GATE ELECTRICAL ENGINEERING Vol 4 of 4 - Nodia and …€¦ · · 2015-08-19GATE ELECTRICAL ENGINEERING Vol 4 of 4. Second Edition GATE ELECTRICAL ENGINEERING ... basic filter concepts;

GATEELECTRICAL ENGINEERING

Vol 4 of 4

Second Edition

GATEELECTRICAL ENGINEERING

Vol 4 of 4

RK Kanodia Ashish Murolia

NODIA & COMPANY

GATE Electrical Engineering Vol 4, 2eRK Kanodia & Ashish Murolia

Copyright © By NODIA & COMPANY

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SYLLABUS

GENERAL ABILITY

Verbal Ability : English grammar, sentence completion, verbal analogies, word groups, instructions, critical reasoning and verbal deduction.

Numerical Ability : Numerical computation, numerical estimation, numerical reasoning and data interpretation.

ENGINEERING MATHEMATICS

Linear Algebra: Matrix Algebra, Systems of linear equations, Eigen values and eigen vectors.

Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series. Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems.

Differential equations: First order equation (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchy’s and Euler’s equations, Initial and boundary value problems, Partial Differential Equations and variable separable method.

Complex variables: Analytic functions, Cauchy’s integral theorem and integral formula, Taylor’s and Laurent’ series, Residue theorem, solution integrals.

Probability and Statistics: Sampling theorems, Conditional probability, Mean, median, mode and standard deviation, Random variables, Discrete and continuous distributions, Poisson,Normal and Binomial distribution, Correlation and regression analysis.

Numerical Methods: Solutions of non-linear algebraic equations, single and multi-step methods for differential equations.

Transform Theory: Fourier transform,Laplace transform, Z-transform.

ELECTRICAL ENGINEERING

Electric Circuits and Fields: Network graph, KCL, KVL, node and mesh analysis, transient response of dc and ac networks; sinusoidal steady-state analysis, resonance, basic filter concepts; ideal current and voltage sources, Thevenin’s, Norton’s and Superposition and Maximum Power Transfer theorems, two-port networks, three phase circuits; Gauss Theorem, electric field and potential due to point, line, plane and spherical charge distributions; Ampere’s and Biot-Savart’s laws; inductance; dielectrics; capacitance.

Signals and Systems: Representation of continuous and discrete-time signals; shifting and scaling operations; linear, time-invariant and causal systems; Fourier series representation of continuous periodic signals; sampling theorem; Fourier, Laplace and Z transforms.

Electrical Machines: Single phase transformer – equivalent circuit, phasor diagram, tests, regulation and efficiency; three phase transformers – connections, parallel operation; auto-transformer; energy conversion principles; DC machines – types, windings, generator characteristics, armature reaction and commutation, starting and speed control of motors; three phase induction motors – principles, types, performance characteristics, starting and speed control; single phase induction motors; synchronous machines – performance, regulation and parallel operation of generators, motor starting, characteristics and applications; servo and stepper motors.

Power Systems: Basic power generation concepts; transmission line models and performance; cable performance, insulation; corona and radio interference; distribution systems; per-unit quantities; bus impedance and admittance matrices; load flow; voltage control; power factor correction; economic operation; symmetrical components; fault analysis; principles of over-current, differential and distance protection; solid state relays and digital protection; circuit breakers; system stability concepts, swing curves and equal area criterion; HVDC transmission and FACTS concepts.

Control Systems: Principles of feedback; transfer function; block diagrams; steady-state errors; Routh and Niquist techniques; Bode plots; root loci; lag, lead and lead-lag compensation; state space model; state transition matrix, controllability and observability.

Electrical and Electronic Measurements: Bridges and potentiometers; PMMC, moving iron, dynamometer and induction type instruments; measurement of voltage, current, power, energy and power factor; instrument transformers; digital voltmeters and multimeters; phase, time and frequency measurement; Q-meters; oscilloscopes; potentiometric recorders; error analysis.

Analog and Digital Electronics: Characteristics of diodes, BJT, FET; amplifiers – biasing, equivalent circuit and frequency response; oscillators and feedback amplifiers; operational amplifiers – characteristics and applications; simple active filters; VCOs and timers; combinational and sequential logic circuits; multiplexer; Schmitt trigger; multi-vibrators; sample and hold circuits; A/D and D/A converters; 8-bit microprocessor basics, architecture, programming and interfacing.

Power Electronics and Drives: Semiconductor power diodes, transistors, thyristors, triacs, GTOs, MOSFETs and IGBTs – static characteristics and principles of operation; triggering circuits; phase control rectifiers; bridge converters – fully controlled and half controlled; principles of choppers and inverters; basis concepts of adjustable speed dc and ac drives.

