Targate-maths Booklet (Non Dowloadable)

(1)

ENGINEERING

MATHEMATICS

Objective Paper –“Topic & Level-wise”

GATE

For “Electrical”, “Mechanical”, “CS/IT” & “Electronics & Comm.”

Engg.

Product of,

TARGATE EDUCATION

(2)

Copyright © TARGATE EDUCATION, Bilaspur-2013

All rights reserved

No part of this publication may be reproduced, stored in retrieval system, or transmitted in any form or by

any means, electronics, mechanical, photocopying, digital, recording or otherwise without the prior

permission of the TARGATE EDUCATION.

Authors:

Subject Experts @TRGATE EDUCATION, BILASPUR

TARGATE EDUCATION

Ground Floor, Below Old Arpa Bridge,Jabdapara, SARKANDA RD. Bilaspur (Chhattisgarh) 495001

Phone No: 07752406380,093004-32128 (01:30 PM - 07:30 PM, Wed-Off) Web Address: www.targate.org, E-Contact: [email protected]

(3)

SYLLABUS: ENGG.

MATHEMATICS

GATE – 2013

EE /ECEC

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.

Mechanical Engineering (ME)

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

Calculus: Functions of single variable, Limit, continuity and differentiability, Mean value theorems, Evaluation of

definite and improper integrals, Partial derivatives, Total derivative, Maxima and minima, Gradient, Divergence and Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems.

Differential equations: First order equations (linear and nonlinear), Higher order linear differential equations with

constant coefficients, Cauchy’s and Euler’s equations, Initial and boundary value problems, Laplace transforms, Solutions of one dimensional heat and wave equations and Laplace equation.

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

Probability and Statistics: Definitions of probability and sampling theorems, Conditional probability, Mean, median,

mode and standard deviation, Random variables, Poisson,Normal and Binomial distributions.

Numerical Methods: Numerical solutions of linear and non-linear algebraic equations Integration by trapezoidal and

(4)

Mathematical Logic: Propositional Logic; First Order Logic.

Probability: Conditional Probability; Mean, Median, Mode and Standard Deviation; Random Variables; Distributions;

uniform, normal, exponential, Poisson, Binomial.

Set Theory & Algebra: Sets; Relations; Functions; Groups; Partial Orders; Lattice; Boolean Algebra.

Combinatorics: Permutations; Combinations; Counting; Summation; generating functions; recurrence relations;

asymptotics.

Graph Theory: Connectivity; spanning trees; Cut vertices & edges; covering; matching; independent sets; Colouring;

Planarity; Isomorphism.

Linear Algebra: Algebra of matrices, determinants, systems of linear equations, Eigen values and Eigen vectors.

Numerical Methods: LU decomposition for systems of linear equations; numerical solutions of non-linear algebraic

equations by Secant, Bisection and Newton-Raphson Methods; Numerical integration by trapezoidal and Simpson’s rules.

Calculus: Limit, Continuity & differentiability, Mean value Theorems, Theorems of integral calculus, evaluation of

definite & improper integrals, Partial derivatives, Total derivatives, maxima & minima.

Expert Comment

Comparing to the ME syllabus EE/EC has an extra topic “Transform Theory”. ME students need not to read this topics. CS students have to refer topics from this booklet which is listed in there syllabus. Remaining topic for CS will be covered in separate booklet.

(5)

Page 5

www.targate.org

LINEAR ALGEBRA 7

1.1PROPERTY BASED PROBLEM 7

1.2DETERMINANTE 10

1.3ADJOINT -INVERSE 11

1.4EIGEN VALUES &EIGEN VECTORS 13

1.5RANK 19

1.6SOLUTION OF LINEAR EQUATION 21

1.7MISCELLANEOUS 26

1.8CALY-HAMILTON 31

CALCULUS 32

2.1MEAN VALUE THEOREM 32

2.2MAXIMA AND MINIMA 32

2.3DIFFERENTIAL CALCULUS 34

2.4INTEGRAL CALCULUS 36

2.5LIMIT AND CONTINUITY 39

2.6SERIES 43

2.7VECTOR CALCULUS 44

2.8AREA/VOLUME 51

DIFFERENTIAL EQUATIONS 55

3.1DEGREE AND ORDER OF DE 55

3.2 HIGHER ORDER DE 56

3.3LEIBNITZ LINEAR EQUATION 61

COMPLEX VARIABLE 66

4.1CAUCHY’S THEOREM 66

PROBABILITY AND STATISTICS 74

5.2COMBINATION 74

5.3PROBABILITY RELATED PROBLEMS 75

5.4BAYS THEOREMS 80

5.5PROBABILITY DISTRIBUTION 80

5.6RANDOM VARIABLE 82

(6)

NUMERICAL METHODS 87 6.1CLUBBED PROBLEM 87 6.

2

NEWTON-RAP SON 89 6.3DIFFERENTIAL 93 6.4INTEGRATION 93 TRANSFORM THEORY 95

(7)

Page 7

www.targate.org

01

Linear Algebra

Complete subtopic in this chapter, is in the scope of “GATE- CS/ME/EC/EE SYLLABUS”

.

1.1 Property Based Problem

Question Level – 0 (Basic Problems)

eE1 / T1 / K1 / L0 / V1 / R11 / AD [GATE – CS – 1994]

(01) If A and B are real symmetric matrices of order n

then which of the following is true.

