GATE ELECTRICAL ENGINEERING Vol 4 of 4 - Nodia and …€¦ · · 2015-08-19GATE ELECTRICAL...
Transcript of GATE ELECTRICAL ENGINEERING Vol 4 of 4 - Nodia and …€¦ · · 2015-08-19GATE ELECTRICAL...
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
MA 10 Linear Algebra MA 1PE 1 Linear Algebra PE 10EF 1 Linear Algebra EF 10
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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 _____
MA 1 Linear Algebra MA 11PE 11 Linear Algebra PE 1EF 11 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 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
MA 12 Linear Algebra MA 1PE 1 Linear Algebra PE 12EF 1 Linear Algebra EF 12
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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
MA 1 Linear Algebra MA 13PE 13 Linear Algebra PE 1EF 13 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 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
MA 14 Linear Algebra MA 1PE 1 Linear Algebra PE 14EF 1 Linear Algebra EF 14
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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
MA 1 Linear Algebra MA 15PE 15 Linear Algebra PE 1EF 15 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 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
MA 16 Linear Algebra MA 1PE 1 Linear Algebra PE 16EF 1 Linear Algebra EF 16
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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
MA 1 Linear Algebra MA 17PE 17 Linear Algebra PE 1EF 17 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 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
************
MA 18 Linear Algebra MA 1PE 1 Linear Algebra PE 18EF 1 Linear Algebra EF 18
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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= .
MA 1 Linear Algebra MA 19PE 19 Linear Algebra PE 1EF 19 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 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= - =-+ ;
MA 20 Linear Algebra MA 1PE 1 Linear Algebra PE 20EF 1 Linear Algebra EF 20
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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
MA 1.13 Correct option is (A).
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=- - -
MA 1 Linear Algebra MA 21PE 21 Linear Algebra PE 1EF 21 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 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 =
MA 1.18 Correct option is (A).
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
MA 22 Linear Algebra MA 1PE 1 Linear Algebra PE 22EF 1 Linear Algebra EF 22
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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 !
MA 1.22 Correct option is (B).
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.
MA 1 Linear Algebra MA 23PE 23 Linear Algebra PE 1EF 23 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 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
MA 24 Linear Algebra MA 1PE 1 Linear Algebra PE 24EF 1 Linear Algebra EF 24
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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= -
MA 1.31 Correct option is (A).
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.
MA 1.36 Correct option is (B).
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
MA 1 Linear Algebra MA 25PE 25 Linear Algebra PE 1EF 25 Linear Algebra EF 1
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GATE EE vol-4Electrical machines, Power systems
Engineering mathematics, General Aptitude
( )A na
cossin
sincos
nn
nn
aa
aa= -> H
MA 1.38 Correct option is (A).
We know that adj(adj A) .A An 2 := -
Here 3n = , and A 3=
So, ( )Aadj adj 3 3A A( )3 2 := =- .
MA 1.39 Correct option is (C).
We have ( )adj adjA A ( )n 1 2
= -
Putting 3n = , we get ( )adj adjA A 4=
MA 1.40 Correct option is (C).
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&= = .
MA 1.43 Correct option is (D).
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
MA 1.44 Correct option is (C).
AAT 13
21
04
120
314
= - -
R
T
SSSS
>
V
X
WWWW
H
MA 26 Linear Algebra MA 1PE 1 Linear Algebra PE 26EF 1 Linear Algebra EF 26
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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.
MA 1.47 Correct option is (C).
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 =
MA 1.49 Correct option is (B).
From linear algebra for An n# triangular matrix det ,aA iii
n
1
==% . The product of
the diagonal entries of A
MA 1.50 Correct option is (C).
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 =
MA 1.52 Correct option is (B).
Given A 351
3= > H
MA 1 Linear Algebra MA 27PE 27 Linear Algebra PE 1EF 27 Linear Algebra EF 1
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GATE EE vol-3Control systems, Signals & systems
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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.
MA 1.56 Correct option is (A).
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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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
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.
MA 1.57 Correct option is (A).
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
#=- =-
MA 1.58 Correct option is (C).
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= -
MA 1.59 Correct option is (B).
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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::: 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=
MA 1.60 Correct option is (B).
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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GATE EE vol-1Electric circuit & Field, Electrical & electronic measurement
GATE EE vol-2Analog electronics, Digital electronics, Power electronics
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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.
MA 1.62 Correct option is (B).
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.
MA 1.63 Correct option is (B).
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.
MA 1.65 Correct option is (D).
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.
MA 1.70 Correct option is (A).
sinx ! !
...x x x3 5
3 5= + + +
cosx ! !
...x x12 4
2 4= + + +
Thus only ( )sin x3 will have odd power of x .
***********