9387.Matrices Mcq

MCQ QUESTION BANK ON MATRICES Type I Rank and Normal Form Q.1)Which of the following matrix is in normal form? 1 2 5 A) 0 1 9     0 0 5

B)

Q.2) Echelon form of matrix

A)

1 4   7

1 4 7

1 4 7

1 0  0  0

1 4 7

1 4  7 

4

3

0

1

1

3

0

0

1 1 1 5 5 5   8 8 8 0 B)  4   7

1 4  8  1

1 5 5  8 8  0 0 1

4 7

1 0 C)   0 1

D)

1 1 

0 0

0 0 

is

4 7

0 4 7

0 4  7 

1

1 0 0

C)  0   0

1 0 0

1 0 0

1 0  0 

1

D)  6   9

1 6 9

1 6 9

1 6 9

1 6  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. 1 2 3 

Q.4) The rank of matrix A =  2 2 2  is equal to    3 3 3 

A) 4

B) 3

C) 2

D) 1

1 2 3 Q5) If A =  3 4 5  and det(A)=0 then rank of a matrix A is    4 5 6 

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?

1 0 0  0 0 0  1 1 1  B) 0 1 0  C) 0 1 0  A) 0 1 0  0 0 0  0 0 1  1 1 1  Q.7) Which of the following matrix is in the Normal form? B) 1 0 0 0 0 1 0 1   0 0 0 1

A) 1 1 0 0 0 1 0 0   0 0 1 0

C)  10

 0  0

0 0 0  D) 0 0 0  0 1 0 

0

0

1 0

0 1

0

0

10 10 10 10 10 

Q.8) The rank of matrix 10 10 10 10 10 is   10 10 10 10 10 

A) 10 B) 5 Q.9)The rank of the matrix 1 1

1  1

A)0

b) 1

c) 2

C)2 1  is ,

−1 0   1 1 

d) 3

D)1.

0 0  0  0

1 1 1

D) 0 1 1   0 0 1

Q.10) For matrix A of order mxn, the rank r of matrix A is B) r ≥ max{m, n} A) r ≥ min{m, n} C) r ≤ min{m, n} D) r ≤ max{m, n} Q.11)For non singular matrix A If PAQ is in normal form then A)PQ

B) QP

C) P+Q

A−1 is equal to

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  2 3 4

Q.13) The rank of the following matrix by determinant method 4 3 1 is   1 2 4

A)2

B) 3

C) 1

D) 0 3

P

P

 P

P

3 

Q.14)If P=3 then the rank of matrix A =  P 3 P  .   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) ρ ([ A / B]) B) ρ ( A) ≠ ρ ([ A / B ]) C) ρ ( A) < ρ ([ A / B ]) D) ρ ( A) = ρ ([ A / B]) 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) ρ ( A) = ρ ([ A / B]) = m B) ρ ( A) = ρ ([ A / B ]) = n D) ρ ( A) = ρ ([ A / B]) < m C) ρ ( A) = ρ ([ A / B]) 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) ρ ( A) = ρ ([ A / B]) = m B) ρ ( A) = ρ ([ A / B ]) < n D) ρ ( A) = ρ ([ A / B]) > m C) ρ ( A) = ρ ([ A / B]) < m 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 ρ ( A) ≠ ρ ( A / B ) 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

1 b −5 is an orthogonal? 13 5 b  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. T −1 Q.33) If A = A then A is ,

Q.31) For what value of b the matrix A =

A)Symmetric B) Skew Symmetric Q.34)If A is an Orthogonal matrix then A) A

B)

AT

C)

A2

C) Orthogonal is equal to ,

D)

− AT

D) None of these

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( A − λ I ) = 0 B) ( A − λ I ) = 0 C) det( A − λ A) = 0 D) ( A − λ A) = 0 2 Q.37) If eigen values of matrix A are 1,2,3 then eigen values of matrix 2A are A)-1,-2,-3 B)1,2,3 C)2,4,9 D) 2,8,18 1 1 1

Q.38) The characteristics roots of the matrix 1 1 1 are   1 1 1

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 2 − 3 4 − 2 A)2

B)4

C) 0

D) 1

 2 − 3

Q.40)Find product of eigenvalues of matrix    4 − 2 A)4 B)8

C) 6 6

Q.41)The product of two eigen values of the matrix A = − 2   2

A)1

B)2

D) 2

2 3 − 1 is 16. Find the third eigenvalue. − 1 3 

−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

2 0 1 Q.43) If 2 & 3 are eigenvalues of A = 0 2 0 find the third eigenvalue 1 0 2 A)2 B)3 C)1  2 0 1