(A) A AT = I (B) A = A-1

(C) AB = BA (D) (AB)T = BTAT

eE1 / T1 / K1 / L0 / V1 / R11 / AA [GATE – PI – 1994]

(02) If for a matrix, rank equals both the number of

rows and number of columns, then the matrix is called

(A) Non-singular (B) singular

(C) Transpose (D) Minor

eE1 / T1 / K1 / L0 / V1 / R11 / A [GATE – CE – 2000]

(03) If A, B, C are square matrices of the same order

then (ABC)1 is equal be

(A) C A B1 1 1 (B) C B A1 1 1

(C) A B C1 1 1 (D) A C B1 1 1

eE1 / T1 / K1 / L0 / V1 / R11 / AB [GATE – CE – 2008]

(04) The product of matrices (PQ)1P is

(A) P1 (B) Q1

(C) P Q P1 1 (D) P Q P1

---00000---Question Level – 01

(01) If A is a real square matrix then AAT is

(A) Un symmetric

(B) Always symmetric

(C) Skew – symmetric

(D) Sometimes symmetric

eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – CE – 1998]

(02) In matrix algebra AS = AT (A, S, T, are matrices

of appropriate order) implies S = T only if

(A) A is symmetric

(B) A is singular

(C) A is non-singular

(8)

Page 8

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

eE1 / T1 / K1 / L1 / V1 / R11 / AA [GATE – IN – 2001]

(03) The necessary condition to diagonalizable a

matrix is that

(A) Its all Eigen values should be distinct

(B) Its Eigen values should be independent

(C) Its Eigen values should be real

(D) The matrix is non-singular

eE1 / T1 / K1 / L1 / V1 / R11 / AD [GATE – EC – 2005]

(04) Given an orthogonal matrix A =

1

0

1

1 





_























1 (AAT) Is ____ (A) 4 1 4I (B) 4 1 2I (C) I (D) 1 ₄ 3I eE1 / T1 / K1 / L1 / V1 / R11 / AA [GATE – ME – 2007]

(05) 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 non-negative

(D) Occur in complex conjugate pairs

eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – EC – 2010]

(06) 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

eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – ME – 2011]

(07) Eigen values of a real symmetric matrix are

always

(A) Positive (B) Negative

(C) Real (D) 162. [A] is square

---00000---Question Level – 02

eE1 / T1 / K1 / L2 / V2 / R11 / AA [GATE – CS – 2001]

(01) Consider the following statements

S1: The sum of two singular matrices may be

singular.

S2: The sum of two singulars may be

non-singular.

This of the following statements is true.

(A) S1 & S2 are both true

(B) S1 & S2 are both false

(C) S1 is true and S2 is false

(9)

www.targate.org

Page 9

eE1 / T1 / K1 / L2 / V2 / R11 / AD [GATE – EE – 2008]

(02) A is m x n full rank matrix with m > n and I is an

identity matrix. Let matrix A(A AT )1AT. then which one of the following statements is false?

(A) AA+A = A (B) (AA+)2 = AA+

(C) A+A = I (D) AA+A = A+

(03) A square matrix B is symmetric if ---

(A) BT = B (B) BT = B

(C) B1= B (D) B1 = BT

---00000---Question Level – 03

eE1 / T1 / K1 / L3 / V2 / R11 / AD [GATE – CE – 1998]

(01) The real symmetric matrix C corresponding to

the quadratic form Q = 4x x_{1 2}5x x_{1 2} is

(A) 1 2 2 5    _    (B) 2 0 0 5    _    (C) 1 1 1 2    _    (D) 0 2 2 5    _    eE1 / T1 / K1 / L3 / V2 / R11 / AB [GATE – CE – 2000]

(02) Consider the following two statements.

(I) The maximum number of linearly independent column vectors of a matrix A is called the rank of A.

(II) If A is

n n



square matrix then it will be non-singular is rank of A = n

(A) Both the statements are false

(B) Both the statements are true

(C) (I) is true but (II) is false

(D) (I) is false but (II) is true

eE1 / T1 / K1 / L3 / V2 / R11 / AA [GATE – CE – 2004]

(03) Real matrices

   

A_{3 1,}_ B _{3 3}_ ,

 

C _{3 5}_ ,

 

D_,

 

E ₅,



 

F 1 are given. Matrices [B] and [E]

are symmetric. Following statements are made with respect to their matrices.

(I) Matrix product [F]T[C]T[B] [C] [F] is a scalar. Matrix product [D]T[F] [D] is always

symmetric. With reference to above statements which of the following applies?

(A) Statement (I) is true but (II) is false

(B) Statement (I) is false but (II) is true

(C) Both the statements are true

(D) Both the statements are false

eE1 / T1 / K1 / L3 / V2 / R11 / AB [GATE – EE – 2008]

(04) Let P be 2x2 real orthogonal matrix and x is a real vector



1 2



T

x x with length || ||x =

2 2 1/ 2

1 2

(x x ) Then which one of the following statement is correct?

(A) ||px|| || || x where at least one vector satisfies ||px|| || || x

(10)

Page 10

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(B) ||px|| || || x for all vectors x

(C) ||px|| || || x when atleast one vector satisfies

|| ||x and ||px||

(D) No relationship can be established between

|| ||x and ||px||

eE1 / T1 / K6 / L3 / V2 / R11 / A [GATE – CS – 2008]

(05) The following system of equations

1 2 2 3 1,

x x  x  x₁2x₁3x₃ ,

1 4 1 3 4

x  x αx  has a unique solution solution. The only possible value(s) for α is/are

(A) 0 (B) either 0 (or) 1

(C) one of 0, 1 (or) – 1 (D) any real number

(06) [A] is a square matrix which is neither symmetric

nor skew-symmetric and [A]T is its transpose. The sum and differences of these matrices are defined as [S] = [A] + [A]T and [D] = [A] – [A]T respectively. Which of the following statements is true?

