9387.Matrices Mcq

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

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

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Q.10) For matrix A of order mxn, the rank r of matrix A is A) r ≥min{ , }m n B) r ≥max{ , }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−1 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

          4 2 1 1 3 4 4 3 2 is A)2 B) 3 C) 1 D) 0

Q.14)If P=3 then the rank of matrix

          = 3 3 3 P P P P P P 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=Axand z=By then composite transformation which Converts vector x in to a vector z is

A) z=BAx B) 1 1

z=A B x− −

C) 1 1

z=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

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A) ( )ρ A >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 C) ( )ρ A =ρ([ / ])A B D) ( )ρ A =ρ([ / ])A B <m

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 C) ( )ρ A =ρ([ / ])A B <m D) ( )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

Q.31) For what value of b the matrix 1 5 5 13 b A 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

_A

T

=

_A

−1 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 ,

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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−λI)= 0 B) (A−λI)= 0 C) det(A−λA)= 0 D) (A−λA)= 0 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 _

     − − 2 4 3 2 A)2 B)4 C) 0 D) 1 Q.40)Find product of eigenvalues of matrix _

     − − 2 4 3 2 A)4 B)8 C) 6 D) 2

Q.41)The product of two eigen values of the matrix

          − − − − = 3 1 2 1 3 2 2 2 6

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

          = 2 0 1 0 2 0 1 0 2

A find the third eigenvalue

A)2 B)3 C)1 D) 4

Q.44) If 1, 2 & 3 are the eigen values of

          = 2 0 0 2 0 1 0 2 a

A , find the value of a?

A)1 B)0 C) 2 D) 3 Q.45) The characteristic equation of the matrix _

     − − = 1 5 10 14 A is , A)

λ

2

+

5 λ

+

21 0

=

B)

λ

2 −₁₃

λ

+₃₆= ₀ C)

λ

2 ₊₁₃

λ

₊₃₆ ₌₀ D)

λ

2 ₊₁₃

λ

₋₃₆₌₀

Q.46) Find the eigen values of the matrix _      − − = 1 5 10 14 A A) 1 4, 2 9

λ

=

λ

= B) 1 5, 2 6

λ

=

λ

= C) 1 18, 2 2

λ

=

λ

= D) 1 10, 2 3

λ

=

λ

=

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Q.47) Two eigen values of the matrix           = 2 2 1 1 3 1 1 2 2

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

          = 2 2 1 1 3 1 1 2 2

A are 1 & 1 find the eigenvalues of A−1

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

_α

₋_5,

_β

₋_5,

_γ

₋₅ where

_{α β γ}

_{, ,}

the eigenvalues of are

          − − − − − = 9 8 7 6 5 4 3 2 1 A A)           − − − − − 9 8 7 6 5 4 3 2 1 B)           − − − − − 4 8 7 6 0 4 3 2 6 C)           − − − 9 8 7 6 5 4 3 2 1 D)           − − − − 14 8 7 6 10 4 3 2 4

Q.50) If the characteristic equation of one matrix is 3 2

4 4 0

λ − 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

          − − 1 1 0 1 2 1 0 1 1 A) 3 2 4 3 0 λ − λ + λ= B) 3 2 4 3 1 0 λ − λ + λ+ = C) 3 2 4 3 0 λ + λ + λ= D) 3 2 4 3 0 λ − 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 λis 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 λis eigen value of matrix A then eigen values of matrix kA is A) kλ B) −λ C) 1

kλ D) λ.

Q.57) If λis eigen value of matrix A then eigen values of matrix A+kI is A) kλ B) λ+k C) 1

kλ D) λ−k.

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A) nλ B) λn C) n

λ D) λ.

Q.59) If λ λ λ1, 2, 3are eigen values of matrix A then eigen values of 1 A− are A) 1 2 3 1 1 1 , , λ λ λ B) λ λ λ1, 2, 3 C) 2 2 2 1 , 2 , 3 λ λ λ D) −λ₁,−λ₂,−λ₃

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

          − − 4 0 0 5 2 0 3 2 1

product of the eigen values is

A)-8 B) 4 C)1 D) -2 Q. 62) The eigen values of the matrix _

     3 2 4 1 are A) 2,3 B)4,5 C) 0,2 D) 5,-1 Q.63) The Characteristic equation of the matrix 3₁ 1₅ 1₁