Q.44) If 1, 2 & 3 are the eigen values of A = 0 2 0 , find the value of a?   a 0 2

A)1

B)0

C) 2

D) 3 14 − 10 is , − 1 

Q.45) The characteristic equation of the matrix A =  5 A)

λ 2 + 5λ + 21 = 0

B)

λ 2 − 13λ + 36 = 0

C)

λ 2 + 13λ + 36 = 0

D)

λ 2 + 13λ − 36 = 0

14 − 10 Q.46) Find the eigen values of the matrix A =    5 −1  A) λ = 4, λ = 9 B) λ = 5, λ = 6 1

C)

2

λ1 = 18, λ2 = 2

1

D)

2

λ1 = 10, λ2 = 3

D) 4

2 2 1  Q.47) Two eigen values of the matrix A = 1 3 1  are 1 &1, find the 3rd eigenvalue of A.   1 2 2

A)1

B)3

C) 5

D)4

2 2 1

Q.48)Two eigenvalues of the matrix A = 1 3 1 are 1 & 1 find the eigenvalues of  

A −1

1 2 2

A)1/1 , 1/1 , 1/5

B)½ , 1 , 5

Q.49) Form the matrix whose eigenvalues are α

C)½ , ½ , 5

D)1 , 1 , 5

− 5, β − 5, γ − 5 whereα , β , γ

 − 1 − 2 − 3 the eigenvalues of are A =  4 5 − 6    7 − 8 9 

 − 1 − 2 − 3 A)  4 5 − 6  7 − 8 9 

 − 6 − 2 − 3  − 1 2 − 3 B)  4 0 − 6 C) − 4 5 6   7 − 8 4   7 8 9 

 4 − 2 − 3 D) 4 10 − 6 7 − 8 14 

Q.50) If the characteristic equation of one matrix is λ3 − 4λ2 − λ + 4 = 0 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 X 3 , 2 and 3 are the eigenvalues. Find its 3rd eigenvalue. A)1 B) 0 C) 2 D) 3 1

− 1 0

Q.52) What is the characteristic equation of the matrix − 1 2 1    0

3

2

A) λ − 4λ +3λ = 0

3

2

B) λ − 4λ + 3λ +1= 0

1

1

3

C) λ + 4λ2 + 3λ = 0

D) λ3 − 4λ2 −3λ = 0

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 1 B) −λ C) D)1. A) λ

λ

Q.56) If λ is eigen value of matrix A then eigen values of matrix kA is 1 A) k λ B) −λ C) D) λ . kλ Q.57) If λ is eigen value of matrix A then eigen values of matrix A+kI is 1 A) k λ B) λ + k C) D) kλ Q.58) If λ is eigen value of matrix A then eigen values of matrix An is

λ −k .

B) λ n

A) n λ

n

C)

D) λ .

λ

Q.59) If λ1 , λ2 , λ3 are eigen values of matrix A then eigen values of A−1 are 1 1 1 B) λ1 , λ2 , λ3 C) λ12 , λ2 2 , λ32 A) , ,