***********

CONTENTS

EM ELECTRICAL MACHINES

EM 1 Transformer 3

EM 2 DC Generator 36

EM 3 DC Motor 57

EM 4 Synchronous Generator 87

EM 5 Synchronous Motor 119

EM 6 Induction Motor 139

EM 7 Single Phase Induction Motor & Special Purpose Machines 166

EM 8 Gate Solved Questions 181

PS POWER SYSTEM

PS 1 Fundamentals of Power System 3

PS 2 Transmission Lines 28

PS 3 Load Flow Studies 66

PS 4 Symmetrical Fault Analysis 82

PS 5 Symmetrical Components and Unsymmetrical Fault Analysis 109

PS 6 Power System Stability and Protection 134

PS 7 Power System Control 162

PS 8 Gate Solved Questions 179

MA ENGINEERING MATHEMATICS

MA 1 Linear Algebra 3

MA 2 Differential Calculus 27

MA 3 Integral Calculus 51

MA 4 Directional Derivatives 73

MA 5 Differential Equation 85

MA 6 Complex Variable 110

MA 7 Probability & Statistics 132

MA 8 Numerical Methods 153

MA 9 Gate Solved Questions 171

VA VERBAL ABILITY

VA 1 Synonyms 3

VA 2 Antonyms 18

VA 3 Agreement 29

VA 4 Sentence Structure 42

VA 5 Spellings 65

VA 6 Sentence Completion 95

VA 7 Word Analogy 123

VA 8 Reading Comprehension 152

VA 9 Verbal Classification 168

VA 10 Critical Reasoning 174

VA 11 Verbal Deduction 190

QA QUANTITATIVE ABILITY

QA 1 Number System 3

QA 2 Surds, Indices and Logarithm 16

QA 3 Sequences and Series 30

QA 4 Averages, Mixture and Alligation 47

QA 5 Ratio, Proportion and Variation 61

QA 6 Percentage 78

QA 7 Interest 92

QA 8 Time, Speed & Distance 102

QA 9 Time, Work & Wages 116

QA 10 Data Interpretation 130

QA 11 Number Series 151

***********

MA 1 Linear Algebra MA 9PE 9 Linear Algebra PE 1EF 9 Linear Algebra EF 1

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GATE EE vol-1Electric circuit & Field, Electrical & electronic measurement

GATE EE vol-2Analog electronics, Digital electronics, Power electronics

GATE EE vol-3Control systems, Signals & systems

GATE EE vol-4Electrical machines, Power systems

Engineering mathematics, General Aptitude

MA 1LINEAR ALGEBRA

MA 1.1 If A012

102

23l

= --

-R

T

SSSS

V

X

WWWW

is a singular matrix, then l is ____

MA 1.2 If A and B are square matrices of order 4 4# such that 5A B= and A Ba= , then a is _____

MA 1.3 If A and B are square matrices of the same order such that AB A= and BA A= , then A and B are both(A) Singular (B) Idempotent

(C) Involutory (D) None of these

MA 1.4 The matrix, A531

852

001

=- -

-

R

T

SSSS

V

X

WWWW

is

(A) Idempotent (B) Involutory

(C) Singular (D) None of these

MA 1.5 Every diagonal element of a skew-symmetric matrix is(A) 1 (B) 0

(C) Purely real (D) None of these

MA 1.6 The matrix, i

i

iA 21

2

2

2

=- -

R

T

SSSSS

V

X

WWWWW

is

(A) Orthogonal (B) Idempotent

(C) Unitary (D) None of these

MA 1.7 Every diagonal elements of a Hermitian matrix is(A) Purely real (B) 0

(C) Purely imaginary (D) 1

MA 1.8 Every diagonal element of a Skew-Hermitian matrix is(A) Purely real (B) 0

(C) Purely imaginary (D) 1

MA 1.9 If A is Hermitian, then iA is(A) Symmetric (B) Skew-symmetric

(C) Hermitian (D) Skew-Hermitian


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MA 1.10 If A is Skew-Hermitian, then iA is(A) Symmetric (B) Skew-symmetric

(C) Hermitian (D) Skew-Hermitian

MA 1.11 If A122

212

221

=- -

-

--

R

T

SSSS

V

X

WWWW

, then adj. A is equal to

(A) A (B) cT

(C) 3AT (D) 3A

MA 1.12 The inverse of the matrix 13

25

--> H is

(A) 53

21> H (B)

52

31> H

(C) 53

21

--

--> H (D) None of these

MA 1.13 Let A153

021

002

=

R

T

SSSS

V

X

WWWW

, then A 1- is equal to

(A) 41

410

1

021

002- -

R

T

SSSS

V

X

WWWW

(B) 21

251

011

002

-- -

R

T

SSSS

V

X

WWWW

(C) 1101

021

002

-- -

R

T

SSSS

V

X

WWWW

(D) None of these

MA 1.14 If the rank of the matrix, A241

174

3

5l=

-R

T

SSSS

V

X

WWWW

is 2, then the value of l is ____

MA 1.15 Let A and B be non-singular square matrices of the same order. Consider the following statements(I) ( )AB A BT T T=

(II) AB B A( ) 11 1=- - -

(III) adj AB A B( ) (adj. )(adj. )=

(IV) ( ) ( ) ( )AB A Br r r=(V) AB A B.=Which of the above statements are false ?(A) I, III & IV (B) IV & V

(C) I & II (D) All the above

MA 1.16 The rank of the matrix A202

134

123

=---

R

T

SSSS

V

X

WWWW

is _____


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Engineering mathematics, General AptitudeMA 1.17 The system of equations 3 0x y z- + = , x y z15 6 5 0- + = , x y z2 2 0l - + = has

a non-zero solution, if l is ____

MA 1.18 The system of equation x y z2 0- + = , 2 3 0x y z- + = , 0x y zl + - = has the trivial solution as the only solution, if l is

(A) 54!l - (B) 3

4l =

(C) 2!l (D) None of these

MA 1.19 The system equations 6x y z+ + = , 2 3 10x y z+ + = , 2 12x y zl+ + = is inconsistent, if l is(A) 3 (B) 3-(C) 0 (D) None of these

MA 1.20 The system of equations 5 3 7 4x y z+ + = , 3 26 2 9x y z+ + = , 7 2 10 5x y z+ + = has(A) a unique solution (B) no solution

(C) an infinite number of solutions (D) none of these

MA 1.21 If A is an n -row square matrix of rank ( 1)n - , then(A) adj 0A = (B) adj 0A !