(A) Both [S] and [D] are symmetric

(B) Both [S] and [D] are skew-symmetric

(C) [S] is skew-symmetric and [D] is symmetric

(D) [S] is symmetric and [D] is skew-symmetric

---00000---

1.2 Determinante

Question Level – 00 (Basic Problem)

eE1 / T1 / K2 / L0 / V1 / R11 / AD [GATE – PI – 1994]

(01) The value of the following determinant

1 4 9 4 9 16 9 16 25 is (A) 8 (B) 12 (C) – 12 (D) – 8

Question Level – 01

eE1 / T1 / K2 / L1 / V2 / R11 / AB [GATE – CS – 1997]

(01) The determinant of the matrix

6 8 1 1 0 2 4 6 0 0 4 8 0 0 0 1               (A) 11 (B) – 48 (C) 0 (D) – 24 eE1 / T1 / K2 / L1 / V1 / R11 / AA [GATE – CE – 1997]

(02) If the determinant of the matrix

1 3 2 0 5 6 2 7 8    _        is

26 then the determinant of the matrix 2 7 8 0 5 6 1 3 2    _        is (A) – 26 (B) 26 (C) 0 (D) 52

(11)

www.targate.org

Page 11

Question Level – 02

eE1 / T1 / K2 / L2 / V2 / R11 / AB [GATE – CS – 1998] (01) If  = 1 1 1 a bc b ca c ab

then which of the following is

a factor of  . (A) a + b (B) a - b (C) abc (D) a + b + c eE1 / T1 / K2 / L2 / V2 / R11 / AA [GATE – PI – 2007] (02) The determinant 1 1 1 1 1 2 1 b b b b b   equals to (A) 0 (B) 2b(b – 1) (C) 2(1 – b)(1 + 2b) (D) 3b(1 + b) eE1 / T1 / K2 / L2 / V2 / R11 / AB [GATE – PI – 2009]

(03) The value of the determinant

1 3 2 4 1 1 2 1 3 is (A) – 28 (B) – 24 (C) 32 (D) 36

---00000---1.3 Adjoint - Inverse

Question Level – 00 (Basic Problem)

(01) The inverse of 2 2 matrix 1 2

5 7       is (A) 1 7 2 5 1 3     _    (B) 1 7 2 5 1 3       (C) 1 7 2 5 1 3    _    (D) 1 7 2 5 1 3     _ _   

Question Level – 01

eE1 / T1 / K3 / L1 / V1 / R11 / AA [GATE – PI – 1994] (01) The matrix 1 4 1 5     _   

is an inverse of the matrix

5 4 1 1     _   

(A) True (B) False

Question Level – 02

(01) The inverse of the matrix S =

1 1 0 1 1 1 0 0 1            is (A) 1 0 1 0 0 0 0 1 1           (B) 0 1 1 1 1 1 1 0 1   _ _       _

(12)

Page 12

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(C) 2 2 2 2 2 2 0 2 2   _ _        (D) 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 0 0 1    _ _        eE1 / T1 / K3 / L2 / V1 / R11 / AA [GATE – CE – 1997] (02) Inverse of matrix 0 1 0 0 0 1 1 0 0           is (A) 0 0 1 1 0 0 0 1 0           (B) 1 0 0 0 0 1 0 1 0          _ (C) 1 0 0 0 1 0 0 0 1           (D) 0 0 1 0 1 0 1 0 0          _ eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – EE – 1998] (03) If A = 5 0 2 0 3 0 2 0 1           then A1 = (A) 1 0 2 0 1 / 3 0 2 0 5            (B) 5 0 2 0 1 / 3 0 2 0 1    _       _ (C) 1 / 5 0 1 / 2 0 1 / 3 0 1 / 2 0 1           (D) 1 / 5 0 1 / 2 0 1 / 3 0 1 / 2 0 1            eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – EE – 1999] (04) If A = 1 2 1 2 3 1 0 5 2              and ad(A) = 11 9 1 4 2 3 10 k 7      _ _        Then k = (A) – 5 (B) 3 (C) – 3 (D) 5 eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – PI – 2008]

(05) The inverse of matrix

0 1 0 1 0 0 0 0 1           is (A) 0 1 0 1 0 0 0 0 1           (B) 0 1 0 1 0 0 0 0 1    _         (C) 0 1 0 0 0 1 1 0 0           (D) 0 1 0 0 0 1 1 0 0     _       _ eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – ME – 2009] (06) For a matrix [M] = 3 / 4 4 / 5 3 / 5 x      . The transpose

of the matrix is equal to the inverse of the matrix,

1

[M]T [M] . The value of x is given by

(A) 4 5  (B) 3 5  (C) 3 5 (D) 4 5 eE1 / T1 / K3 / L2 / V2 / R11 / AB [GATE – CE – 2010]

(07) The inverse of the matrix 3 2

3 2 i i i i     _ _    is (A) 1 3 2 3 2 2 i i i i      _    (B) 1 3 2 3 2 12 i i i i      _   

(13)

www.targate.org

Page 13

(C) 1 3 2 3 2 14 i i i i      _    (D) 1 3 2 3 2 14 i i i i      _   _ ---00000---

1.4 Eigen Values & Eigen Vectors

Question Level – 00 (Basic Problem)

eE1 / T1 / K4 / L0 / V1 / R11 / AA [GATE – EE – 1994]

(01) The Eigen values of the matrix 1

1 a a       are (A) (a 1),0 (B) , 0a (C) (a 1),0 _{(D) 0, 0} eE1 / T1 / K4 / L0 / V1 / R11 / AA [GATE – EE – 1998] (02) A = 2 0 0 1 0 1 0 0 0 0 3 0 1 0 0 4              

the sum of the Eigen

Values of the matrix A is

(A) 10 (B) – 10

(C) 24 (D) 22

eE1 / T1 / K4 / L0 / V1 / R11 / AB [GATE – ME – 2004]

(03) The sum of the eigen values of the matrix given

below is 1 1 3 1 5 1 3 1 1           (A) 5 (B) 7 (C) 9 (D) 18 eE1 / T1 / K4 / L0 / V1 / R11 / AC [GATE – CE – 2004]

(04) The eigen values of the matrix 4 2

2 1    _    are (A) 1, 4 (B) – 1, 2 (C) 0, 5 (D) cannot be determined eE1 / T1 / K4 / L0 / V1 / R11 / AB [GATE – CS – 2005]

(05) What are the Eigen values of the following 2 x 2

matrix? 2 1 4 5    _    (A) – 1, 1 (B) 1, 6 (C) 2, 5 (D) 4, -1 eE1 / T1 / K4 / L0 / V1 / R11 / AC [GATE – PI – 2005]

