of orthogonal matrix is always
A) Greater than -1 B) less than +1 C) Equal to 0 D) Equal to +1 or -1.
Q.33) If 1T
A A−= then A is ,
A)Symmetric B) Skew Symmetric C) Orthogonal D) None of these
Q.34)If A is an Orthogonal matrix then is equal to ,
A) A B) T
A C) 2
A D) T
A−
Type III] Eigen Values, Eigen Vectors,Cayley Hamilton Theorem.
Q.35) If x is eigen vector of matrix A corresponding to eigen value λ then x and kx ,k < 0
A) has same direction as that of x B) has opposite direction
C) x is orthogonal to kx D) x is parallel to kx
Q.36) If A is any square matrix then its characteristic equation is given by
A) det( ) 0A Iλ− = B) ( ) 0A Iλ− = C) det( ) 0A Aλ− = D) ( ) 0A Aλ− =
Q.37) If eigen values of matrix A are 1,2,3 then eigen values of matrix 2A2 are
A)-1,-2,-3 B)1,2,3 C)2,4,9 D) 2,8,18
Q.38) The characteristics roots of the matrix 1 1 1
1 1 1
1 1 1
are
A) (0,0,0) B)(0,0,3) C) (0,0,1) D) (1,1,1)
Q.39)Find sum of the eigenvalues of the matrix
−
−
24
32
A)2 B)4 C) 0 D) 1
Q.40)Find product of eigenvalues of matrix
−
−
24
32
A)4 B)8 C) 6 D) 2
Q.41)The product of two eigen values of the matrix
−
−−
−
=
312
132
226
A is 16. Find the third eigenvalue.
A)1 B)2 C) 4 D) 3
Q.42) For a given matrix A of order 3×3, det(A)=32 & two of its eigenvalues are 8 & 2. Find sum of
eigenvalues
A)12 B)8 C) 10 D) 2
Q.43) If 2 & 3 are eigenvalues of
=
201
020
102
A find the third eigenvalue
A)2 B)3 C)1 D) 4
Q.44) If 1, 2 & 3 are the eigen values of
=
20
020
102
a
A , find the value of a?
A)1 B)0 C) 2 D) 3
Q.45) The characteristic equation of the matrix
−
−=
15
1014A is ,
A) 2 5 21 0λ λ+ + = B)
2 13 36 0λ λ− + =
C) 2 13 36 0λ λ+ + = D)
2 13 36 0λ λ+ − =
Q.46) Find the eigen values of the matrix
−
−=
15
1014A
A) 1 24, 9λ λ= = B)
1 25, 6λ λ= =
C) 1 218, 2λ λ= = D)
1 210, 3λ λ= =
Q.47) Two eigen values of the matrix
=
221
131
122
A are 1 &1, find the 3rd
eigenvalue of A.
A)1 B)3 C) 5 D)4
Q.48)Two eigenvalues of the matrix
=
221
131
122
A are 1 & 1 find the eigenvalues of 1
A−
A)1/1 , 1/1 , 1/5 B)½ , 1 , 5 C)½ , ½ , 5 D)1 , 1 , 5
Q.49) Form the matrix whose eigenvalues are 5, 5, 5α β γ− − − where , ,α β γ
the eigenvalues of are
−
−
−−−
=
987
654
321
A
A)
−
−
−−−
987
654
321
B)
−
−
−−−
487
604
326
C)
−
−−
987
654
321
D)
−
−
−−
1487
6104
324
Q.50) If the characteristic equation of one matrix is 3 24 4 0λ λ λ− − + = then find the Eigen values of that
matrix
A)1 , 2 , 3 B)1 , 1 , 4 C) -1 , 1 , 4 D) 1 , 1 , 5
Q.51) For a singular matrix of order 3 3X , 2 and 3 are the eigenvalues. Find its 3rd
eigenvalue.
A)1 B) 0 C) 2 D) 3
Q.52) What is the characteristic equation of the matrix
−
−
110
121
011
A) 3 24 3 0λ λ λ− + = B) 3 2
4 3 1 0λ λ λ− + + = C) 3 24 3 0λ λ λ+ + = D) 3 2
4 3 0λ λ λ− − =
Q.53) Determinant of square matrix is equal to
A) Sum of all elements B) Product of diagonal elements
C) Product of its eigen values D) Sum of its eigen values .
Q.54) If 1,2,3 are eigen values of matrix A then eigen values of matrix A3 are
A)1,8,27 B) 1,4,9, C) 2,3,4, D) 4,5,6
Q.55) If λ is eigen value of matrix A then eigen values of matrix A-1
is
A) λ B) λ− C) 1
λ D)1.
Q.56) If λ is eigen value of matrix A then eigen values of matrix kA is
A) k λ B) λ− C) 1
kλ D) λ .