(C) adj IA n= (D) None of these

MA 1.22 The system of equations 4 7 14x y z- + = , 3 8 2 13x y z+ - = , 7 8 26 5x y z- + = has(A) a unique solution

(B) no solution

(C) an infinite number of solution

(D) none of these

MA 1.23 The eigen values of A39

45= -> H are

(A) 1! (B) 1, 1

(C) 1, 1- - (D) None of these

MA 1.24 The eigen values of A862

674

243

= --

--

R

T

SSSS

V

X

WWWW

are

(A) 0,3, 15- (B) 0, 3, 15- -(C) 0,3,15 (D) 0, 3,15-

MA 1.25 If the eigen values of a square matrix be 1, 2- and 3, then the eigen values of the matrix 2A are(A) , 1,2

123- (B) 2, 4,6-

(C) 1, 2,3- (D) None of these


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MA 1.26 If A is a non-singular matrix and the eigen values of A are , ,2 3 3- then the eigen values of A 1- are(A) 2,3, 3- (B) , ,2

131

31-

(C) 2 ,3 , 3A A A- (D) None of these

MA 1.27 If , ,1 2 3- are the eigen values of a square matrix A then the eigen values of A2 are(A) 1,2,3- (B) 1,4,9

(C) 1,2,3 (D) None of these

MA 1.28 If ,2 4- are the eigen values of a non-singular matrix A and 4A = , then the eigen values of adj A are

(A) , 121 - (B) 2, 1-

(C) 2, 4- (D) 8, 16-

MA 1.29 If 2 and 4 are the eigen values of A then the eigen values of AT are(A) ,2

141 (B) 2, 4

(C) 4, 16 (D) None of these

MA 1.30 If 1 and 3 are the eigen values of a square matrix A then A3 is equal to(A) 13( )A I2- (B) A I13 12 2-(C) A I12( )2- (D) None of these

MA 1.31 If A is a square matrix of order 3 and 2A = then A(adjA) is equal to

(A) 200

020

002

R

T

SSSS

V

X

WWWW

(B) 21

0

0

0

21

0

0

0

21

R

T

SSSSSS

V

X

WWWWWW

(C) 100

010

001

R

T

SSSS

V

X

WWWW

(D) None of these

MA 1.32 The sum of the eigenvalues of A842

250

395

=

R

T

SSSS

V

X

WWWW

is equal to ____

MA 1.33 If 1, 2 and 5 are the eigen values of the matrix A then A is equal to ____

MA 1.34 If the product of matrices

A cos

cos sincos sin

sin

2

2

qq q

q qq= > H and B

coscos sin

cos sinsin

2

2

ff f

f ff= > H

is a null matrix, then q and f differ by(A) an odd multiple of p (B) an even multiple of p(C) an odd multiple of /2p

(D) an even multiple /2p


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Engineering mathematics, General AptitudeMA 1.35 If A and B are two matrices such that A B+ and AB are both defined, then A

and B are(A) both null matrices

(B) both identity matrices

(C) both square matrices of the same order

(D) None of these

MA 1.36 If A31

41=

--> H, then for every positive integer ,n An is equal to

(A) n

nn

n1 2 4

1 2+

+> H (B) n

nnn

1 2 41 2

+ --> H

(C) n

nn

n1 2 4

1 2-

+> H (D) None of these

MA 1.37 If cossin

sincosA

aa

aa= -a > H, then consider the following statements :

I. A A A: =a b ba II. A A A( ): =a b a b+

III. ( )cossin

sincos

A nn

n

n

n

aa

aa= -a > H IV. ( )

cossin

sincos

nn

nnA n

aa

aa= -a > H

Which of the above statements are true ?(A) I and II (B) I and IV

(C) II and III (D) II and IV

MA 1.38 If A is a 3-rowed square matrix such that 3A = , then adj(adjA) is equal to :(A) 3A (B) 9A

(C) 27A (D) none of these

MA 1.39 If A is a 3-rowed square matrix, then adj(adj A) is equal to(A) A 6 (B) A 3

(C) A 4 (D) A 2

MA 1.40 If A is a 3-rowed square matrix such that 2A = , then Aadj(adj )2 is equal to(A) 24 (B) 28

(C) 216 (D) None of these

MA 1.41 If A121

211

=

R

T

SSSS

V

X

WWWW

then A 1- is

(A) 132

425

R

T

SSSS

V

X

WWWW

(B) 121

212

--R

T

SSSS

V

X

WWWW

(C) 232

317

R

T

SSSS

V

X

WWWW

(D) Undefined


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MA 1.42 If Axx x2 0

= > H and A11

02

1 = --

> H, then the value of x is ____

MA 1.43 If A119

82

22

10515

=--

--

-R

T

SSSS

V

X

WWWW

and B13

24

50=

- -> H then AB is

(A) 119

8222

105

15

--

--

-R

T

SSSS

V

X

WWWW

(B) 010

0221

10515

- ----

R

T

SSSS

V

X

WWWW

(C) 119

8222

10515

- --

--

R

T

SSSS

V

X

WWWW

(D) 019

8221

10515

--

--

R

T

SSSS

V

X

WWWW

MA 1.44 If A13

21

04= -> H, then AAT is

(A) 11

34-> H (B)

11

02

13-> H

(C) 51

126> H (D) Undefined

MA 1.45 The matrix, that has an inverse is

(A) 36

12> H (B)

52

21> H

(C) 69

23> H (D)

84

21> H

MA 1.46 The skew symmetric matrix is

(A) 025

206

560-

-

-

R

T

SSSS

V

X

WWWW

(B) 162

534

210

R

T

SSSS

V

X

WWWW

(C) 013

105

350

R

T

SSSS

V

X

WWWW

(D) 021

301

320

R

T

SSSS

V

X

WWWW

MA 1.47 If 11

10

01A = > H and

101

B =

R

T

SSSS

V

X

WWWW

, the product of A and B is

(A) 10> H (B)

10

01> H

(C) 12= G (D)

10

02= G

MA 1.48 Matrix D is an orthogonal matrix AC

B0D = > H. The value of B is

(A) 21 (B)

21

(C) 1 (D) 0


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Engineering mathematics, General AptitudeMA 1.49 If An n# is a triangular matrix then det A is

(A) ( )a1 iii

n

1

-=% (B) aii

i

n

1=%

(C) ( )a1 iii

n

1

-=/ (D) aii

i

n

1=/

MA 1.50 If cossin

te

tt

A t

2

= > H, then dtdA will be

(A) sinsin

te

ttt

2

> H (B) cossin

te

tt

2t> H

(C) sincos

te

tt

2t

-> H (D) Undefined

MA 1.51 If , 0detA R An n !! # , then(A) A is non singular and the rows and columns of A are linearly independent.