1 1 3     − −    ₋    is A) 3 2 11 38 40 0 λ − λ + λ− = B) 3 2 11 38 40 0 λ − λ + λ+ = C) λ3 +11λ2 +38λ+40= 0 D) λ3+11λ2 +38λ+40= 0

Q.64) If the Characteristic equation of the matrix A of order 3x3 is λ3−3λ2 +3λ− = then by Cayley 1 0 Hamilton

Theorem A−1is equal to

A) A3 −3A2 +3A− I B) A2 −3A−3I C) 3A2 −3A− I D) A2 −3A−3I

Q.65) If λ2−S₁λ+S₂ =0 is a characteristic equation of 2x2 matrix A then A) S₁= Sum of principle diagonal elements, S₂ =Sum of all elements

B) S₁= Sum of principle diagonal elements, S₂ = Product of principle diagonal elements C) S₁=Trace of matrix A, S₂ = Product of principle diagonal elements

D) S₁=Trace of matrix A, S₂ = Product of Eigen values of matrix A.

Q.66) If λ3−S₁λ2+S₂λ−S₃ =0 is a characteristic equation of 3x3 matrix A then A) S₁= Sum of principle diagonal elements,S₂ =Sum of all elements,S3 = A

B) S₁= Sum of principle diagonal elements,S₂ = Product of principle diagonal elements,

3

S = A

C)S₁=Trace of matrix A,S₂ = sum of minors of Principle diagonal elements,S3 = A

D) S₁=Trace of matrix A,S₂ = Product of Eigen values of matrix A,S3= A

Q.67) The characteristic equation of matrix 14 10

5 1 −     −   is A) 2 13 36 0 λ − λ+ = B) 2 13 36 0 λ − λ− = C) 2 4 64 0 λ − λ− = D) 2 36 0 λ − +λ =

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Q.68) The characteristic equation of matrix 4₁ 6₃ 6₂ 1 4 3       ₋ ₋ ₋    is A) λ3−4λ2+λ− = B) 4 0 3 2 4 4 0 λ + λ − + = λ C) λ3−λ2+λ− = D) 4 0 3 2 4 4 0 λ − λ − + = λ

Q.69) If all eigen values of matrix A_{3 3}x are distinct then which of the following is true

A) MatrixA_{3 3}x has three equal Eigen vectors

B) MatrixA_{3 3}x has three distinct eigen vectors

C) MatrixA_{3 3}x has three distinct linearly independent eigen vectors

D) ) MatrixA_{3 3}x 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

          1 1 1 1 1 1 1 1 1 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 7 A_{= } _

 

then value of k for which A2 =8A+ kI is

A) 5 B)7 C)-5 D)-7 Q.77) If 1 2 2 1 A_{= } _ −   then 8 A is A) A8 =5I B) A8 =25I C) A8 =65I D) A8 =625I 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

2

A are

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Q.79) If 1 2 3 0 3 2 0 0 2 A −     =    ₋   

then the eigen values of 3A3+5A2−6A+2I 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 4 A_{= } _  then A) 1 1 3 2 4 A− _{= } _   B) 1 1 1 2 1 1 3 4 A−     =         C) 1 2 1 3 1 2 2 A− −     =_ ₋ _   D) 1

A− 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

2 A + I are A) 20,39,228 B)10,29,218 C) 0,19,208 D) 3,5,3 Q.83)Eigenvalues of matrix _      = 3 2 2 3

A are 5 &1 .what are the eigen values of matrix A2

A) 1 & 25 B) 2 & 10 C) 6& 4 D) 5 & 1 GENERAL: Q84)If 1 2 3 ( , , ,.... _n) D=diag d d d d where ₀ i

d ≠ for all i=1,2,3,….n ,then D−1 is equal to ,

A)D _B) 1 1 1 1

1 2 3

( , , ,.... _n )

diag d − d − d − d − C)

I

n D) None of these

Q85)If

1 2 3

( , , ,.... _n)

A=diag d d d d then Anis equal to ,

A) 1 1 1 1 1 2 3 ( n , n , n ,.... _nn ) diag d − d − d − d − B)diag d( ₁n,d₂n,d₃n,....d_nn) C) A D)None of these. Q86)If           − = 9 8 1 9 7 0 7 5 1

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

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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.