Q.57) If λ is eigen value of matrix A then eigen values of matrix A+kI is
A) k λ B) kλ + C) 1
kλ D) kλ− .
Q.58) If λ is eigen value of matrix A then eigen values of matrix An is
A) n λ B) nλ C) n
λ D) λ .
Q.59) If 1 2 3, ,λ λ λ are eigen values of matrix A then eigen values of 1A− are
A) 1 2 3
1 1 1, ,
λ λ λ B) 1 2 3, ,λ λ λ C)
2 2 2
1 2 3, ,λ λ λ D) 1 2 3, ,λ λ λ− − −
Q.60) The sum & product of the eigen values of matrix 2 3
4 2
− −
are
A)0,0 B)4,8 C)0,8 D) 2,-2
Q.61) For the matrix
−
−
400
520
321product of the eigen values is
A)-8 B) 4 C)1 D) -2
Q. 62) The eigen values of the matrix
32
41are
A) 2,3 B)4,5 C) 0,2 D) 5,-1
Q.63) The Characteristic equation of the matrix 3 1 1
1 5 1
1 1 3
− − −
is
A) 3 211 38 40 0λ λ λ− + − = B) 3 211 38 40 0λ λ λ− + + =
C) 3 211 38 40 0λ λ λ+ + + = D) 3 211 38 40 0λ λ λ+ + + =
Q.64) If the Characteristic equation of the matrix A of order 3x3 is 3 23 3 1 0λ λ λ− + − = then by Cayley
Hamilton
Theorem 1A− is equal to
A) 3 23 3A A A I− + − B) 2 3 3A A I− −
C) 23 3A A I− − D) 2 3 3A A I− −
Q.65) If 2
1 2 0S Sλ λ− + = is a characteristic equation of 2x2 matrix A then
A) 1S = Sum of principle diagonal elements, 2S = Sum of all elements
B) 1S = Sum of principle diagonal elements, 2S = Product of principle diagonal elements
C) 1S = Trace of matrix A, 2S = Product of principle diagonal elements
D) 1S = Trace of matrix A, 2S = Product of Eigen values of matrix A.
Q.66) If 3 2
1 2 3 0S S Sλ λ λ− + − = is a characteristic equation of 3x3 matrix A then
A) 1S = Sum of principle diagonal elements, 2S = Sum of all elements, 3S A=
B) 1S = Sum of principle diagonal elements, 2S = Product of principle diagonal elements,
3S A=
C) 1S = Trace of matrix A, 2S = sum of minors of Principle diagonal elements, 3S A=
D) 1S = Trace of matrix A, 2S = Product of Eigen values of matrix A, 3S A=
Q.67) The characteristic equation of matrix 14 10
5 1
− −
is
A) 2 13 36 0λ λ− + = B) 2 13 36 0λ λ− − = C) 2 4 64 0λ λ− − = D) 2 36 0λ λ− + =
Q.68) The characteristic equation of matrix 4 6 6
1 3 2
1 4 3
− − −
is
A) 3 24 4 0λ λ λ− + − = B) 3 24 4 0λ λ λ+ − + = C) 3 2 4 0λ λ λ− + − = D) 3 24 4 0λ λ λ− − + =
Q.69) If all eigen values of matrix 3 3xA are distinct then which of the following is true
A) Matrix 3 3xA has three equal Eigen vectors
B) Matrix 3 3xA has three distinct eigen vectors
C) Matrix 3 3xA has three distinct linearly independent eigen vectors
D) ) Matrix 3 3xA has more than three Eigen vectors.
Q.70) If two or more eigen values are equal then corresponding eigen vectors
A) always Linearly Dependent B) always Linearly Independent
C) may or may not be Linearly Independent D) none of these
Q.71) The eigen vectors corresponding to distinct eigen values of a real symmetric matrix are
A) Linearly Dependent B)Linearly Independent C) Orthogonal D) Orthonormal
Q.72) The eigen values of matrix
1 2 3 4
0 1 2 3
0 0 5 3
0 0 0 4
−
−
are
A)-1,1,-5,4 B)1,-1,5,-4 C)0,0,0,0 D) -1,-1,-5,-4
Q.73) The eigenvalues of matrix
111
111
111 are
A)0,0,0 B) 0,0,1 C) 0,0,3 D) 1,1,1
Q.74) Consider the two statements
(i) a particular eigen value may be zero. (ii) a particular eigen vector may be zero.
which of the above correct.
A) only (i) B) only (ii) C) Both (i) and (ii) D) Both are incorrect.
Q.75) Cayley Hamilton Theorem is
A) Every symmetric matrix satisfies its own characteristic equation
B) Every square matrix satisfies its own characteristic equation.