(B) A is non singular and the rows A are linearly dependent.

(C) A is non singular and the A has one zero rows.

(D) A is singular.

MA 1.52 For the matrix 3A51

3= > H, ONE of the normalized eigen vectors given as

(A) 21

23> H (B) 2

1

21-> H

(C) 103

101-> H (D) 5

1

52> H

MA 1.53 The system of algebraic equations x y z2+ + 4= , x y z2 2+ + 5= and x y z- + 1= has(A) a unique solution of 1, 1 1andx y z= = = .

(B) only the two solutions of ( 1, 1, 1) ( 2, 1, 0)andx y z x y z= = = = = =(C) infinite number of solutions

(D) no feasible solution

MA 1.54 Eigen values of a real symmetric matrix are always(A) positive (B) negative

(C) real (D) complex

MA 1.55 Consider the following system of equations

x x x2 1 2 3+ + 0= x x2 3- 0= x x1 2+ 0=This system has(A) a unique solution (B) no solution

(C) infinite number of solutions (D) five solutions


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MA 1.56 One of the eigen vectors of the matrix A21

23= > H is

(A) 21-> H (B)

21> H

(C) 41> H (D)

11-> H

MA 1.57 For a matrix M x53

54

53=6 >@ H, the transpose of the matrix is equal to the

inverse of the matrix, M MT 1= -6 6@ @ . The value of x is given by

(A) 54- (B) 5

3-

(C) 53 (D) 5

4

MA 1.58 The matrix 4

p

131

201

6

R

T

SSSS

V

X

WWWW

has one eigen value equal to 3. The sum of the

other two eigen value is(A) p (B) 1p -(C) 2p - (D) 3p -

MA 1.59 For what value of a, if any will the following system of equation in , andx y z have a solution ?

2 3 4x y+ = 4x y z+ + = 3 2x y z a+ - =(A) Any real number (B) 0

(C) 1 (D) There is no such value

MA 1.60 The eigen vector of the matrix 210

2> H are written in the form anda b

1 1> >H H. What

is a b+ ?

(A) 0 (B) 21

(C) 1 (D) 2

MA 1.61 If a square matrix A is real and symmetric, then the eigen values(A) are always real

(B) are always real and positive

(C) are always real and nonnegative

(D) occur in complex conjugate pairs

MA 1.62 The number of linearly independent eigen vectors of 20

12

> H is (A) 0 (B) 1

(C) 2 (D) infinite


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Engineering mathematics, General AptitudeMA 1.63 The system of equations

4 6 20

4

x y z

x y y

x y z

6

ml

+ + =+ + =+ + =

has NO solution for values of l and μ given by(A) 6, 20ml = = (B) 6, 20ml = =Y(C) 6, 20ml = =Y (D) 6, 20ml = =Y

MA 1.64 The eigen values of a skew-symmetric matrix are(A) always zero (B) always pure imaginary

(C) either zero or pure imaginary (D) always real

MA 1.65 The Taylor series expansion of sinx

xp-

at x p= is given by

(A) !

( )...

x1

3

2p+ - + (B) !

( )...

x1

3

2p- - - +

(C) !

( )...

x1

3

2p- - + (D) !

( )...

x1

3

2p- + - +

MA 1.66 The Eigen values of following matrix are

130

310

563

-- -

R

T

SSSS

V

X

WWWW

(A) 3, 3 5 , 6j j+ - (B) 6 5 , 3 , 3j j j- + + -(C) 3 , 3 , 5j j j+ - + (D) 3, 1 3 , 1 3j j- + - -

MA 1.67 All the four entries of the 2 2# matrix Ppp

pp

11

21

12

22= = G are nonzero, and one of its

eigenvalue is zero. Which of the following statements is true?(A) p p p p 111 12 12 21- = (B) p p p p 111 22 12 21- =-(C) p p p p 011 22 12 21- = (D) p p p p 011 22 12 21+ =

MA 1.68 The system of linear equations x y4 2+ 7= , x y2 + 6= has(A) a unique solution (B) no solution

(C) an infinite number of solutions (D) exactly two distinct solutions

MA 1.69 The equation ( )sin z 10= has(A) no real or complex solution(B) exactly two distinct complex solutions(C) a unique solution(D) an infinite number of complex solutions

MA 1.70 Which of the following functions would have only odd powers of x in its Taylor series expansion about the point x 0= ?(A) ( )sin x3 (B) ( )sin x2

(C) ( )cos x3 (D) ( )cos x2

************


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MA 1.1 Correct answer is 2- .A is singular if 0A =

& 012

102

23l

--

-R

T

SSSS

V

X

WWWW

0=

& ( 1) 2 012

2 10

23

02

3l l- - -

-+

-+ - 0=

& ( 4) 2(3)l- + 0=& 4 6l- + 0= 2& l =-

MA 1.2 Correct answer is 625.If k is a constant and A is a square matrix of order n n# then k kA An- .

A B A B B B5 5 5 6254&= = = =& a 625=

MA 1.3 Correct option is (B).A is singular, if A 0=A is Idempotent, if A A2 =A is Involutory, if IA2 =Now, A2 AA AB A A BA AB A( ) ( )= = = = =and B2 BB BA B AB BA B( ) ( )= = = = =& A2 A= and B B2 = ,Thus A B& both are Idempotent.