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_{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+i 3 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 ( ) 2i z+z C) 1 ( ) 2 z−z 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 z1z2 is equal to

A) z1+ z2 B) z1− z2

C) z₁ z₂ D) None of these.

Q.8. If z1 and z2 are two complex nos. then arg z z

(

1 2

)

=---

A) arg( ) arg( )z1 z2 B) arg( ) arg( )z1 + z2

C) arg( ) arg( )z1 − z2 D) None of these

Q.9. If z1 and z2 are two complex nos. then       2 1 arg z z is

Q.10.Modulus of complex number z=x+ iy is A) 2 2 y x + B) 2 2 y x − C) x y 1

tan− D) None of these

Q.11. Argument of complex number z=x+iy is A) 2 2 y x + B) y x 1 tan− C) x y 1

tan− D) None of these

Q.12 The complex numbers z1 and z2 are comparable if

A) z1 and z2 are real numbers.

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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 --- A) 2(cos sin ) 4 4 z= π +i π B) 2(cos sin ) 4 4 z= π +i π C) 2(cos sin ) 4 4 z= π −i π D) 2( cos sin ) 4 4 z= − π −i π Q.14.If zz =0then A)Re(z)=0 B)Im(z)=0 C) )z=0 D)z≠0 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+i 3 B) 1−i 3 C) − +1 i 3 D− −1 i 3

Q.17. If z =acosθ +iasinθ then

z z z z

+ is equal to

A)2sin2θ B) 2cos2θ C) 2tan2θ D) 2sinθ

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) 2 , 4 π D) 2 , 2 π

Q.20.The exponential form of 1-i is A) ₂ 4 π e B) ₂ 4 π − e C) 4 π e D) 4 3π − e

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

2

π

C) π D)2arg

( )

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 5 2

π i e z= + A) 5 e B) C)5 D)0

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

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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)cosx i− sinx B) cosx i+ sinx C) sinx i+ cosx D) sinx i− cosx Q.30. The value of ix

e 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 ix e +e− B) 2 ix ix e e i − + C) 2 ix ix e −e− D) 2 ix ix e e i − −

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

(13)

A) B) C) D) Q.36. If 2 z i z +

+ is purely real then locus of z is ---- ---

A) 2 2 2 0 x +y + x+y= B) 2 2 2 0 x +y − x−y= C) x+2y+ =2 0 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 zsatisfying 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 zsatisfying z−2 =3 is --- A) Parabola B) hyperbola C) Straight Line D) circle

Q.40.The locus of zsatisfying 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 8 0, 6 z i z −   =   +

  then z lies on the curve ---

A) 2 2 8 0 x +y − = B) 2 2 6 8 0 x +y + x− y= C) 4x−3y+24=0 D) none of these Q.43.By rotating the vector in anticlockwise through an angle we get

A)

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C)

D)

TypeIII: Problems on DeMoivers theorems & application. Q.44. DeMoiver’s theorem states that

(

cosθ +isinθ

)

n =---

A)

(

cosnθ +isinnθ

)

B) cos isin

n n θ θ   +     C)

(

cos isin

)

n θ+ θ D) None of these Q.45. The value of

(

) (

)

(

) (

)

2 5 4 3

cos 3 sin 3 cos sin cos 5 sin 5 cos 4 sin 4

i i i i θ θ θ θ θ θ θ θ − − + + is---

A)

(

cos 43θ +isin 43θ

)

B)

(

cos 4θ +isin 4θ

)

C)

(

cos 43θ−isin 43θ

)

D)

(

cos 4θ −isin 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 is equal to

A) B) C) D)

Q.50. If ,then is

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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 2 2 1 x +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 ix e +e− B) 2 ix ix e e i − − C) 2 x x e +e− D) 2 x x e −e− Q.57. The cosh x is --- A) 2 ix ix e +e− B) 2 ix ix e e i − − C) 2 x x e +e− D) 2 x x e −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 zare 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 2 cosh x+sinh x=1 B) 2 2 cosh x−sinh x=1 C) 2 2 cosh x=sinh x−1 D) 2 2 sinh x−cosh x=1

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Q.62. which of the following identity is correct A)sinh(x+y)=sinh coshx y+cosh sinhx y

B) sinh(x+y)=sinh coshx y−cosh sinhx y C) sinh(x+y)=sinh coshx y+icosh sinhx y

D)sinh(x+y)=sinh coshx y−ixcosh sinhx y

Q.63. which of the following identity is correct A)cosh(x+y)=cosh coshx y+sinh sinhx y