λ1 λ2 λ3

D) −λ1 , −λ2 , −λ3

Q.60) The sum & product of the eigen values of matrix  2 −3 are 4 −2 

A)0,0

B)4,8

Q.61) For the matrix A)-8

C)0,8



D) 2,-2

1 − 2 3  0 − 2 5 product of the eigen values is   0 0 4

B) 4

C)1

D) -2

Q. 62) The eigen values of the matrix 1 4 are 2 3 A) 2,3

B)4,5

C) 0,2

D) 5,-1

3 1 1 Q.63) The Characteristic equation of the matrix  −1 5 −1 is    1 −1 3 

A) λ 3 − 11λ 2 + 38λ − 40 = 0 B) λ 3 − 11λ 2 + 38λ + 40 = 0 C) λ 3 + 11λ 2 + 38λ + 40 = 0 D) λ 3 + 11λ 2 + 38λ + 40 = 0 Q.64) If the Characteristic equation of the matrix A of order 3x3 is λ 3 − 3λ 2 + 3λ − 1 = 0 then by Cayley Hamilton Theorem A−1 is equal to A) A3 − 3 A2 + 3 A − I B) A2 − 3 A − 3I C) 3 A2 − 3 A − I D) A2 − 3 A − 3I Q.65) If λ 2 − S1λ + S2 = 0 is a characteristic equation of 2x2 matrix A then A) S1 = Sum of principle diagonal elements, S 2 = Sum of all elements B) S1 = Sum of principle diagonal elements, S 2 = Product of principle diagonal elements C) S1 = Trace of matrix A, S 2 = Product of principle diagonal elements D) S1 = Trace of matrix A, S 2 = Product of Eigen values of matrix A. Q.66) If λ 3 − S1λ 2 + S 2λ − S3 = 0 is a characteristic equation of 3x3 matrix A then A)

S1 = Sum of principle diagonal elements, S 2 = Sum of all elements, S3 = A

B)

S1 = Sum of principle diagonal elements, S 2 = Product of principle diagonal elements,

S3 = A C) S1 = Trace of matrix A, S 2 = sum of minors of Principle diagonal elements, S3 = A D) S1 = Trace of matrix A, S 2 = Product of Eigen values of matrix A, S3 = A Q.67) The characteristic equation of matrix 14 −10 is 5 −1 

2

A) λ − 13λ + 36 = 0

2

B) λ − 13λ − 36 = 0



C) λ 2 − 4λ − 64 = 0

D) λ 2 − λ + 36 = 0

4

6 3 2  −1 −4 −3

Q.68) The characteristic equation of matrix  1 

6

is

A) λ 3 − 4λ 2 + λ − 4 = 0 B) λ 3 + 4λ 2 − λ + 4 = 0 C) λ 3 − λ 2 + λ − 4 = 0 D) λ 3 − 4λ 2 − λ + 4 = 0 Q.69) If all eigen values of matrix A3 x 3 are distinct then which of the following is true A) Matrix A3 x 3 has three equal Eigen vectors B) Matrix A3 x 3 has three distinct eigen vectors C) Matrix A3 x 3 has three distinct linearly independent eigen vectors D) ) Matrix A3 x 3 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  0 −1  0 0  0 0

4 3  5 3  0 −4 

3 2

A)-1,1,-5,4 B)1,-1,5,-4 Q.73) The eigenvalues of matrix 1 1 1 are

are

C)0,0,0,0

D) -1,-1,-5,-4

1 1 1 1 1 1

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 2 Q.76)If 1 0  then value of k for which A = 8 A + kI is A=  1 7  A) 5 B)7 C)-5 D)-7  Q.77) If A =    2 −1 A) A8 = 5 I 1

2

then A8 is B) A8 = 25 I

C) A8 = 65 I

D) A8 = 625I

4 6 6 Q.78) Two eigen values of A =  1 3 2  are equal and are double the third then the eigen values    −1 −5 −2 

A2 are

A) 1,4,4

B) 3,2,1

C) 4,9,16

D) 25,4,9

1 2 −3 3 2 2  then the eigen values of 3 A + 5 A − 6 A + 2 I are 0 0 −2 

Q.79) If A = 0 3  A)5,-6,2

B)1,3,-2

C) 3,5,6

D) 4,110,10  −1

Q.80) Sum and product of the eigen values of matrix A =  1   1

A) -3,-1 1 2  Q.81) If A =   then 3 4  1 3 A) A =   2 4 −1

B)-3,4

1 1 −1 1  is 1 −1

C) 4,3

 1 B) A−1 =  1  3

D)1,-3

 −2 C) A =  3  2

1 2  1 4 

−1

3

1 −1  2

D) A−1 does not exist.

1 −1 then the eigen values of matrix −1 3 

−1

Q.82) If 2,3,6 are the eigen values of matrix A =  −1 5   1

A3 + 2 I

are A) 20,39,228

B)10,29,218

C) 0,19,208

D) 3,5,3

3 2

Q.83)Eigenvalues of matrix A =  are 5 &1 .what are the eigen values of matrix A2  2 3 A) 1 & 25

B) 2 & 10

C) 6& 4

D) 5 & 1

GENERAL: Q84)If D = diag (d , d , d ,....d ) where d ≠ 0 for all i=1,2,3,….n ,then D −1 is equal to , 1 2 3 n i B) diag (d −1 , d −1 , d −1 ,....d −1 ) 1 2 3 n

A)D

C)

In

D) None of these

Q85)If A = diag (d , d , d ,....d ) then An is equal to , 1 2 3 n n −1 1

A) diag (d

, d 2 n −1 , d3n−1 ,....d n n −1 )

C) A

B) diag (d n , d n , d n ,....d n ) 1 2 3 n D)None of these.