MA 1.4 Correct option is (B).

Since, A2 531

852

001

531

852

001

=- -

-

- -

-

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

100

010

001

I= =

R

T

SSSS

V

X

WWWW

, IA A2 &= is involutory.

MA 1.5 Correct option is (B).Let aA ij= be a skew-symmetric matrix, then

AT A=- , a aij ij& =- ,if i j= then 2 0 0a a a aii ii ii ii& &=- = =Thus diagonal elements are zero.

MA 1.6 Correct option is (C).A is orthogonal if AA IT =A is unitary if AA IQ = , where AQ is the conjugate transpose of A i.e., ( )A AQ T= .


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Engineering mathematics, General AptitudeHere,

AAQ i

i

i

i2

1

2

2

21

21

2

2

21

10

01 I2=

- - - -= =

R

T

SSSSS

R

T

SSSSS

>

V

X

WWWWW

V

X

WWWWW

H

Thus A is unitary.

MA 1.7 Correct option is (A).A square matrix A is said to Hermitian if A AQ = . So a aij ji= . If i j= then a aii ii= i.e. conjugate of an element is the element itself and aii is purely real.

MA 1.8 Correct option is (C).A square matrix A is said to be Skew-Hermitian if A AQ =- . If A is Skew-Hermitian then A AQ =-& a ji aij=- ,If i j= then 0a a a aii ii ii ii&=- + = it is only possible when aii is purely imaginary.

MA 1.9 Correct option is (D).A is Hermitian then A AQ =Now, (iA)Q ( )i i i iA A A, A A(i )Q Q Q&= =- =- =-Thus iA is Skew-Hermitian.

MA 1.10 Correct option is (C).A is Skew-Hermitian then A AQ =-Now, ( ) ( )i i iA A A AQ Q= =- - = then iA is Hermitian.

MA 1.11 Correct option is (C).If [ ]aA ij n n= # then det [ ]cA ij n n

T= # where cij is the cofactor of aij

Also ( 1)c Miji j

ij= - + , where Mij is the minor of aij , obtained by leaving the row and the column corresponding to aij and then take the determinant of the remaining matrix.

Now, M11 = minor of a11 i.e. 1 312

21- = -

-=-

Similarly

M12 622

21=

-= ; 6M

22

1213 = - =-

M21 622

21=

--

-=- ; 3M

12

2122 =

- -= ;

M23 612

22=

- -- = ; 6M

21

2231 =

- -- = ;

M32 612

22=

- -- = ; 3M

12

2133 =

- -=

C11 ( 1) 3;M1 111= - =-+ ( 1) 6;C M12

1 212= - =-+

C13 ( 1) 6;M1 313= - =-+ ( 1) 6;C M21

2 121= - =+

C22 ( 1) 6;M3 131= - =+ ( 1) 6C M23

2 323= - =-+ ;


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C31 ( 1) 6;M3 131= - =+ ( 1) 6;C M32

3 232= - =-+

C33 ( 1) 3M3 333= - =+

detA CCC

CCC

CCC

T11

21

31

12

22

32

13

23

33

=

R

T

SSSS

V

X

WWWW

3 3A366

636

663

122

212

221

T=- -

-

-- =

- -

-

-- =

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

MA 1.12 Correct option is (A).

Since A 1- A A1 adj=

Now, Here A 113

25=

-- =-

Also, adj A A52

31

53

21adj

T

&=--

-- =

--

--> >H H

A 1- 11 5

321

53

21= -

--

-- => >H H


Since, A 1- A A1 adj=

A 4 0,153

021

002

!= =

adj A 400

1020

112

4101

021

002

T

=-- =

- -

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

A 1- 41

4101

021

002

=- -

R

T

SSSS

V

X

WWWW

MA 1.14 Correct answer is 13.A matrix A( )m n# is said to be of rank r if(i) it has at least one non-zero minor of order r , and(ii) all other minors of order greater than r ,if any; are zero. The rank of A is denoted by ( )Ar . Now, given that ( ) 2A "r = minor of order greater than 2 i.e., 3 is zero.

Thus A 0241

174

3

5l=

-=

R

T

SSSS

V

X

WWWW

& 2(35 4 ) 1(20 ) 3(16 7)l l- + - + - 0=& 70 8 20 27l l- + - + 0= ,

& 9 117 &l l= 13=

MA 1.15 Correct option is (A).The correct statements are

( )AB T B AT T= , ( ) ,AB B A1 1 1=- - -


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Engineering mathematics, General Aptitude adj ( )AB ( ) ( )B Aadj adj= ( )ABr ( ) ( ),A B AB A . B! r r =Thus statements I, III, and IV are wrong.

MA 1.16 Correct answer is 2.Since

A 2( 9 8) 2( 2 3) 2 2 0= - + + - + =- + = & ( ) 3A <r

Again, one minor of order 2 is 6 020

13 != & ( )Ar 2=

MA 1.17 Correct answer is 6.

Here, the coefficient matrix A315

162

152l

=---

R

T

SSSS

V

X

WWWW

For a non-trivial (non-zero) solution ( ) 3A <r& A 0=

& 315

162

152l

---

0=

& 315

010

152l

- 0= ( )C C C2 1 2&+

& 1(6 )l- - 0= 6& l =


Here, coefficient matrix A12

211

131l

=--

-

R

T

SSSS

V

X

WWWW

for trivial solution ( ) 3Ar = i.e., 0A !

& 12

211

131l

--

- 0! ,

& 1(1 3) 2( 2 3 ) 1(2 )l l- + - - + + 0!

& 2 4 6 2l l- - - + + 0!