B)cosh(x+y)=cosh coshx y−sinh sinhx y C)cosh(x+y)=cosh coshx y−isinh sinhx y D)cosh(x+y)=cosh coshx y+isinh sinhx y Q.64. s in iz = --- A) c o s h z B) is in hiz C) is in h z D) ic o s h z Q.65. 1 sinh− x=--- A)

(

2

)

log x+ x +1 B)

(

2

)

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

(

2

)

log x+ x +1 B)

(

2

)

log x± x −1 C) 1log 1 2 1 x x +     −   D) 1 1 log 2 1 x x +     −   Q.67. 1 tanh− x=--- A)

(

2

)

log x+ x +1 B)

(

2

)

log x+ x −1 C) 1log 1 2 1 x x +     −   D) 1 1 log 2 1 x x +     −   Q.68. If i

z=reθbe any complex number then principal value of log z is

A)log r+iθ B) log r−iθ C) 1log

2 r+iθ D) 1

log

2 r−iθ

Q.69.If i

z=reθbe any complex number then general value of value of log z is

---- A)1log (2 ) 2 r+i nπ θ+ B) 1 log (2 ) 2 r+i nπ θ− C) 1 log 2 r+iθ D) 1 log 2 r−iθ

Q.70. The principal value of log

( )

−5 is---

A)log 5 i− π B) log 5 i+ π C) log 5+iπ D) log 5−iπ

Q.71. The principal value of log 1 i

(

+

)

is--- A)log 2 4 iπ − B) log 2 4 iπ + C) 1log 2 2 i 4 π + D) 1log 2 2 i4 π −

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A)log 2 6 iπ + B) og 2 (2 ) 6 L +i nπ+π C) og 4 (2 ) 6 L +i nπ +π D) log 4 6 iπ +

Q.73. The general value of 1 i+ is--- A)log 2 4 iπ + B) og 2 (2 ) 4 L +i nπ+π C) og 2 (2 ) 4 L +i nπ+π D) log 2 4 iπ +

Q.74. The value of i is---

A) 4 i e π B) 4 i e π − C) -eiπ D) eiπ Q.75. The value of

( )

2i i is--- A)eπ B) e−π C) e2iπ D) eiπ

Q.76.The value of sin log

(

ii

)

is ---

A) 1 B)-1 C) 0 D) none of these Q.77. If ωis cube root of unity then

(

2

)

7

1+ω ω− is --- A) 128ω B) −128ω C) 2

128ω D) 2

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

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.

_{A) 135 B)i C)-1 D)-i}

_{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

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Q.4. If 2+i 3 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 ( ) 2i z+z C) 1 ( ) 2 z−z 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 is equal to

A) B) C) D) None of these.

Q.8. If z1 and z2 are two complex nos. then arg z z

(

1 2

)

=---

Q.9. If z1 and z2 are two complex nos. then is

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 z= −1 i is --- A) 2(cos sin )

4 4

z= π +i π B) 2(cos sin )

4 4

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C) 2(cos sin ) 4 4 z= π −i π D) 2( cos sin ) 4 4 z= − π −i 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+i 3 B) 1−i 3 C) − +1 i 3 D− −1 i 3 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

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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)cosx i− sinx B) cosx i+ sinx C) sinx i+ cosx D) sinx i− cosx Q.30. The value of ix

e 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 ix e +e− B) 2 ix ix e e i − + C) 2 ix ix e −e− D) 2 ix ix e e i − −

Q.32.By Euler’s Formula the value of sinx is --- A) 2 ix ix e +e− B) 2 ix ix e e i − + C) 2 ix ix e −e− D) 2 ix ix e 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 0 x +y + x+y= B) 2 2 2 0 x +y − x−y= C) x+2y+ =2 0 D) None of these

Q.37.If z is complex number such that then is

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A) B) C) D)

Q.38.The locus of zsatisfying 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 zsatisfying z−2 =3 is --- A) Parabola B) hyperbola C) Straight Line D) circle

Q.40.The locus of zsatisfying 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 8 0, 6 z i z −   =   +

  then z lies on the curve ---

A) 2 2 8 0 x +y − = B) 2 2 6 8 0 x +y + x− y= C) 4x−3y+24=0 D) none of these Q.43.By rotating the vector in anticlockwise through an angle we get E) F) G) H)