(06) The Eigen values of the matrix M given below

are 15, 3 and 0. M = 8 6 2 6 7 4 2 4 3    _ _         , the value of

the determinant of a matrix is

(A) 20 (B) 10

(C) 0 (D) – 10

eE1 / T1 / K4 / L0 / V1 / R11 / A [GATE – CS – 2008]

(07) How many of the following matrices have an

Eigen value 1? 1 0 0 1 1 1 1 0 , , & 0 0 0 0 1 1 0 1                  _         

(14)

Page 14

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(A) One (B) Two

(C) Three (D) Four

eE1 / T1 / K4 / L0 / V1 / R11 / AC [GATE – EE – 2009]

(08) The trace and determinant of a 2x2 matrix are

shown to be -2 and -35 respectively. Its eigen values are (A) -30, -5 (B) -37, -1 (C) -7, 5 (D) 17.5, -2 ---00000---

Question Level – 01

eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – –1993]

(01) The eigen vector (s) of the matrix

0 0 0 0 0 , 0 0 0 0 α α     _       Is (are); (A)



0, 0, α



(B)



α, 0, 0



(C)



0, 0,1



(D)



0, , 0α



(02) The eigen values of

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 eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – EC – 1998]

(03) The eigen values of the matrix A = 0 1

1 0       are (A) 1, 1 (B) -1, -1 (C) ,j  j (D) 1, 1_ eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – CE – 2001 ]

(04) The eigen values of the matrix 5 3

2 9       are (A) (5.13,9.42) (B) (3.85,2.93) (C) (9.00,5.00) (D) (10.16,3.84) eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – CE – 2002]

(05) Eigen values of the following matrix are

1 4 4 1     _    (A) 3, -5 (B) -3, 5 (C) -3, -5 (D) 3, 5 eE1 / T1 / K4 / L1 / V1 / R11 / A [GATE – IN – 2005]

(06) Identify which one of the following is an eigen

vector of the matrix A = 1 0

1 2   _ _    (A)

_

1 1

_

T (B)

_

3 1

_

T (C)

_

1 1

_

T (D)

_

2 1

_

T

(15)

www.targate.org

Page 15

(07) The minimum and maximum Eigen values of

Matrix 1 1 3 1 5 1 3 1 1          

are -2 and 6 respectively.

What is the other Eigen value?

(A) 5 (B) 3

(C) 1 (D) -1

eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – EC – 2008]

(08) All the four entries of 2 x 2 matrix P =

11 12 21 22 p p p p      

are non-zero and one of the Eigen values is zero. Which of the following statement is true?

(A) P P_{11 22}P P_{12 21}1 (B) P P_{11 22}P P_{12 21} 1

(C) P P_{11 22}P P_{21 12}0 (D) P P_{11 22}P P_{12 21}0

(09) The eigen values of the matrix [P] = 4 5

2 5    _   

are

(A) – 7 and 8 (B) – 6 and 5

(C) 3 and 4 (D) 1 and2

eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – ME – 2010]

(10) One of the eigen vector of the matrix A =

2 2 1 3       is (A) 2 1   _    (B) 2 1       (C) 4 1       (D) 1 1   _    eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – CS – 2010]

(11) Consider the following matrix A = 2 3 .

x y

     

If the eigen values of A are 4 and 8 then

(A) x = 4, y = 10 (B) x = 5, y = 8

(C) x = -3, y = 9 (D) x = -4, y = 10

(12) Obtain the eigen values of the matrix A =

1 2 34 49 0 2 43 94 0 0 2 104 0 0 0 1               (A) 1,2,-2,-1 (B) -1,-2,-1,-2 (C) 1,2,2,1 (D) None ---00000---

Question Level – 02

eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – EE – 1998] (01) The vector 1 2 1           is an eigen vector of A =

(16)

Page 16

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

2 2 3 2 1 6 1 2 0      _        

one of the eigen value of A is

(A) 1 (B) 2

(C) 5 (D) -1

eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – EC – 2000]

(02) The eigen values of the matrix

2 1 0 0 0 3 0 0 0 0 2 0 0 0 1 4                are (A) 2, -2, 1, -1 (B) 2, 3, -2, 4 (C) 2, 3, 1, 4 (D) None eE1 / T1 / K4 / L2 / V2 / R11 / AD [GATE – EE – 2005]

(03) For the matrix P =

3 2 2 0 2 1 0 0 1     _       

, one of the Eigen

A value is – 2. Which of the following is an Eigen vector? (A) 3 2 1            (B) 3 2 1           _ (C) 1 2 3   _        (D) 2 5 0           eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – ME – 2006]

(04) Eigen values of a matrix S = 3 2

2 3      

are 5 and 1. What are the Eigen values of the matrix S2 = SS?

(A) 1 and 25 (B) 6, 4

(C) 5, 1 (D) 2, 10

(05) The number of linearly independent eigen vectors

of 2 1 0 2       is (A) 0 (B) 1 (C) 2 (D) Infinite eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – ME – 2008] (06) The matrix 1 2 4 3 0 6 1 1 p          

has one eigen value to 3.

The sum of the other two eigen values is

(A) p (B) p – 1

(C) p – 2 (D) p – 3

(07) The eigen vectors of the matrix 1 2

0 2       are

written in the form 1 & 1

a b

           

(17)

www.targate.org

Page 17

(A) 0 (B) 1/2

(C) 1 (D) 2

(08) The eigen vector pair of the matrix 3 4

4 3    _    is (A) 2 1 1 2       _      (B) 2 1 1 2             (C) 2 1 1 2        _      (D) 2 1 1 2              eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – EC – 2006]

(09) For the matrix 4 2 .