C) Every orthogonal matrix satisfies its own characteristic equation
D) Every real symmetric matrix satisfies its own characteristic equation
Q.76)If 1 0
1 7A
=
then value of k for which 2 8A A kI= + is
A) 5 B)7 C)-5 D)-7
Q.77) If 1 2
2 1A
= −
then 8A is
A) 8 5A I= B) 8 25A I= C) 8 65A I= D) 8 625A I=
Q.78) Two eigen values of 4 6 6
1 3 2
1 5 2
A
= − − −
are equal and are double the third then the eigen values
2A are
A) 1,4,4 B) 3,2,1 C) 4,9,16 D) 25,4,9
Q.79) If 1 2 3
0 3 2
0 0 2
A
− = −
then the eigen values of 3 23 5 6 2A A A I+ − + are
A)5,-6,2 B)1,3,-2 C) 3,5,6 D) 4,110,10
Q.80) Sum and product of the eigen values of matrix 1 1 1
1 1 1
1 1 1
A
− = − −
is
A) -3,-1 B)-3,4 C) 4,3 D)1,-3
Q.81) If 1 2
3 4A
=
then
A) 11 3
2 4A
− =
B) 1
11
2
1 1
3 4
A−
=
C) 1
2 1
3 1
2 2
A−
− = −
D) 1A− does not exist.
Q.82) If 2,3,6 are the eigen values of matrix 3 1 1
1 5 1
1 1 3
A
− = − − −
then the eigen values of matrix 3 2A I+
are
A) 20,39,228 B)10,29,218 C) 0,19,208 D) 3,5,3
Q.83)Eigenvalues of matrix
=
32
23A are 5 &1 .what are the eigen values of matrix
2A
A) 1 & 25 B) 2 & 10 C) 6& 4 D) 5 & 1
GENERAL:
Q84)If 1 2 3( , , ,.... )
nD diag d d d d= where 0
id ≠ for all i=1,2,3,….n ,then
1D
− is equal to ,
A)D B) 1 1 1 1
1 2 3( , , ,.... )n
diag d d d d− − − − C) n
I D) None of these
Q85)If 1 2 3( , , ,.... )
nA diag d d d d= then
nA is equal to ,
A) 1 1 1 1
1 2 3( , , ,.... )n n n n
ndiag d d d d
− − − − B)1 2 3( , , ,.... )n n n n
ndiag d d d d
C) A D)None of these.
Q86)If
−
=
981
970
751
A , then Trace of the matrix A is ,
A)17 B) 25 C) 10 D) 63
ANSWERS
Que Ans Que Ans Que Ans Que Ans Que Ans
1 C 21 B 41 B 61 A 81 C
2 C 22 D 42 A 62 D 82 B
3 D 23 B 43 C 63 A 83 A
4 C 24 B 44 A 64 D 84 B
5 B 25 A 45 B 65 D 85 B
6 A 26 B 46 A 66 C 86 A
7 C 27 B 47 C 67 A
8 D 28 C 48 A 68 D
9 C 29 A 49 B 69 C
10 C 30 B 50 C 70 B
11 B 31 C 51 B 71 C
12 C 32 D 52 D 72 B
13 B 33 C 53 C 73 C
14 A 34 B 54 A 74 A
15 C 35 B 55 C 75 B
16 C 36 A 56 A 76 D
17 B 37 D 57 B 77 D
18 A 38 B 58 B 78 A
19 B 39 C 59 A 79 D
20 A 40 B 60 C 80 B
MCQ Of Complex Numbers
Type I: Problems on Basic Definition.
Q.1.What is the value of complex number i135. A) 135 B)i C)-1 D)-i
Q.2.What is the value of complex number i90. A) 1 B)i C)-1 D)-i
Q.3.If Z is purely imaginary number then its real part is ------- A) = 0 B) 0≥ C) 0≤ D) 1
Q.4. If 2 3i+ is a root of the quadratic equation 2 0, , ,x ax b where a b R+ + = ∈
then the values of a and b are respectively A) 4, 7 B) -4,-7 C) -4, 7 D) 4,-7.
Q.5. If z x iy= + and z x iy= − then ( )Re z = ---------
A) 1
( )2
z z+ B) 1
( )2
z zi
+ C) 1
( )2
z z− D) ( )i z z+
Q.6.If 1z and 2z are two complex nos. then 1 2z z+ is
A) 1 2z z+ = 1 2z z+ B) 1 2z z+ 1 2z z≤ +
C) 1 2z z+ 1 2z z≥ + D) None of the above
Q.7. If 1z and 2z are two complex nos. then 21zz is equal to
A) 21 zz + B) 21 zz −
C) 21 zz D) None of these.