& 5 4l- - 0! 54& !l -

MA 1.19 Correct option is (A).Equation xA B= is consistent only if ( ) ( )A A B:r r=Otherwise system is said to be inconsistent i.e. possesses no solution. Now,

[ : ]A B :::

111

122

13

61012l

=

R

T

SSSS

V

X

WWWW

111

112

12

1

642l

=-

R

T

SSSS

V

X

WWWW

R RR R

RR

2 1

3 1

2

3

&

&

--f p


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100

110

12

3

642l

=-

R

T

SSSS

V

X

WWWW

( )R R R3 2 3&-

& ( : )A Br 3=

As one of the minor 0100

110

642

!

Now, system is inconsistent if

( )Ar ( : )A B! r i.e. ( ) 3A !r It is possible only when 3 0l- = i.e. 3l =

MA 1.20 Correct option is (B).The system xA B= is consistent (has solution) if ( ) ( : )A A Br t= Also if

( ) ( )A ABr r= .no= of unknowns, then system has a unique solution and if ( ) ( : ) .noA A B <r r= of unknowns, then system has an infinite no. of solution.

Now, here [ : ]A B :::

537

3262

7210

495

=

R

T

SSSS

V

X

WWWW

::

:

50

0

3

5121

511

7

511

51

4

533

53

=-

-

-

R

T

SSSSS

V

X

WWWWW

R R

R R

R

R53

57

2 1

3 1

2

3

&

&

-

-

J

L

KKKK

N

P

OOOO

:::

500

3

5121

0

7

511

0

4

533

0= -

R

T

SSSS

V

X

WWWW

R R R1122

3&+b l

& ( )Ar 2 ( : )A Br= =i.e. ( )Ar ( : ) 2A B <r= = no. of unknowns (3)Thus system has an infinite no. of solutions.

MA 1.21 Correct option is (B).Since ( ) 1nAr = - , at least one ( 1)n - rowed minor of A is non-zero, so at least one minor and therefore the corresponding co-factor is non-zero.So, adj 0A !


Here [ : ]A B :::

137

488

72

26

14135

=-

--

R

T

SSSS

V

X

WWWW

:::

107

42020

72323

142993

=-

--

--

R

T

SSSS

V

X

WWWW

R RR R

RR3

2 1

3 1

2

3

&

&

--f p

:::

107

4200

7230

142964

=-

- --

R

T

SSSS

V

X

WWWW

R R R3 2 3&-^ h

( : )A Br 3& ( ) 2Ar= = ,

( )Ar ( : )A B! rThus system is inconsistent i.e. has no solution.


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Engineering mathematics, General AptitudeMA 1.23 Correct option is (C).

The characteristic equation of a matrix A is given as 0A Il- = .The roots of the characteristic equation are called Now, here 0A Il- =

& 3

45

5l

l-- - - 0=

& (3 )( 5 ) 16l l- - - + 0 15 2 16 02& l l= - + + + =& 2 12l l+ + 0 ( 1) 0 1, 12& &l l= + = =- -Thus eigen values are 1, 1- -

MA 1.24 Correct option is (C).Characteristic equation is 0A Il- =

& 8

62

67

4

24

3

ll

l

--

---

--

0=

& 18 452 2l l l- + 0=

& ( 3)( 15)l l l- - 0= , ,0 3 15& l =

MA 1.25 Correct option is (B).If eigen values of A are , ,1 2 3l l l then the eigen values of kA are , ,k k k1 2 3l l l . So the eigen values of 2A are 2, 4- and 6

MA 1.26 Correct option is (B).If , , ...., n1 2l l l are the eigen values of a non-singular matrix A, then A 1- has the

eigen values , , ....,1 1 1n1 2l l l . Thus eigen values of A 1- are , ,2

131

31- .

MA 1.27 Correct option is (B).If , , ..., n1 2l l l are the eigen values of a matrix A, then A2 has the eigen values

, , ..., n12

22 2l l l . So, eigen values of A2 are 1, 4, 9.

MA 1.28 Correct option is (B).If , , ..., n1 2l l l are the eigen values of A then the eigen values adj A eigen values

adj A are , , ..., ; 0A A A

An1 2

!l l l . Thus eigen values of

adj A are ,24

44- i.e. 2 and 1- .

MA 1.29 Correct option is (B). Since, the eigen values of A and AT are square so the eigen values of AT are 2 and 4.

MA 1.30 Correct option is (B).Since 1 and 3 are the eigen values of A so the characteristic equation of A is

( 1)( 3)l l- - 0= 4 3 02& l l- + =Also, by Cayley-Hamilton theorem, every square matrix satisfies its own


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characteristic equation so

4 3A A I22- + 0=

& A2 4 3A I2= -& A3 4 3 4(4 3 ) 3A A A I A2= - = - -& A3 13 12A I2= -


Since A (adj A) A I3=

& A (adj A) 2100

010

001

200

020

002

= =

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

MA 1.32 Correct answer is 18.Since the sum of the eigen values of an n-square matrix is equal to the trace of the matrix (i.e. sum of the diagonal elements)So, required sum 8 5 5 18= + + =

MA 1.33 Correct answer is 10.Since the product of the eigen values is equal to the determinant of the matrix so 1 2 5 10A # #= =

MA 1.34 Correct option is (C).

AB ( )( )

( )( )

cos cos coscos sin cos

cos sin cossin sin cos

q f q ff q q f

q f q fq f q f=

--

--> H

Null matrix when ( ) 0cos q f- =This happens when ( )q f- is an odd multiple of /2p .

MA 1.35 Correct option is (C).Since A B+ is defined, A and B are matrices of the same type, say m n# . Also, AB is defined. So, the number of columns in A must be equal to the number of rows in B, i.e. n m= . Hence, A and B are square matrices of the same order.