1 − 5 7  Q86)If A = 0 7 9 , then Trace of the matrix A is ,   1 8 9 

A)17

B) 25

C) 10

D) 63

ANSWERS Que 1 2 3

Ans C C D

Que 21 22 23

Ans B D B

Que 41 42 43

Ans B A C

Que 61 62 63

Ans A D A

Que 81 82 83

Ans C B A

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

C B A C D C C B C B A C C B A B A

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

B A B B C A B C D C B B A D B C B

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

A B A C A B C B D C A C A B B A C

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

D D C A D C B C B C A B D D A D 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

84 85 86

B B A

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 ------C) ≤ 0 D) 1 A) = 0 B) ≥ 0 Q.4. If 2 + i 3 is a root of the quadratic equation x 2 + ax + b = 0, 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.

z = x + iy and

Q.5. If A)

1 ( z + z) 2

z = x − iy then Re ( z ) = ---------

B)

1 ( z + z) 2i

C)

1 ( z − z) 2

D) i ( z + z )

Q.6.If z1 and z2 are two complex nos. then z1 + z2 is A) z1 + z2 = z1 + z2 B) z1 + z2 ≤ z1 + z2 C) z1 + z2 ≥ z1 + z2 D) None of the above Q.7. If z1 and z2 are two complex nos. then z1 z 2 B) z1 − z 2 A) z1 + z 2 C) z1 z 2 D) None of these.

is equal to

Q.8. If z1 and z2 are two complex nos. then arg ( z1 z2 ) = ---------A) arg( z1 ) arg( z2 ) B) arg( z1 ) + arg( z2 ) C) arg( z1 ) − arg( z2 ) D) None of these z 

Q.9. If z1 and z2 are two complex nos. then arg 1  is  z2  B) arg( z1 ) + arg( z2 ) A) arg( z1 ) arg( z2 ) C) arg( z1 ) − arg( z2 ) D) None of these

Q.10.Modulus of complex number z=x+ iy is A) x 2 + y 2

B)

x2 − y2

C) tan −1

y x

D) None of these

Q.11. Argument of complex number z=x+iy is A) x 2 + y 2

B) tan −1

x y

C) tan −1

y x

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 z =1 − i is --------π

π

4

4

π

π

4

4

A) z = 2(cos + i sin )

B)

C) z = 2(cos − i sin )

z = 2(cos

π

π

+ i sin ) 4 4

π

π

4

4

D) z = 2(− cos − i sin )

Q.14.If z z = 0 then A)Re(z)=0 B)Im(z)=0

C) )z=0 D) z ≠ 0 2

1+ i    is ------ 1− i 

Q.15.If i is square root of -1 then the value of A) 1

B) -1

Q.16 The value of A) 1 + i 3

C) 2i

D) -2i

2 + i6 3 5+i 3 B) 1 − i 3

C)

D −1 − i 3

−1 + i 3

z z + is equal to z z B) 2 cos 2θ C) 2 tan 2θ

Q.17. If z = a cos θ + ia sin θ then A) 2 sin 2θ

D) 2 sin θ

Q.18.What will be modulus and principal argument of -4 A) 2, 2π B) 4, 2π C) 4, π D) − 4, 2π Q.19. What will be modulus and principal argument of 2i A) 2, π

B) 4, π

C) 4,

π 2

D) 2,

π 2

Q.20.The exponential form of 1-i is π

A) 2e

4

B)

2e

−π 4

π

C) e

4

D) e

−3π 4

Q.21For any nonzero complex number z, arg z + arg z is equal to A)0

B)

π 2

C) π

D) 2 arg(z )

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 z = e A) e 5 B) C)5

5+ i

π 2

Q.24.The imaginary part of the complex number A) B)

D)0 is

C)

D)

Q.25. Polar form of complex number A)

B)

C)

D)

Q.26. The argument of B) A) Q.27. The argument of A) B) Q.28. If A)

is

is C)

D) is

C) D) then arg(z) is B)

C)

D)