Q.8. If 1z and 2z are two complex nos. then ( )1 2 arg z z = ----------
A) 1 2 arg( ) arg( )z z B) 1 2 arg( ) arg( )z z+
C) 1 2 arg( ) arg( )z z− D) None of these
Q.9. If 1z and 2z are two complex nos. then
2
1argz
zis
A) 1 2 arg( ) arg( )z z B) 1 2 arg( ) arg( )z z+
C) 1 2 arg( ) arg( )z z− D) None of these
Q.10.Modulus of complex number z=x+ iy is
A) 22yx + B) 22
yx − C) x
y1tan − D) None of these
Q.11. Argument of complex number z=x+iy is
A) 22yx + B)
y
x1tan − C) x
y1tan − D) None of these
Q.12 The complex numbers z1 and z2 are comparable if A) z1 and z2 are real numbers. B) z1 and z2 are purely imaginary numbers.
C) z1 is real and z2 is imaginary numbers. D) z1 is imaginary and z2 real is numbers. Q.13. The polar form of 1z i= − is ---------
A) 2(cos sin )4 4
z iπ π
= + B) 2(cos sin )4 4
z iπ π
= +
C) 2(cos sin )4 4
z iπ π
= − D) 2( cos sin )4 4
z iπ π
= − −
Q.14.If 0=zz then
A)Re(z)=0 B)Im(z)=0 C) )z=0 D) 0≠z
Q.15.If i is square root of -1 then the value of 2
1
1
i
i
+
− is -------
A) 1 B) -1 C) 2i D) -2i
Q.16 The value of 2 6 3
5 3
i
i
+
+
A) 1 3i+ B) 1 3i− C) 1 3i− + D 1 3i− −
Q.17. If θθ sincos iaaz += then z
z
z
z+ is equal to
A) θ2sin2 B) θ2cos2 C) θ2tan2 D) θsin2 Q.18.What will be modulus and principal argument of -4 A) π2,2 B) π2,4 C) π,4 D) π2,4−
Q.19. What will be modulus and principal argument of 2i
A) π,2 B) π,4 C) 2
,4π D)
2,2π
Q.20.The exponential form of 1-i is
A) 42
π
e B) 42
π−
e C) 4
π
e D) 4
3π−
e
Q.21For any nonzero complex number z, zz argarg + is equal to
A)0 B) 2
π C) π D) ( )zarg2
Q.22.Which of the following is correct? A)1+i > 2+ i B)2+ i > 1+ i C)i > -i D) None of these.
Q.23.The real part of complex number 25
πi
ez+
=
A) 5e B) C)5 D)0
Q.24.The imaginary part of the complex number is
A) B)
C) D)
Q.25. Polar form of complex number is
A) B)
C) D)
Q.26. The argument of is
A) B) C) D)
Q.27. The argument of is
A) B) C) D)
Q.28. If then arg(z) is
A) B)
C) D)
Q.29. By Euler’s Formula the value of ix
e is ---------- A) cos sinx i x− B) cos sinx i x+ C) sin cosx i x+ D) sin cosx i x−
Q.30. The value of ixe is
A) Equal to One B) Equal to Zero C) Greater than zero D) less than zero Q.31. By Euler’s Formula the value of cos x is ----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−+ C)
2
ix ixe e
−− D)
2
ix ixe e
i
−−
Q.32.By Euler’s Formula the value of sinx is ----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−+ C)
2
ix ixe e
−− D)
2
ix ixe e
i
−−
Q.33.If The real part of is A)6 B)-6 C)12 D)-12 Type II Locus Problem
Q.34.If is purely imaginary number then locus of z is
A) B)
C) D)None of these
Q.35. By rotating vector in anticlockwise direction through an angle
We get
A) B) C) D)
Q.36. If 2
z i
z
+
+is purely real then locus of z is ---- --------
A) 2 2 2 0x y x y+ + + = B) 2 2 2 0x y x y+ − − =
C) 2 2 0x y+ + = D) None of these
Q.37.If z is complex number such that then is
equal to
A) B)
C) D)
Q.38.The locus of z satisfying 1z z i+ = − is ------------
A) Straight Line y =-x B) Straight Line y = x C) Circle with center (1, 1) & radius is 1 D) None of these
Q.39. The locus of z satisfying 2 3z − = is ------------
A) Parabola B) hyperbola C) Straight Line D) circle Q.40.The locus of z satisfying 2 3z i+ = is ------------
A) Circle with center (2, 0) & radius is 3 B) Circle with center (0,-2) & radius is 3
C) Circle with center (2, 0) & radius is 3
D) Circle with center (2,2) & radius is 3
Q.41. The locus of is A) Circle with center (1, 0) & radius is 6 B) Circle with center(0,-1) & radius is 3 C) Circle with center (0,1) & radius is6 D) ) Circle with center(0,-1) & radius is 3
Q.42) If 8
Re 0,6
z i
z
− =
+ then z lies on the curve ------
A) 2 2 8 0x y+ − = B) 2 2 6 8 0x y x y+ + − =
C) 4 3 24 0x y− + = D) none of these
Q.43.By rotating the vector in anticlockwise through an
angle we get
A)
B)
C)
D)
TypeIII: Problems on DeMoivers theorems & application.