2 4      

The eigen value

corresponding to the eigen vector 101

101       is (A) 2 (B) 4 (C) 6 (D) 8 eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – CE – 2006]

(10) For a given matrix A =

2 2 3 2 1 6 1 2 0    _ _        , one of the

eigen value is 3. The other two eigen values are

(A) 2, -5 (B) 3, -5 (C) 2, 5 (D) 3, 5 eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – EE – 2010] (11) An eigen vector of p = 1 1 0 0 2 2 0 0 3           is (A)

_

1 1 1

_

T (B)

_

1 2 1

_

T (C)



1 1 2



T (D)



2 1 1



T eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – PI – 2011]

(12) The Eigen values of the following matrix

10 4 18 12     _    are (A) 4, 9 (B) 6, - 8 (C) 4, 8 (D) – 6, 8 eE1 / T1 / K4 / L2 / V2 / R11 / AD [GATE – EC – 2009]

(13) The eigen values of the following matrix

1 3 5 3 1 6 0 0 3    _ _        are (A) 3, 3 5 ,6 j j (B)  6 5 ,3j j,3j (C) 3j,3j,5j (D) 3, 1 3 , 1 3  j   j eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – IN – 2011] (14) The matrix M = 2 2 3 2 1 6 1 2 0             

has eigen values

-3, -3, 5. An eigen vector corresponding to the eigen value 5 is

_

1 2 1 .

_

T One of the eigen vector of the matrix M3 is

(18)

Page 18

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(A)

_

1 8 1

_

T (B)

_

1 2 1

_

T (C) _₁ 3₂ _₁_T   (D)



1 1 1



T  eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – CS – 2011]

(15) Consider the matrix as given below

1 2 3 0 4 7 0 0 3           .

Which one of the following options provides the correct values of the eigen values of the matrix?

(A) 1, 4, 3 (B) 3, 7, 3

(C) 7, 3, 2 (D) 1, 2, 3

(16) If {1,0, 1} T is an eigen vector of the following

matrix 1 1 0 1 2 1 0 1 1    _ _        

then the corresponding

eigen value is

(A) 1 (B) 2

(C) 3 (D) 5

eE1 / T1 / K4 / L2 / V3 / R11 / AC [GATE – IN – 2009]

(17) The eigen values of a 2 2 matrix X are 2 and -3. The eigen values of matrix (X I) (1 X 5 )I

are (A) – 3, - 4 (B) -1, -2 (C) -1, -3 (D) -2, -4

---00000---Question Level – 03

eE1 / T1 / K4 / L3 / V2 / R11 / AB [GATE – PI – 2007]

(01) If A is square symmetric real valued matrix of

dimension 2n, then the eigen values of A are

(A) 2n distinct real values

(B) 2n real values not necessarily distinct

(C) n distinct pairs of complex conjugate

numbers

(D) n pairs of complex conjugate numbers, not

necessarily distinct

eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – EC – 2006]

(02) The eigen values and the corresponding eigen

vectors of a 2x2 matrix are given by

Eigen Value Eigen Vector

1 8 λ  1 1 1 V _{  }    2 4 λ  2 1 1 V _{  }     The matrix is (A) 6 2 2 6       (B) 4 6 6 4       (C) 2 4 4 2       (D) 4 8 8 4       eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – ME – 2005]

(19)

www.targate.org

Page 19

the matrix 5 0 0 0 0 5 0 0 0 0 2 1 0 0 3 1             is (A)

_

1 2 0 0

_

T (B)

_

0 0 1 0

_

T (C)

_

1 0 0 2

_

T (D)

_

1 1 2 1

_

T eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – IN – 2010]

(04) A real nxn matrix A = _aij_ is defined as follows

, 0, ij a i i j otherwise         

The sum of all n eigen values of A is

(A) ( 1) 2 n n  (B) ( 1) 2 n n  (C) ( 1)(2 1) 2 n n n (D) n2 eE1 / T1 / K4 / L3 / V2 / R11 / A [GATE – EE – 2011]

(05) The two vectors

_

1 1 1

_

and 1 a a2

 

where 1 3

2 2

a   j and j   are 1

(A) Orthonormal (B) Orthogonal

(C) Parallel (D) Collinear

(06) q q q₁, ₂, ₃,...q_m are n-dimensional vectors with m < n. This set of vectors is linearly dependent. Q is the matrix with q q q1, 2, 3,...qm as the

columns. The rank of Q is

(A) Less than m (B) m

(C) Between m and n (D) n

(07) X =

_

x x_{1 2}...x_n

_

T is an n – tuple non zero vector. The n x n matrix V = XXT

(A) has rank zero (B) has rank 1

(C) is orthogonal (D) has rank n

---00000---1.5 Rank

Question Level – 00 (Basic Problem)

eE1 / T1 / K5 / L0 / V1 / R11 / AA [GATE – EC – 1994]

(01) The rank of (m x n) matrix (m < n) cannot be

more than

(A) m (B) n

(C) mn (D) None

eE1 / T1 / K5 / L0 / V1 / R11 / AC [GATE – CS – 2002]

(02) The rank of the matrix 1 1

0 0       is (A) 4 (B) 2 (C) 1 (D) 0 eE1 / T1 / K5 / L0 / V1 / R11 / AC [GATE – EE – 1994]

(03) A 5x7 matrix has all its entries equal to -1. Then

the rank of a matrix is

(A) 7 (B) 5

(20)

Page 20

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

Question Level – 01

(01) The number of Linearly independent solutions of

the system of equations

1 0 2 1 1 0 2 2 0    _         1 2 3 x x x           =0 is equal to (A) 1 (B) 2 (C) 3 (D) 0 eE1 / T1 / K5 / L1 / V1 / R11 / AC [GATE – CS – 1994]

(02) The rank of matrix

0 0 3 9 3 5 3 1 1            is (A) 0 (B) 1 (C) 2 (D) 3 eE1 / T1 / K5 / L1 / V1 / R11 / AB [GATE – EC –]

(03) Rank of the matrix

0 2 2 7 4 8 7 0 4            is 3

(A) True (B) False

eE1 / T1 / K5 / L1 / V1 / R11 / AC [GATE – IN – 2000]