Q.29. By Euler’s Formula the value of eix is ---------A) cos x − i sin x B) cos x + i sin x C) sin x + i cos x D) sin x − i cos x Q.30. The value of eix is A) Equal to One C) Greater than zero

B) Equal to Zero D) less than zero

Q.31. By Euler’s Formula the value of cos x is ---------eix + e − ix A) 2

eix + e − ix B) 2i

eix − e − ix C) 2

eix − e − ix D) 2i

Q.32.By Euler’s Formula the value of sinx is ---------A)

eix + e − ix 2

Q.33.If A)6

B)

eix + e − ix 2i

The real part of B)-6 C)12

C)

eix − e − ix 2

D)

eix − e − ix 2i

is 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)

z +i is purely real then locus of z is ----------z+2 A) x 2 + y 2 + 2 x + y = 0 B) x 2 + y 2 − 2 x − y = 0

Q.36. If

D) None of these

C) x + 2 y + 2 = 0

Q.37.If z is complex number such that equal to A)

B)

C)

D)

then

is

Q.38.The locus of z satisfying z + 1 = 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 z − 2 = 3 is -----------A) Parabola B) hyperbola C) Straight Line D) circle Q.40.The locus of z satisfying z + 2i = 3 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 Re   = 0, then z lies on the curve -----z + 6   2 2 A) x + y − 8 = 0 B) x 2 + y 2 + 6 x − 8 y = 0 C) 4 x − 3 y + 24 = 0 D) none of these Q.43.By rotating the vector in anticlockwise through an angle we get z − 8i

A) B)

C) D)

TypeIII: Problems on DeMoivers theorems & application. n

Q.44. DeMoiver’s theorem states that ( cos θ + i sin θ ) = ---------θ θ B)  cos + i sin 

A) ( cos nθ + i sin nθ ) C)



( cos θ + i sin θ )

n

n

D) None of these

n 2

5

( cos 3θ − i sin 3θ ) ( cos θ − i sin θ ) is-------Q.45. The value of 4 3 ( cos 5θ + i sin 5θ ) ( cos 4θ + i sin 4θ ) A) ( cos 43θ + i sin 43θ ) B) ( cos 4θ + i sin 4θ ) C) ( cos 43θ − i sin 43θ ) D) ( cos 4θ − i sin 4θ ) 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 equal to A) B) C) D) Q.50. If A)

,then B)

is C)

D) None of the above

is

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) z + 1 = 1 B) z − 1 = 1 C) z = 1 D) z − 1 = 2 Q.53.If x 2 + y 2 = 1 ,then A) x + iy C) y + ix

1 + x + iy is equal to---1 + x − iy B) x − iy

D) y − ix then

Q.54.If

B) D)None of these

A) C)

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)

eix + e − ix 2

B)

eix − e − ix 2i

C)

e x + e− x 2

D)

e x − e− x 2

C)

e x + e− x 2

D)

e x − e− x 2

Q.57. The cosh x is ----------A)

eix + e − ix 2

B)

eix − e − ix 2i

Q.58. The tanh x is ----------e x + e− x eix + e − ix eix − e − ix e x − e− x B) C) D) eix − e − ix eix + e − ix e x + e− x i (e x − e − x ) Q.59. sin z & cos z are periodic functions of period --------

A)

A) π

B) 2π

C) 2π i

D) 1

Q.60. sinh z & cosh z are periodic functions of period -------A) π

B) 2π

C) 2π i

D) 1

Q.61. which of the following identity is correct A) cosh 2 x + sinh 2 x = 1 B) cosh 2 x − sinh 2 x = 1 C) cosh 2 x = sinh 2 x − 1 D) sinh 2 x − cosh 2 x = 1

Q.62. which of the following identity is correct A) sinh( x + y ) = sinh x cosh y + cosh x sinh y B) sinh( x + y ) = sinh x cosh y − cosh x sinh y C) sinh( x + y ) = sinh x cosh y + i cosh x sinh y D) sinh( x + y ) = sinh x cosh y − ix cosh x sinh y Q.63. which of the following identity is correct A) cosh( x + y ) = cosh x cosh y + sinh x sinh y B) cosh( x + y ) = cosh x cosh y − sinh x sinh y C) cosh( x + y ) = cosh x cosh y − i sinh x sinh y D) cosh( x + y ) = cosh x cosh y + i sinh x sinh y Q.64. sin iz = -----------A) co sh z B) i sin h iz