A2 31

41

31

41

52

83=

--

-- =

--> > >H H H

,n

nnn

1 2 41 2=

+ --> H where 2n = .

MA 1.37 Correct option is (D).

A A:a b cossin

sincos

cossin

sincos

aa

aa

bb

bb= - -> >H H

cossin

sincos

nn

nn

aa

aa= -> H A= a b+

Also, it is easy to prove by induction that


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( )A na

cossin

sincos

nn

nn

aa

aa= -> H


We know that adj(adj A) .A An 2 := -

Here 3n = , and A 3=

So, ( )Aadj adj 3 3A A( )3 2 := =- .


We have ( )adj adjA A ( )n 1 2

= -

Putting 3n = , we get ( )adj adjA A 4=


Let B ( )Aadj adj 2= .Then, B is also a 3 3# matrix.

( )}Aadj{adj adj 2 adj B BB 3 3 1 2= = =-

( )adj adjA 2 2= 2A A( )2 3 1 2 16 162

= = =-9 C

... A28 A 2= B

MA 1.41 Correct option is (D).Inverse matrix defined for square matrix only.

MA 1.42 Correct answer is 0.5.

xx x2 0 1

102-> >H H

10

01= > H

& x

x20

02> H ,

10

01= > H So, 2 1x x 2

1&= = .


AB 213

104

13

24

50=

-

- - -R

T

SSSS

>

V

X

WWWW

H

( )( ) ( )( )

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( )( )( ) ( )( )( )( ) ( )

( ) ( )( )( )( ) ( )( )( )( ) ( )( )

2 1 1 31 1 0 33 1 4 3

2 2 1 41 2 0 4

3 2 4 4

2 5 1 01 5 0 0

3 5 4 0=

+ -+

- +

- + -- +

- - +

- + -- +

- - +

R

T

SSSS

V

X

WWWW

119

82

22

10515

=- -

---

R

T

SSSS

V

X

WWWW


AAT 13

21

04

120

314

= - -

R

T

SSSS

>

V

X

WWWW

H


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( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )

1 1 2 2 0 03 1 1 2 4 0

1 3 2 1 0 43 3 1 1 4 4=

+ ++ - +

+ - ++ - - +> H

51

126= > H

MA 1.45 Correct option is (B).If A is zero, A 1- does not exist and the matrix A is said to be singular. Only (B) satisfy the condition.

A (5)(1) (2)(2) 152

21= = - =

MA 1.46 Correct option is (A).A skew symmetric matrix An n# is a matrix with A AT =- . The matrix of (A) satisfy this condition.


AB ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )

11

10

01

101

1 1 1 0 0 11 1 0 0 1 1

12= =

+ ++ + =

R

T

SSSS

> > >

V

X

WWWW

H H H

MA 1.48 Correct option is (C).For orthogonal matrix det 1M = and M MT1 =- , therefore Hence D DT1 =-

DT AB

CBC C

BAD0

1 01= = = - -

--> >H H

This implies 1B BCC B B B1& & != -

- = =

Hence 1B =


From linear algebra for An n# triangular matrix det ,aA iii

n

1

==% . The product of

the diagonal entries of A


dtdA

( )

( )

( )

( )

cos

sinsin

cosdt

d t

dtd e

dtd t

dtd t

te

tt

2t t

2

= =-

R

T

SSSSS

>

V

X

WWWWW

H

MA 1.51 Correct option is (A).If det 0A ! , then An n# is non-singular, but if An n# is non-singular, then no row can be expressed as a linear combination of any other. Otherwise det 0A =


Given A 351

3= > H


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Engineering mathematics, General AptitudeFor finding eigen values, we write the characteristic equation as

A Il- 0=

5

13

3l

l-

- 0=

& ( )( )5 3 3l l- - - 0= 8 122l l- + 0= & l ,2 6=Now from characteristic equation for eigen vector.

xA Il-6 @" , 0= 6 @

For 2l =

XX

5 21

33 2

1

2

--> >H H

00= > H

& 1XX

31

3 1

2> >H H

00= > H

X X1 2+ 0= X X1 2& =-

So eigen vector 11= -* 4

Magnitude of eigen vector ( ) ( )1 1 22 2= + =

Normalized eigen vector 21

21

=-

R

T

SSSSS

V

X

WWWWW

MA 1.53 Correct option is (C).For given equation matrix form is as follows

A 121

211

121

=-

R

T

SSSS

V

X

WWWW

, B451

=

R

T

SSSS

V

X

WWWW

The augmented matrix is

:A B8 B :::

121

211

121

451

=-

R

T

SSSS

V

X

WWWW

,R R R22 2 1" - R R R3 3 1" -

:::

100

233

100

433

+ --

--

R

T

SSSS

V

X

WWWW

R R R3 3 2" -

:::

100

230

100

430

+ - -

R

T

SSSS

V

X

WWWW

/ 3R R2 2" -

:::

100

210

100

410

+

R

T

SSSS

V

X

WWWW

This gives rank of A, ( )Ar 2= and Rank of : : 2A B A Br= =8 8B B

Which is less than the number of unknowns (3)

Ar6 @ :A B 2 3<r= =8 B

Hence, this gives infinite No. of solutions.

MA 1.54 Option (C) is correctLet a square matrix


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A xy

yx= > H

We know that the characteristic equation for the eigen values is given by

A Il- 0=

x

yy

xl

l-

- 0=

( )x y2 2l- - 0= ( )x 2l- y2= x l- y!= & l x y!=So, eigen values are real if matrix is real and symmetric.

MA 1.55 Correct option is (C).Given system of equations are,

x x x2 1 2 3+ + 0= ...(i)

x x2 3- 0= ...(ii)

x x1 2+ 0= ...(iii)Adding the equation (i) and (ii) we have

x x2 21 2+ 0= x x1 2+ 0= ...(iv)We see that the equation (iii) and (iv) is same and they will meet at infinite points. Hence this system of equations have infinite number of solutions.