(04) The rank of matrix A =

1 2 3 3 4 5 4 6 8           is (A) 0 (B) 1 (C) 2 (D) 3 eE1 / T1 / K5 / L1 / V1 / R11 / AA [GATE – EE – 1995]

(05) The rank of the following (n+1) x (n+1) matrix,

where ‘a’ is a real number is

2 2 2 1 . . . 1 . . . . . 1 . . . n n n a a a a a a a a a                 (A) 1 (B) 2

(C) n (D) depends on the value of a

----00000---Question Level – 02

(01) The rank of the matrix

1 4 8 7 0 0 3 0 4 2 3 1 3 12 24 2             is (A) 3 (B) 1 (C) 2 (D) 4 eE1 / T1 / K5 / L2 / V2 / R11 / AC [GATE – EC – 2006]

(02) The rank of the matrix

1 1 1 1 1 0 1 1 1    _        is (A) 0 (B) 1 (C) 2 (D) 3

(21)

www.targate.org

Page 21

Question Level – 03

(01) Given matrix [A] =

4 2 1 3 6 3 4 7 2 1 0 1           , the rank of the matrix is (A) 4 (B) 3 (C) 2 (D) 1 eE1 / T1 / K5 / L3 / V2 / R11 / AB [GATE – IN – 2007]

(02) Let A = [aij],1i j,  with n n  and 3 aij i j. .

Then the rank of A is

(A) 0 (B) 1

(C) n – 1 (D) n

(03) If the rank of a 5x6 matrix Q is 4 then which one

of the following statements is correct?

(A) Q will have four linearly independent rows

and four linearly independent columns

(B) Q will have four linearly independent rows

and five linearly independent columns

(C) QQT will be invertible.

(D) QT Q will be invertible.

---00000---1.6 Solution of Linear Equation

Question Level – 01

(01) Solve the following system

1 2 3 3 x x x  1 3 0 x x  1 2 3 1 x x x 

(A) Unique solution

(B) No solution

(C) Infinite number of solutions

(D) Only one solution

(02) In the Gauss – elimination for a solving system of

linear algebraic equations, triangularization leads to

(A) diagonal matrix

(B) lower triangular matrix

(C) upper triangular matrix

(D) singular matrix

eE1 / T1 / K6 / L1 / V1 / R11 / AB [GATE – IN – 2005]

(03) Let A be 3 3 matrix with rank 2. Then AX = O has

(A) Only the trivial solution X = 0

(22)

Page 22

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(C) Two independent solutions

(D) Three independent solutions

eE1 / T1 / K6 / L1 / V1 / R11 / A [GATE – CS – 2004]

(04) How many solutions does the following system

of linear equations have

5 1 x y     2 x y  3 3 x y

(A) Infinitely many

(B) Two distinct solutions

(C) Unique

(D) None

eE1 / T1 / K6 / L1 / V1 / R11 / AC [GATE – EC –]

(05) The value of q for which the following set of

linear equations 2x + 3y = 0, 6x + qy = 0 can have non-trival solution is

(A) 2 (B) 7

(C) 9 (D) 11

---00000---

Question Level – 02

(01) A system of equations represented by AX = 0

where X is a column vector of unknown and A is a matrix containing coefficient has a non-trivial solution when A is.

(A) non-singular (B) singular

(C) symmetric (D) Hermitian

(02) Consider the following set of equations

2 5,

x y 4x8y12, 3x6y3z15. This set

(A) has unique solution

(B) has no solution

(C) has infinite number of solutions

(D) has 3 solutions

(03) Consider the following system of equations in

three real variable x x and x₁, ₂ ₃:

1 2 3 2x x 3x 1 1 2 3 3x 2x 5x 2 1 4 2 3 3 x x x    

This system of equations has

(A) No solution

(23)

www.targate.org

Page 23

(C) More than one but a finite number of

solutions.

(D) An infinite number of solutions.

(04) Consider a non-homogeneous system of linear

equations represents mathematically an over determined system. Such a system will be

(A) Consistent having a unique solution (B) Consistent having many solutions. (C) Inconsistent having a unique solution. (D) Inconsistent having no solution.

(05) In the matrix equation PX = Q which of the

following is a necessary condition for the existence of at least one solution one solution for the unknown vector X.

(A) Augmented matrix [P|Q] must have the same

rank as matrix P.

(B) Vector Q must have only non-zero elements.

(C) Matrix P must be singular

(D) Matrix p must be square

(06) A is a 3 4 matrix and AX = B is an inconsistent system of equations. The highest possible rank of A is

(A) 1 (B) 2

(C) 3 (D) 4

eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – IN – 2006]

(07) A system of linear simultaneous equations is

given as AX = b Where A = 1 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1             & b = 0 0 0 1             Then the rank of matrix A is

(A) 1 (B) 2

(C) 3 (D) 4

eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC –]

(08) A system of linear simultaneous equations is

given as Axb Where A = 1 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1             & b = 0 0 0 1            

Which of the following statement is true?

(A) x is a null vector

(B) x is unique

(C) x does not exist

(D) x has infinitely many values

(09) Solution for the system defined by the set of

equations 4y3z8,2x z 2 & 3x2y5

is

(A) x0,y1,z4 / 5

(24)

Page 24

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(C) x1,y1/ 2,z2

(D) Non existent

(10) For what values of α and β the following simultaneous equations have an infinite number of solutions x  y z 5, x3y3z9, 2 x y αz = β (A) 2, 7 (B) 3, 8 (C) 8, 3 (D) 7, 2 eE1 / T1 / K6 / L2 / V2 / R11 / A [GATE – CE – 2008]

(11) The following system of equations x  y z 3, 2 3 4,

x y z x4y k 6 will not have a unique solution for k equal to

(A) 0 (B) 5

(C) 6 (D) 7

(12) For the set of equations x₁2x₂x₃4x₄2,

1 2 3 4

3x 6x 3x 12x 6. The following statement is true

(A) Only the trivial solution x1x2x3x40

exist

(B) There are no solutions

(C) A unique non-trivial solution exist

(D) Multiple non-trivial solution exist

eE1 / T1 / K6 / L2 / V2 / R11 / A [GATE – IN – 2010]

(13) X and Y are non-zero square matrices of size

nxn. If XY = Onxn then (A) |X | 0 and | | 0Y  (B) |X | 0 and | | 0Y  (C) |X | 0 and | | 0Y  (D) |X | 0 and | | 0Y  eE1 / T1 / K6 / L2 / V2 / R11 / AC [GATE – ME – 2011]

(14) Consider the following system of equations

1 2 3 2 3

2x x x 0,x x 0 and x1x20.