C) i sin h z

Q.65. sinh −1 x = -----------A) log x + x 2 + 1

log x + x 2 − 1

)

(

(

B)

D) i c o sh z

)

1 1 1+ x   x +1  D) log  log    2 2  x −1  1− x  Q.66. cosh −1 x = ------------

C)

(

A) log x + x 2 + 1

)

1  x +1  log   2  x −1  Q.67. tanh −1 x = ------------

(

)

log x ± x 2 − 1

)

 D) log   2 1− x  1

C)

A) log x + x 2 + 1

(

B)

1+ x

(

B)

log x + x 2 − 1

)

C)

1  x +1  log   2  x −1 

 D) log   2 1− x  1

1+ x

Q.68. If z = reiθ be any complex number then principal value of log z is A) log r + iθ

B) log r − iθ

C)

1 log r + iθ 2

D)

1 log r − iθ 2

Q.69.If z = reiθ be any complex number then general value of value of log z is ---1 2

A) log r + i (2nπ + θ ) B)

1 log r + i (2nπ − θ ) 2

C)

1 log r + iθ 2

Q.70. The principal value of log ( −5) is---------A) log 5 − iπ B) log 5 + iπ C) log 5 + iπ Q.71. The principal value of log (1 + i ) is---------A) log 2 − i

π 4

B) log 2 + i

Q.72. The general value of

π

C)

1 π log 2 + i 2 4

4 3 + i is----------

D)

1 log r − iθ 2

D) log 5 − iπ D)

1 π log 2 − i 2 4

A) log 2 + i

π

π

B) L og 2 + i(2nπ + )

6

π

π

6

6

C) L og 4 + i(2nπ + ) D) log 4 + i

6

Q.73. The general value of 1 + i is---------A) log 2 + i

π

π

B) L og 2 + i(2nπ + )

4

π

π

4

4

C) L og 2 + i (2nπ + ) D) log 2 + i

4

Q.74. The value of i is---------A) e

iπ 4

B) e

− iπ 4

C) - e

iπ

iπ D) e

2i

Q.75. The value of ( i ) is---------π

A) e

−π B) e

2iπ C) e

Q.76.The value of sin ( log i i ) is -------A) 1 B)-1 C) 0

iπ D) e

D) none of these 7

Q.77. If ω is cube root of unity then (1 + ω − ω 2 ) is --------B) −128ω C) 128ω 2 D) −128ω 2 A) 128ω Q.78.If tan(x+iy)=p+iq then tan(x-iy) = A)p-iq B)p+iq C)q-pi D)q+ip is real, is Q.79.The least positive integer n for which 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 Q.2.What is the value of complex number i90. A) 1 B)i C)-1

D)-i

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 + i 3 is a root of the quadratic equation x 2 + ax + b = 0, 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 A)

z = x + iy and 1 ( z + z) 2

z = x − iy then Re ( z ) = ---------

1 ( z + z) 2i

B)

C)

1 ( z − z) 2

D) i ( z + z )

Q.6.If z1 and z2 are two complex nos. then z1 + z2 is A) z1 + z2 = z1 + z2 B) z1 + z2 ≤ z1 + z2 C) z1 + z2 ≥ z1 + z2 D) None of the above Q.7. If z1 and z2 are two complex nos. then A) B) C) D) None of these.

is equal to

Q.8. If z1 and z2 are two complex nos. then arg ( z1 z2 ) = ---------B) arg( z1 ) + arg( z2 ) A) arg( z1 ) arg( z2 ) C) arg( z1 ) − arg( z2 ) D) None of these Q.9. If z1 and z2 are two complex nos. then A) arg( z1 ) arg( z2 ) C) arg( z1 ) − arg( z2 )

is

B) arg( z1 ) + arg( z2 ) D) None of these

Q.10.Modulus of complex number z=x+ iy is A) B) C) Q.11. Argument of complex number z=x+iy is A) B) C)

D) None of these 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 z =1 − i is --------π

π

4

4

A) z = 2(cos + i sin )

B)

z = 2(cos

π

π

+ i sin ) 4 4

π

π

4

4

C) z = 2(cos − i sin )

π

π

4

4

D) z = 2(− cos − i sin )

Q.14.If then A)Re(z)=0 B)Im(z)=0

C) )