Q.44. DeMoiver’s theorem states that ( )cos sinn
iθ θ+ = ----------
A) ( )cos sinn i nθ θ+ B) cos sinin n
θ θ +
C) ( )cos sini
n
θ θ+ D) None of these
Q.45. The value of ( ) ( )
( ) ( )
2 5
4 3
cos3 sin 3 cos sin
cos5 sin 5 cos 4 sin 4
i i
i i
θ θ θ θ
θ θ θ θ
− −
+ + is--------
A) ( )cos 43 sin 43iθ θ+ B) ( )cos 4 sin 4iθ θ+
C) ( )cos 43 sin 43iθ θ− D) ( )cos 4 sin 4iθ θ−
Q.46)The roots of are
A)
B)
C)
D) None of these Q.47. All the nth root of unity form a A) arithmetic progression B) geometric progression C)Mean D)None of these
Q.48.Using Demoivre’s Theorem , simplified form of is equal to
A) B) C) D)
Q.49. Using Demoivre’s Theorem , simplified form of is equal to
A) B) C) D)
Q.50. If ,then is
A) B) C) D) None of the above
Q.51. The sum of all nth roots of unity is -------- A) 0 B) 1 C) -1 D) i Q.52. The cube root of unity lies on a circles --------
A) 1 1z + = B) 1 1z − = C) 1z = D) 1 2z − =
Q.53.If 2 2 1x y+ = ,then 1
1
x iy
x iy
+ +
+ −is equal to----
A) x iy+ B) x iy−
C) y ix+ D) y ix−
Q.54.If then
A) B)
C) D)None of these Type IV Problems on Hyperbolic Functions & logarithmic of complex nos. Q.55. Hyperboilic functions sinhx and coshx are respectively A)even and odd B) odd and even C) even and even D)odd and odd Q.56. The sinh x is -----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−− C)
2
x xe e
−+ D)
2
x xe e
−−
Q.57. The cosh x is -----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−− C)
2
x xe e
−+ D)
2
x xe e
−−
Q.58. The tanh x is -----------
A)ix ix
ix ix
e e
e e
−
−
+
− B)
ix ix
ix ix
e e
e e
−
−
−
+ C)
x x
x x
e e
e e
−
−
−
+ D)
( )
x x
x x
e e
i e e
−
−
+
−
Q.59. sin & cosz z are periodic functions of period --------
A)π B) 2π C) 2 iπ D) 1 Q.60. sinh & coshz zare periodic functions of period --------
A)π B) 2π C) 2 iπ D) 1 Q.61. which of the following identity is correct A) 2 2cosh sinh 1x x+ = B) 2 2cosh sinh 1x x− = C) 2 2cosh sinh 1x x= − D) 2 2sinh cosh 1x x− =
Q.62. which of the following identity is correct A) sinh( ) sinh cosh cosh sinhx y x y x y+ = +
B) sinh( ) sinh cosh cosh sinhx y x y x y+ = −
C) sinh( ) sinh cosh cosh sinhx y x y i x y+ = +
D) sinh( ) sinh cosh cosh sinhx y x y ix x y+ = −
Q.63. which of the following identity is correct A) cosh( ) cosh cosh sinh sinhx y x y x y+ = +
B) cosh( ) cosh cosh sinh sinhx y x y x y+ = −
C) cosh( ) cosh cosh sinh sinhx y x y i x y+ = −
D) cosh( ) cosh cosh sinh sinhx y x y i x y+ = +
Q.64. sin iz = ------------ A) cosh z B) sinhi iz C) sinhi z D) coshi z Q.65. 1sinh x
− = ------------
A) ( )2log 1x x+ + B) ( )2log 1x x+ −
C) 1 1
log2 1
x
x
+
− D)
1 1log
2 1
x
x
+
−
Q.66. 1cosh x− =------------
A) ( )2log 1x x+ + B) ( )2log 1x x± −
C) 1 1
log2 1
x
x
+
− D)
1 1log
2 1
x
x
+
−
Q.67. 1tanh x− = ------------
A) ( )2log 1x x+ + B) ( )2log 1x x+ − C) 1 1
log2 1
x
x
+
− D)
1 1log
2 1
x
x
+
−
Q.68. If i
z reθ= be any complex number then principal value of log z is
A) log r iθ+ B) log r iθ− C) 1
log2
r iθ+ D) 1
log2
r iθ−
Q.69.If iz re
θ= be any complex number then general value of value of log z is
----
A)1
log (2 )2
r i nπ θ+ + B) 1
log (2 )2
r i nπ θ+ − C) 1
log2
r iθ+ D) 1
log2
r iθ−