This system has

(A) A unique solution

(B) No solution

(C) Infinite number of solution

(D) Five solutions

(15) The system of linear equations 4 2 7

2 6 x y x y       _ has

(A) A unique solution

(B) No solution

(C) An infinite no. of solution

(25)

www.targate.org

Page 25

eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC –]

(16) The value of x3 obtained by solving the following

system of linear equations is

1 2 2 2 3 4 x  x  x  1 2 3 2x x x  2 1 2 3 2 x x x     (A) – 12 (B) - 2 (C) 0 (D) 12 eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC – 2011]

(17) The system of equations x  y z 6 ,

4 6 20,

x y z and x4yλzμ has no solution for values of λ and μ given by

(A) λ6,μ20 (B) λ6,μ20

(C) λ6,μ =20 (D) λ6,μ20

eE1 / T1 / K6 / L2 / V2 / R11 / AC [GATE – EC –]

(18) For the following set of simultaneous equations

1.5x0.5y z 2 4x2y3z0 7x y 5z10

(A) the solution is unique

(B) infinitely many solutions exist

(C) the equations are incompatible

(D) finite many solutions exist

(19) In the solution of the following set of linear

equations by Gauss-elimination using partial pivoting 5x y 2z34, 4y3z12 and

10x2y z  4. The pivots for elimination of x and y are

(A) 10 and 4 (B) 10 and 2

(C) 5 and 4 (D) 5 and – 4

Question Level – 03

(01) Let AX = B be a system of linear equations

where A is an mn matrix B is an m  column 1 matrix which of the following is false?

(A) The system has a solution, if ρ A( )ρ A B( / )

(B) If m = n and B is a non – zero vector then the

system has a unique solution

(C) If m < n and B is a zero vector then the

system has infinitely many solutions.

(D) The system will have a trivial solution when

m = n , B is the zero vector and rank of A is n.

(02) A set of linear equations is represented by the

matrix equations Ax = b. The necessary condition for the existence of a solution for this system is

(A) must be invertible

(B) b must be linearly dependent on the columns

of A

(C) b must be linearly independent on the

(26)

Page 26

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(D) None

eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – IN – 2007]

(03) Let A be an n x n real matrix such that A2 = I and Y be an n-dimensional vector. Then the linear system of equations Ax = Y has

(A) No solution

(B) unique solution

(C) More than one but infinitely many dependent

solutions.

(D) Infinitely many dependent solutions

(04) Let x and y be two vectors in a 3 – dimensional space and x y,  denote their dot product. Then the determinant det , ,

, , x x x y y x y y       _ _{ } _  =_____

(A) Is zero when x and y are linearly independent

(B) Is positive when x and y are linearly

independent

(C) Is non-zero for all non-zero x and y

(D) Is zero only when either x(or) y is zero

(05) For what values of ‘a’ if any will the following

system of equations in x, y are z have a solution?

2x3y4,x  y z 4,x2y z a

(A) Any real number

(B) 0

(C) 1

(D) There is no such value

1.7 Miscellaneous

Question Level – 00 (Basic Problem)

(01) Let A, B,C, D be

n n



matrices, each with non-zero determinant. ABCD = I then B1 =

(A) D C A1 1 1 (B) CDA

(C) ABC (D) Does not exist

(02) If A and B are two matrices and if AB exist then

BA exists.

(A) Only if A has as many rows as B has

columns

(B) Only if both A and B are square matrices

(C) Only if A and B are skew matrices

(D) Only if both A and B are symmetric

---00000---Question Level – 01

eE1 / T1 / K7 / L1 / V1 / R11 / AC [GATE – CS – 1997]

(01) Let Anxn be matrix of order n and I12 be the matrix

obtained by interchanging the first.

(A) Row is the same as its second row

(27)

www.targate.org

Page 27

(C) column is the same as the second column of

(D) Row is a zero row.

(02) If A is any

n n



matrix and k is a scalar then

|kA|α A| | where α is

(A) kn (B) nk

(C) kn (D) k

n

(03) The number of terms in the expansion of general

determinant of order n is

(A) n2 (B) !n

(C) n (D) (n 1)2

(04) The determinant of the following matrix

5 3 2 1 2 6 3 5 10           (A) – 76 (B) – 28 (C) 28 (D) 72 eE1 / T1 / K7 / L1 / V1 / R11 / AA [GATE – CE – 2001]

(05) The product [P] [Q]T of the following two matrices [P] and [Q] is where [P] = 2 3 ,

4 5       4 8 [ ] 9 2 Q _{ } _   (A) 32 24 56 46       (B) 46 56 24 32       (C) 35 22 61 42       (D) 32 56 24 46       eE1 / T1 / K7 / L1 / V1 / R11 / AC [GATE – CS – 2004]

(06) The number of different

n n



symmetric matrices with each elements being either 0 or 1 is

(A) 2n (B) 2 2n (C) 2 2 2 n n (D) 2 2 2 n n eE1 / T1 / K7 / L1 / V1 / R11 / AD [GATE – EC – 2005]

(07) Given the matrix 4 2 ,

4 3       

the eigen vector is

(A) 3 2       (B) 4 3       (C) 2 1   _    (D) 2 1       

Question Level – 02

eE1 / T1 / K7 / L2 / V1 / R11 / A [GATE – PI – 1994]

(01) For the following matrix 1 1

2 3        the number of positive roots is

(A) One (B) Two

(C) Four (D) Cannot be found

(28)