D) 2

1+ i    is ------ 1− i 

Q.15.If i is square root of -1 then the value of A) 1

B) -1

Q.16 The value of A) 1 + i 3 Q.17. If A)

C) 2i

D) -2i

2 + i6 3 5+i 3 B) 1 − i 3

C) −1 + i 3 is equal to C)

then B)

D −1 − i 3 D)

Q.18.What will be modulus and principal argument of -4 B) C) D) A) 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 A) B) C) D) Q.25. Polar form of complex number A)

B)

C)

D)

Q.26. The argument of A) B)

is

is C)

D)

is

Q.27. The argument of A) B) Q.28. If A)

is C) D) then arg(z) is B)

C)

D)

Q.29. By Euler’s Formula the value of eix is ---------A) cos x − i sin x B) cos x + i sin x C) sin x + i cos x D) sin x − i cos x Q.30. The value of eix is A) Equal to One C) Greater than zero

B) Equal to Zero D) less than zero

Q.31. By Euler’s Formula the value of cos x is ---------A)

eix + e − ix 2

B)

eix + e − ix 2i

C)

eix − e − ix 2

D)

eix − e − ix 2i

Q.32.By Euler’s Formula the value of sinx is ---------eix + e − ix A) 2

Q.33.If A)6

eix + e − ix B) 2i

The real part of B)-6 C)12

eix − e − ix C) 2

eix − e − ix D) 2i

is D)-12

Type II Locus Problem is purely imaginary number then locus of z is Q.34.If A) B) D)None of these C) Q.35. By rotating vector in anticlockwise direction through an angle We get A)

B)

C)

D)

z +i is purely real then locus of z is ----------z+2 A) x 2 + y 2 + 2 x + y = 0 B) x 2 + y 2 − 2 x − y = 0

Q.36. If

C) x + 2 y + 2 = 0

D) None of these

Q.37.If z is complex number such that equal to

then

is

A)

B)

C)

D)

Q.38.The locus of z satisfying z + 1 = 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 z − 2 = 3 is -----------A) Parabola B) hyperbola C) Straight Line D) circle Q.40.The locus of z satisfying z + 2i = 3 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 Re 

z − 8i   = 0, then z lies on the curve ----- z+6  A) x 2 + y 2 − 8 = 0 B) x 2 + y 2 + 6 x − 8 y = 0

C) 4 x − 3 y + 24 = 0 Q.43.By rotating the vector angle we get

D) none of these in anticlockwise through an

E) F) G) H)

TypeIII: Problems on DeMoivers theorems & application.

n

Q.44. DeMoiver’s theorem states that ( cos θ + i sin θ ) = ---------θ θ B)  cos + i sin 

A) ( cos nθ + i sin nθ ) C)



( cos θ + i sin θ )

n

n

D) None of these

n 2

5

( cos 3θ − i sin 3θ ) ( cos θ − i sin θ ) is-------Q.45. The value of 4 3 ( cos 5θ + i sin 5θ ) ( cos 4θ + i sin 4θ ) A) ( cos 43θ + i sin 43θ ) B) ( cos 4θ + i sin 4θ ) C) ( cos 43θ − i sin 43θ ) D) ( cos 4θ − i sin 4θ ) are

Q.46)The roots of 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

B) C) D) A) Q.49. Using Demoivre’s Theorem , simplified form of equal to A) B) C) D) Q.50. If A)

,then B)

is 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) z + 1 = 1 B) z − 1 = 1 C) z = 1 D) z − 1 = 2 Q.53.If x 2 + y 2 = 1 ,then

1 + x + iy is equal to---1 + x − iy

is

A) x + iy C) y + ix

B) x − iy D) y − ix

Q.54.If

then

A) C)

B) 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 ----------eix + e − ix A) 2

B)

eix − e − ix 2i

e x + e− x C) 2

e x − e− x D) 2

Q.57. The cosh x is ----------A)

eix + e − ix 2

B)

eix − e − ix 2i

C)

e x + e− x 2

D)

e x − e− x 2

C)

e x − e− x e x + e− x

D)

e x + e− x i (e x − e − x )

Q.58. The tanh x is ----------A)

eix + e − ix eix − e − ix

B)

eix − e − ix eix + e − ix

Q.59. sin z & cos z are periodic functions of period -------A) π B) 2π C) 2π i D) 1 Q.60. sinh z & cosh z are periodic functions of period -------A) π