Q.70. The principal value of ( )log 5− is----------
A) log 5 iπ− B) log 5 iπ+ C) log 5 iπ+ D) log 5 iπ−
Q.71. The principal value of ( )log 1 i+ is----------
A) log 24
iπ
− B) log 24
iπ
+ C) 1
log 22 4
iπ
+ D) 1
log 22 4
iπ
−
Q.72. The general value of 3 i+ is----------
A) log 26
iπ
+ B) og 2 (2 )6
L i nπ
π+ + C) og 4 (2 )6
L i nπ
π+ + D) log 46
iπ
+
Q.73. The general value of 1 i+ is----------
A) log 24
iπ
+ B) og 2 (2 )4
L i nπ
π+ + C) og 2 (2 )4
L i nπ
π+ + D) log 24
iπ
+
Q.74. The value of i is----------
A) 4
i
e
π
B) 4
i
e
π−
C) -ie π D)
ie π
Q.75. The value of ( )2i
i is----------
A) eπ B) e
π− C)
2ie π D)
ie π
Q.76.The value of ( )sin log ii is --------
A) 1 B)-1 C) 0 D) none of these
Q.77. If ω is cube root of unity then ( )7
21 ω ω+ − is ---------
A) 128ω B) 128ω− C) 2128ω D) 2128ω− Q.78.If tan(x+iy)=p+iq then tan(x-iy) = A)p-iq B)p+iq C)q-pi D)q+ip
Q.79.The least positive integer n for which is real, is
A)2 B)4 C)1 D)None of these Q.80.The value of ii is
A)1 B)i C) D)None of these
MCQ on Complex Numbers
Type I: Problems on Basic Definition.
Q.1.What is the value of complex number i135. A) 135 B)i C)-1 D)-i
Q.2.What is the value of complex number i90. A) 1 B)i C)-1 D)-i
Q.3.If Z is purely imaginary number then its real part is ------- A) = 0 B) 0≥ C) 0≤ D) 1
Q.4. If 2 3i+ is a root of the quadratic equation 2 0, , ,x ax b where a b R+ + = ∈
then the values of a and b are respectively A) 4, 7 B) -4,-7 C) -4, 7 D) 4,-7.
Q.5. If z x iy= + and z x iy= − then ( )Re z = ---------
A) 1
( )2
z z+ B) 1
( )2
z zi
+ C) 1
( )2
z z− D) ( )i z z+
Q.6.If 1z and 2z are two complex nos. then 1 2z z+ is
A) 1 2z z+ = 1 2z z+ B) 1 2z z+ 1 2z z≤ +
C) 1 2z z+ 1 2z z≥ + D) None of the above
Q.7. If 1z and 2z are two complex nos. then is equal to
A) B)
C) D) None of these.
Q.8. If 1z and 2z are two complex nos. then ( )1 2 arg z z = ----------
A) 1 2 arg( ) arg( )z z B) 1 2 arg( ) arg( )z z+
C) 1 2 arg( ) arg( )z z− D) None of these
Q.9. If 1z and 2z are two complex nos. then is
A) 1 2 arg( ) arg( )z z B) 1 2 arg( ) arg( )z z+
C) 1 2 arg( ) arg( )z z− D) None of these
Q.10.Modulus of complex number z=x+ iy is
A) B) C) D) None of these
Q.11. Argument of complex number z=x+iy is
A) B) C) D) None of these
Q.12 The complex numbers z1 and z2 are comparable if A) z1 and z2 are real numbers. B) z1 and z2 are purely imaginary numbers. C) z1 is real and z2 is imaginary numbers. D) z1 is imaginary and z2 real is numbers. Q.13. The polar form of 1z i= − is ---------
A) 2(cos sin )4 4
z iπ π
= + B) 2(cos sin )4 4
z iπ π
= +
C) 2(cos sin )4 4
z iπ π
= − D) 2( cos sin )4 4
z iπ π
= − −
Q.14.If then
A)Re(z)=0 B)Im(z)=0 C) ) D)
Q.15.If i is square root of -1 then the value of 2
1
1
i
i
+
− is -------
A) 1 B) -1 C) 2i D) -2i
Q.16 The value of 2 6 3
5 3
i
i
+
+
A) 1 3i+ B) 1 3i− C) 1 3i− + D 1 3i− − Q.17. If then is equal to
A) B) C) D) Q.18.What will be modulus and principal argument of -4
A) B) C) D) Q.19. What will be modulus and principal argument of 2i
A) B) C) D)
Q.20.The exponential form of 1-i is
A) B) C) D) Q.21For any nonzero complex number z, is equal to
A)0 B) C) D)
Q.22.Which of the following is correct?
A) B)
C) D) None of these.