Let, A 321

2= > H

And 1l and 2l are the eigen values of the matrix A.The characteristic equation is written as

A Il- 0=

21

23

10

01l-> >H H 0=

2

12

3l

l-

- 0= ...(i)

( )( )2 3 2l l- - - 0= 5 42l l- + 0= & l &1 4=Putting 1l = in equation (i),

xx

2 11

23 1

1

2

--> >H H

00= > H where

xx

1

2> H is eigen vector

xx

11

22

1

2> >H H

00= > H

x x21 2+ 0= or x x21 2+ 0=Let x2 K=Then x K21+ 0= & x1 K2=-So, the eigen vector is


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KK

2-> H or

21

-> H

Since option A21-> H is in the same ratio of x1 and x2. Therefore option (A) is an

eigen vector.


Given : M x53

54

53= > H

And [ ]M T [ ]M 1= -

We know that when A A 1=T -6 6@ @ then it is called orthogonal matrix.

M T6 @

MI=

6 @

M MT6 6@ @ I=

Substitute the values of M and M T , we get

x

x53

54

53

53

54

53.> >H H 1

10

0= > H

x

x

x53

53

54

53

53

53

54

53

54

54

53

53

2#

#

#

# #

+

+

+

+

b

b

b

b b

l

l

l

l l

R

T

SSSS

V

X

WWWW

110

0= > H

xx

x1

259 2

2512

53

2512

53+

++

> H 110

0= > H

Comparing both sides a12 element,

x2512

53+ 0= " x 25

1235

54

#=- =-


Let, A 2 4

p

131

01

6=

R

T

SSSS

V

X

WWWW

Let the eigen values of this matrix are , &1 2 3l l lHere one values is given so let 31l =We know that

Sum of eigen values of matrix = Sum of the diagonal element of matrix A

1 2 3l l l+ + p1 0= + + 2 3l l+ p1 1l= + - p1 3= + - p 2= -


Given : x y2 3+ 4= x y z+ + 4= x y z2+ - a=It is a set of non-homogenous equation, so the augmented matrix of this system is


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am

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::: a

211

312

011

44=

-

R

T

SSSS

V

X

WWWW

::: a

202

313

020

44

4+ -

+

R

T

SSSS

V

X

WWWW R3 R R3 2" + , R R R2 22 1" -

::: a

200

310

020

44+ -

R

T

SSSS

V

X

WWWW

R3 R R3 1" -So, for a unique solution of the system of equations, it must have the condition

[ : ]A Br [ ]Ar=So, when putting a 0=We get [ : ]A Br [ ]Ar=


Let A 22

10= > H 1l and 2l is the eigen values of the matrix.

For eigen values characteristic matrix is,

A Il- 0=

22

01

10

10l-> >H H 0=

( )

( )1

02

2l

l-

- 0= ...(i)

( )( )1 2l l- - 0= & l &1 2=So, Eigen vector corresponding to the 1l = is,

1 a00

2 1> >H H 0=

a a2 + 0= 0a& =Again for 2l =

20 b

10

1-> >H H 0=

b1 2- + 0= b 21=

Then sum of &a b a b 0 21

21& + = + =

MA 1.61 Option (A) is correctLet square matrix

A xy

yx= > H

The characteristic equation for the eigen values is given by

A Il- 0=

x

yy

xl

l-

- 0=

( )x y2 2l- - 0= ( )x 2l- y2=


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Engineering mathematics, General Aptitude x l- y!= l x y!=So, eigen values are real if matrix is real and symmetric.


Let, A 20

12= > H

Let l is the eigen value of the given matrix then characteristic matrix is

A Il- 0= Here I10

01= > H = Identity matrix

2

01

2l

l-

- 0=

( )2 2l- 0= l 2= , 2So, only one eigen vector.


Writing :A B we have

:::

111

144

16

620

l m

R

T

SSSS

V

X

WWWW

Apply R R R3 3 2" -

::: 20

110

140

16

6

620

l m- -

R

T

SSSS

V

X

WWWW

For equation to have solution, rank of A and :A B must be same. Thus for no solution; 6, 20!ml =

MA 1.64 Correct option is (C).Eigen value of a Skew-symmetric matrix are either zero or pure imaginary in conjugate pairs.


We have ( )f x sinx

xp

=-

Substituting x p- y= ,we get

( )f y p+ ( )sin sinyy

yyp= + =- ( )sin

yy1= -

! !

...y

y y y13 5

3 5

= - - + -c m

or ( )f y p+ ! !

...y y13 5

2 4

=- + - +

Substituting x yp- = we get

( )f x !

( )!

( )...

x x1

3 5

2 4p p=- + - - - +


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MA 1.66 Correct option is (D).Sum of the principal diagonal element of matrix is equal to the sum of Eigen values. Sum of the diagonal element is 1 1 3 1- - + = .In only option (D), the sum of Eigen values is 1.

MA 1.67 Correct option is (C).The product of Eigen value is equal to the determinant of the matrix. Since one of the Eigen value is zero, the product of Eigen value is zero, thus determinant of the matrix is zero.

Thus p p p p11 22 12 21- 0=

MA 1.68 Correct option is (B).The given system is

xy

42

21= =G G

76= = G

We have A 42

21= = G

and A 42

21 0= = Rank of matrix ( )A 2<r

Now C 42

21

76= = G Rank of matrix ( )C 2r =

Since ( ) ( )A C!r r there is no solution.

MA 1.69 Correct option is (A).sin z can have value between 1- to 1+ . Thus no solution.


sinx ! !

...x x x3 5

3 5= + + +

cosx ! !

...x x12 4

2 4= + + +

Thus only ( )sin x3 will have odd power of x .

***********

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