B) 2π

C) 2π i

D) 1

Q.61. which of the following identity is correct A) cosh 2 x + sinh 2 x = 1 B) cosh 2 x − sinh 2 x = 1 C) cosh 2 x = sinh 2 x − 1 D) sinh 2 x − cosh 2 x = 1 Q.62. which of the following identity is correct A) sinh( x + y ) = sinh x cosh y + cosh x sinh y B) sinh( x + y ) = sinh x cosh y − cosh x sinh y C) sinh( x + y ) = sinh x cosh y + i cosh x sinh y D) sinh( x + y ) = sinh x cosh y − ix cosh x sinh y Q.63. which of the following identity is correct

A) cosh( x + y ) = cosh x cosh y + sinh x sinh y B) cosh( x + y ) = cosh x cosh y − sinh x sinh y C) cosh( x + y ) = cosh x cosh y − i sinh x sinh y D) cosh( x + y ) = cosh x cosh y + i sinh x sinh y Q.64. sin iz = -----------A) co sh z B) i sin h iz

C) i sin h z

Q.65. sinh −1 x = -----------A) log x + x 2 + 1

log x + x 2 − 1

)

(

(

B)

D) i c o sh z

)

1 1 1+ x   x +1  D) log  log    2 2  x −1  1− x  Q.66. cosh −1 x = ------------

C)

(

A) log x + x 2 + 1

)

1  x +1  log   2  x −1  Q.67. tanh −1 x = ------------

(

)

log x ± x 2 − 1

)

D) log 

1+ x   1− x 

1 2

C)

A) log x + x 2 + 1

(

B)

(

B)

log x + x 2 − 1

)

C)

1  x +1  log   2  x −1 

 D) log   2 1 − x   1+ x

1

Q.68. If z = reiθ be any complex number then principal value of log z is A) log r + iθ

B) log r − iθ

1 log r + iθ 2

C)

D)

1 log r − iθ 2

Q.69.If z = reiθ be any complex number then general value of value of log z is ---1 2

A) log r + i (2nπ + θ ) B)

1 log r + i (2nπ − θ ) 2

C)

1 log r + iθ 2

D)

Q.70. The principal value of log ( −5) is---------A) log 5 − iπ B) log 5 + iπ C) log 5 + iπ Q.71. The principal value of log (1 + i ) is---------A) log 2 − i

π

B) log 2 + i

4

Q.72. The general value of A) log 2 + i

π 6

π

C)

D) log 5 − iπ

1 π log 2 + i 2 4

D)

4 3 + i is----------

π

B) L og 2 + i(2nπ + )

1 log r − iθ 2

1 π log 2 − i 2 4

π

π

6

6

C) L og 4 + i(2nπ + ) D) log 4 + i

6

Q.73. The general value of 1 + i is---------A) log 2 + i

π 4

π

B) L og 2 + i(2nπ + ) 4

Q.74. The value of i is---------A) e

iπ 4

B) e

− iπ 4

C) - e

π

π

4

4

C) L og 2 + i (2nπ + ) D) log 2 + i iπ

iπ D) e

2i

Q.75. The value of ( i ) is---------π

A) e

−π B) e

2iπ C) e

iπ D) e

Q.76.The value of sin ( log i i ) is -------A) 1 B)-1 C) 0 D) none of these 7 Q.77. If ω is cube root of unity then (1 + ω − ω 2 ) is --------A) 128ω B) −128ω C) 128ω 2 D) −128ω 2 Q.78.If tan(x+iy)=p+iq then tan(x-iy) = A)p-iq B)p+iq C)q-pi D)q+ip is real, is Q.79.The least positive integer n for which 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 6) B 11) B 16) A 21) A 26) D 31)A 41)C 46)C 51)A 56) 61) B 66) B 71) C

2)C 7) C 12) A 17)B 22)D 27)A 32)C 42) B 47) B 52) C 57) C 62)A 67)D 72) B

3)A 8) B 13)C 18)C 23)D 28)A 33)D 43)C 48) C 53)A 58)C 63)A 68)C 73) C

4)A 9)C 14)C 19)D 24) A 29)B 34)A 44)A 49)B 54)A 59)B 64)B 69)A 74) A

76)B

77)D

78) A

79)A

5)A 10)A 15)B 20)B 25)A 30)A 35)C 45)C 50)C 55)B 60)C 65)A 70)B 75)A 80)C