Q.23.The real part of complex number
A) B) C)5 D)0
Q.24.The imaginary part of the complex number is
A) B)
C) D)
Q.25. Polar form of complex number is
A) B)
C) D)
Q.26. The argument of is
A) B) C) D)
Q.27. The argument of is
A) B) C) D)
Q.28. If then arg(z) is
A) B)
C) D)
Q.29. By Euler’s Formula the value of ix
e is ---------- A) cos sinx i x− B) cos sinx i x+ C) sin cosx i x+ D) sin cosx i x−
Q.30. The value of ixe is
A) Equal to One B) Equal to Zero C) Greater than zero D) less than zero Q.31. By Euler’s Formula the value of cos x is ----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−+ C)
2
ix ixe e
−− D)
2
ix ixe e
i
−−
Q.32.By Euler’s Formula the value of sinx is ----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−+ C)
2
ix ixe e
−− D)
2
ix ixe e
i
−−
Q.33.If The real part of is A)6 B)-6 C)12 D)-12 Type II Locus Problem
Q.34.If is purely imaginary number then locus of z is
A) B)
C) D)None of these
Q.35. By rotating vector in anticlockwise direction through an angle
We get
A) B) C) D)
Q.36. If 2
z i
z
+
+is purely real then locus of z is ---- --------
A) 2 2 2 0x y x y+ + + = B) 2 2 2 0x y x y+ − − =
C) 2 2 0x y+ + = D) None of these
Q.37.If z is complex number such that then is
equal to
A) B)
C) D)
Q.38.The locus of z satisfying 1z z i+ = − is ------------
A) Straight Line y =-x B) Straight Line y = x C) Circle with center (1, 1) & radius is 1 D) None of these Q.39. The locus of z satisfying 2 3z − = is ------------
A) Parabola B) hyperbola C) Straight Line D) circle
Q.40.The locus of z satisfying 2 3z i+ = is ------------
A) Circle with center (2, 0) & radius is 3 B) Circle with center (0,-2) & radius is 3
C) Circle with center (2, 0) & radius is 3
D) Circle with center (2,2) & radius is 3
Q.41. The locus of is A) Circle with center (1, 0) & radius is 6 B) Circle with center(0,-1) & radius is 3 C) Circle with center (0,1) & radius is6 D) Circle with center(0,-1) & radius is 3
Q.42) If 8
Re 0,6
z i
z
− =
+ then z lies on the curve ------
A) 2 2 8 0x y+ − = B) 2 2 6 8 0x y x y+ + − =
C) 4 3 24 0x y− + = D) none of these
Q.43.By rotating the vector in anticlockwise through an
angle we get
E)
F)
G)
H)
TypeIII: Problems on DeMoivers theorems & application.
Q.44. DeMoiver’s theorem states that ( )cos sinn
iθ θ+ = ----------
A) ( )cos sinn i nθ θ+ B) cos sinin n
θ θ +
C) ( )cos sini
n
θ θ+ D) None of these
Q.45. The value of ( ) ( )
( ) ( )
2 5
4 3
cos3 sin 3 cos sin
cos5 sin 5 cos 4 sin 4
i i
i i
θ θ θ θ
θ θ θ θ
− −
+ + is--------
A) ( )cos 43 sin 43iθ θ+ B) ( )cos 4 sin 4iθ θ+
C) ( )cos 43 sin 43iθ θ− D) ( )cos 4 sin 4iθ θ−
Q.46)The roots of are
A)
B)
C)
D) None of these Q.47. All the nth root of unity form a A) arithmetic progression B) geometric progression C)Mean D)None of these
Q.48.Using Demoivre’s Theorem , simplified form of is equal to
A) B) C) D)
Q.49. Using Demoivre’s Theorem , simplified form of is equal to
A) B) C) D)
Q.50. If ,then is
A) B) C) D) None of the above Q.51. The sum of all nth roots of unity is -------- A) 0 B) 1 C) -1 D) i Q.52. The cube root of unity lies on a circles --------
A) 1 1z + = B) 1 1z − = C) 1z = D) 1 2z − =
Q.53.If 2 2 1x y+ = ,then 1
1
x iy
x iy
+ +
+ −is equal to----
A) x iy+ B) x iy−
C) y ix+ D) y ix−
Q.54.If then
A) B)