(22)

Q.44. DeMoiver’s theorem states that

(

cosθ +isinθ

)

n =--- A)

(

cosnθ +isinnθ

)

B) cos isin

n n θ θ   +     C)

(

cos isin

)

n θ+ θ D) None of these Q.45. The value of

(

) (

)

(

) (

)

2 5 4 3

cos 3 sin 3 cos sin cos 5 sin 5 cos 4 sin 4

i i i i θ θ θ θ θ θ θ θ − − + + is---

A)

(

cos 43θ +isin 43θ

)

B)

(

cos 4θ +isin 4θ

)

C)

(

cos 43θ−isin 43θ

)

D)

(

cos 4θ−isin 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 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) z+ =1 1 B) z− =1 1 C) z =1 D) z− =1 2 Q.53.If 2 2 1 x +y = ,then 1 1 x iy x iy + + + − is equal to----

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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 ix e +e− B) 2 ix ix e e i − − C) 2 x x e +e− D) 2 x x e −e− Q.57. The cosh x is --- A) 2 ix ix e +e− B) 2 ix ix e e i − − C) 2 x x e +e− D) 2 x x e −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 zare 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 2 cosh x+sinh x=1 B) 2 2 cosh x−sinh x=1 C) 2 2 cosh x=sinh x−1 D) 2 2 sinh x−cosh x=1

Q.62. which of the following identity is correct A)sinh(x+y)=sinh coshx y+cosh sinhx y

B) sinh(x+y)=sinh coshx y−cosh sinhx y

C) sinh(x+y)=sinh coshx y+icosh sinhx y D)sinh(x+y)=sinh coshx y−ixcosh sinhx y

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A)cosh(x+y)=cosh coshx y+sinh sinhx y B)cosh(x+y)=cosh coshx y−sinh sinhx y C)cosh(x+y)=cosh coshx y−isinh sinhx y D)cosh(x+y)=cosh coshx y+isinh sinhx y

Q.64. s in iz = --- A) c o s h z B) is in hiz C) is in h z D) ic o s h z Q.65. 1 sinh− x=--- A)

(

2

)

log x+ x +1 B)

(

2

)

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

(

2

)

log x+ x +1 B)

(

2

)

log x± x −1 C) 1log 1 2 1 x x +     −   D) 1 1 log 2 1 x x +     −   Q.67. 1 tanh− x=--- A)

(

2

)

log x+ x +1 B)

(

2

)

log x+ x −1 C) 1log 1 2 1 x x +     −   D) 1 1 log 2 1 x x +     −   Q.68. If i

z=reθbe any complex number then principal value of log z is

A)log r+iθ B) log r−iθ C) 1log

2 r+iθ D) 1

log

2 r−iθ

Q.69.If i

z=reθbe any complex number then general value of value of log z is

---- A)1log (2 ) 2 r+i nπ θ+ B) 1 log (2 ) 2 r+i nπ θ− C) 1 log 2 r+iθ D) 1 log 2 r−iθ

Q.70. The principal value of log

( )

−5 is---

A)log 5 i− π B) log 5 i+ π C) log 5+iπ D) log 5−iπ

Q.71. The principal value of log 1 i

(

+

)

is--- A)log 2 4 iπ − B) log 2 4 iπ + C) 1log 2 2 i 4 π + D) 1log 2 2 i4 π −

Q.72. The general value of 3+i is--- A)log 2 6 iπ + B) og 2 (2 ) 6 L +i nπ+π C) og 4 (2 ) 6 L +i nπ +π D) log 4 6 iπ +

Q.73. The general value of 1 i+ is--- A)log 2 4 iπ + B) og 2 (2 ) 4 L +i nπ+π C) og 2 (2 ) 4 L +i nπ+π D) log 2 4 iπ +

Q.74. The value of i is---

A) 4 i e π B) 4 i e π − C) -eiπ D) eiπ

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Q.75. The value of

( )

2i

i is---

A)eπ B) e−π C) e2iπ D) eiπ

Q.76.The value of sin log

(

ii

)

is ---

A) 1 B)-1 C) 0 D) none of these Q.77. If ωis cube root of unity then

(

2

)

7

1+ω ω− is --- A) 128ω B) −128ω C) 2

128ω D) 2

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

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