C) D)None of these Type IV Problems on Hyperbolic Functions & logarithmic of complex nos. Q.55. Hyperboilic functions sinhx and coshx are respectively A)even and odd B) odd and even C) even and even D)odd and odd Q.56. The sinh x is -----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−− C)
2
x xe e
−+ D)
2
x xe e
−−
Q.57. The cosh x is -----------
A)2
ix ixe e
−+ B)
2
ix ixe e
i
−− C)
2
x xe e
−+ D)
2
x xe e
−−
Q.58. The tanh x is -----------
A)ix ix
ix ix
e e
e e
−
−
+
− B)
ix ix
ix ix
e e
e e
−
−
−
+ C)
x x
x x
e e
e e
−
−
−
+ D)
( )
x x
x x
e e
i e e
−
−
+
−
Q.59. sin & cosz z are periodic functions of period --------
A)π B) 2π C) 2 iπ D) 1 Q.60. sinh & coshz zare periodic functions of period --------
A)π B) 2π C) 2 iπ D) 1 Q.61. which of the following identity is correct A) 2 2cosh sinh 1x x+ = B) 2 2cosh sinh 1x x− = C) 2 2cosh sinh 1x x= − D) 2 2sinh cosh 1x x− = Q.62. which of the following identity is correct A) sinh( ) sinh cosh cosh sinhx y x y x y+ = +
B) sinh( ) sinh cosh cosh sinhx y x y x y+ = −
C) sinh( ) sinh cosh cosh sinhx y x y i x y+ = +
D) sinh( ) sinh cosh cosh sinhx y x y ix x y+ = −
Q.63. which of the following identity is correct
A) cosh( ) cosh cosh sinh sinhx y x y x y+ = +
B) cosh( ) cosh cosh sinh sinhx y x y x y+ = −
C) cosh( ) cosh cosh sinh sinhx y x y i x y+ = −
D) cosh( ) cosh cosh sinh sinhx y x y i x y+ = +
Q.64. sin iz = ------------ A) cosh z B) sinhi iz C) sinhi z D) coshi z Q.65. 1sinh x
− = ------------
A) ( )2log 1x x+ + B) ( )2log 1x x+ −
C) 1 1
log2 1
x
x
+
− D)
1 1log
2 1
x
x
+
−
Q.66. 1cosh x− =------------
A) ( )2log 1x x+ + B) ( )2log 1x x± −
C) 1 1
log2 1
x
x
+
− D)
1 1log
2 1
x
x
+
−
Q.67. 1tanh x− = ------------
A) ( )2log 1x x+ + B) ( )2log 1x x+ − C) 1 1
log2 1
x
x
+
− D)
1 1log
2 1
x
x
+
−
Q.68. If iz re θ= be any complex number then principal value of log z is
A) log r iθ+ B) log r iθ− C) 1
log2
r iθ+ D) 1
log2
r iθ−
Q.69.If iz re θ= be any complex number then general value of value of log z is
----
A)1
log (2 )2
r i nπ θ+ + B) 1
log (2 )2
r i nπ θ+ − C) 1
log2
r iθ+ D) 1
log2
r iθ−
Q.70. The principal value of ( )log 5− is----------
A) log 5 iπ− B) log 5 iπ+ C) log 5 iπ+ D) log 5 iπ−
Q.71. The principal value of ( )log 1 i+ is----------
A) log 24
iπ
− B) log 24
iπ
+ C) 1
log 22 4
iπ
+ D) 1
log 22 4
iπ
−
Q.72. The general value of 3 i+ is----------
A) log 26
iπ
+ B) og 2 (2 )6
L i nπ
π+ + C) og 4 (2 )6
L i nπ
π+ + D) log 46
iπ
+
Q.73. The general value of 1 i+ is----------
A) log 24
iπ
+ B) og 2 (2 )4
L i nπ
π+ + C) og 2 (2 )4
L i nπ
π+ + D) log 24
iπ
+
Q.74. The value of i is----------
A) 4
i
e
π
B) 4
i
e
π−
C) -ie π D)
ie π
Q.75. The value of ( )2i
i is----------
A) eπ B) e
π− C)
2ie π D)
ie π
Q.76.The value of ( )sin log ii is --------
A) 1 B)-1 C) 0 D) none of these
Q.77. If ω is cube root of unity then ( )7
21 ω ω+ − is ---------
A) 128ω B) 128ω− C) 2128ω D) 2128ω− Q.78.If tan(x+iy)=p+iq then tan(x-iy) = A)p-iq B)p+iq C)q-pi D)q+ip
Q.79.The least positive integer n for which is real, is
A)2 B)4 C)1 D)None of these Q.80.The value of ii is
A)1 B)i C) D)None of these
Answers 1) D 2)C 3)A 4)A 5)A
6) B 7) C 8) B 9)C 10)A 11) B 12) A 13)C 14)C 15)B
16) A 17)B 18)C 19)D 20)B 21) A 22)D 23)D 24) A 25)A 26) D 27)A 28)A 29)B 30)A 31)A 32)C 33)D 34)A 35)C 41)C 42) B 43)C 44)A 45)C
46)C 47) B 48) C 49)B 50)C 51)A 52) C 53)A 54)A 55)B 56) 57) C 58)C 59)B 60)C
61) B 62)A 63)A 64)B 65)A 66) B 67)D 68)C 69)A 70)B
71) C 72) B 73) C 74) A 75)A
76)B 77)D 78) A 79)A 80)C