Page 28

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(02) Given matrix L = 2 1 3 2 4 5           and M = 3 2 0 1       then L x M is (A) 8 1 13 2 22 5           (B) 6 5 9 8 12 13           (C) 1 8 2 13 5 22           (D) 6 2 9 4 0 5           eE1 / T1 / K7 / L2 / V2 / R11 / AC [GATE – ME – 1995] (03) Among the following, the pair of the vector

orthogonal to each other is

(A)



3, 4, 7 , 3,

 

4, 7



(B)



1, 0, 0 , 1, 1,

 

0



(C)



1, 0, 2 , 0,

 

5, 0



(D)



1, 1, 1 ,

 

1, 1, 1



eE1 / T1 / K7 / L2 / V2 / R11 / AC [GATE – EE – 2002]

(04) The determinant of the matrix

1 0 0 0 100 1 0 0 100 200 1 0 100 200 300 1             is (A) 100 (B 200 (C) 1 (D) 300 eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – CS – 2000]

(05) The determinant of the matrix

2 0 0 0 8 1 7 2 2 0 2 0 9 0 6 1             is (A) 4 (B) 0 (C) 15 (D 20 eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – CE – 2005]

(06) Consider the matrices X4x3, Y4x3, and P2x3. The

order of 1 ( T ) T T P X Y  P     will be (A) 2x2 (B) 3x3 (C) 4x3 (D) 3x4 eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – EC –2005]

(07) The determinant of the matrix given below is

0 1 0 2 1 1 1 3 0 0 0 1 1 2 0 1   _           (A) -1 (B) 0 (C) 1 (D) 2 eE1 / T1 / K7 / L2 / V2 / R11 / AB [GATE – EE – 2005] (08) If R = 1 0 1 2 1 1 2 3 2     _       

then the top row of R1 is

(A)

_

5 6 4

_

(B)

_

5 3 1

_

(C)



2 0 1



(D)



2 1 0



eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – EE – 2005] (09) If A = 2 0.1 0 3        and 1 1 / 2 0 a A b         then __________ a b 

(29)

www.targate.org

Page 29

(A) 7 20 (B) 3 20 (C) 19 60 (D) 11 20 eE1 / T1 / K7 / L2 / V2 / R11 / AC [GATE – IN – 2006]

(10) For a given 2x2 matrix A, it is observed that

1 1 1 1 1 A_ _  _ _       and 1 2 A_ _    and 1 1 2 2 2 A_ _  _ _      

then the matrix A is

(A) 2 1 1 0 1 1 1 1 0 2 1 2 A_{ }  _{ }  _{ } _            (B) 1 1 1 0 2 1 1 2 1 2 1 1 A_{ }  _{ }  _{ } _            (C) 1 1 1 0 2 1 1 2 0 2 1 1 A_{ }  _{ }  _{ } _            (D) 0 2 1 3 A_{ }  _    eE1 / T1 / K7 / L2 / V2 / R11 / AD [GATE – ME – 2011] (11) If a matrix A = 2 4 1 3       and matrix B = 4 6 5 9      

the transpose of product of these two matrices i.e., (AB)T is equal to (A) 28 19 34 47       (B) 19 34 47 28       (C) 48 33 28 19       (D) 28 19 48 33       eE1 / T1 / K7 / L2 / V2 / R11 / AB [GATE – EE – 2011]

(12) The matrix [A] = 2 1

4 1

 

 _ 

 

is decomposed into a product of lower triangular matrix [L] and an upper triangular [U]. The property decomposed [L] and [U] matrices respectively are

(A) 1 0 4 1    _    and 1 1 0 2    _    (B) 1 0 2 1       and 2 1 0 3    _    (C) 1 0 4 1       and 2 1 0 1    _    (D) 2 0 4 3    _    and 1 0.5 0 1      _

---00000---Question Level – 03

(01) The matrices cos sin

sin cos θ θ θ θ        and 0 0 a b      

commute under multiplication.

(A) If a = b (or) θnπ, n is an integer

(B) Always

(C) never

(D) If a cosθbsinθ

(30)

Page 30

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

(02) The equation 2 2 1 1 1 1 1 0 y x x   represents a

parabola passing through the points.

(A) (0,1), (0,2),(0,-1) (B) (0,0), (-1,1),(1,2)

(C) (1,1), (0,0), (2,2) (D) (1,2), (2,1), (0,0)

eE1 / T1 / K7 / L3 / V2 / R11 / AC [GATE – EC –2004]

(03) What values of x, y, z satisfy the following

system of linear equations

1 2 3 6 1 3 4 8 2 2 3 12 x y z          _                    (A) x = 6, y = 3, z = 2 (B) x = 12, y = 3, z = -4 (C) x = 6, y= 6, z = -4 (D) x = 12, y = -3, z = 4 eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – EC –2004] (04) If matrix X = 2 1 1 1 a a a a           and 2 0.

X X I Then the inverse of X is

(A) 1 ₂a 1 a a         (B) ₂1 1 1 a a a a           (C) ₂ 1 1 1 a a a a            (D) 2 ₁ 1 1 a a a a         _ eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – CE – 2005]

(05) Consider the system of equations,

1 1

n n n n

A_ X _ λX _ where λ is a scalar. Let



λ X_i, _i



be an eigen value and its corresponding eigen vector for real matrix A. Let Inxn be unit

matrix. Which one of the following statement is not correct?

(A) For a homogeneous nxn system of linear

equations (A- λ I) is less than n.

(B) For matrix Am, m being a positive integer, ( ,

m i

λ X_im) will be eigen pair for all i.

(C) If AT A1 then |λ _i| 1 for all i.

(D) If AT A then λ_i are real for all i.

(06) Multiplication of matrices E and F is G. Matrices

E and G are E = cos sin 0 sin cos 0 0 0 1 θ θ θ θ            and G = 1 0 0 0 1 0 0 0 1          

. What is the matrix F?

(A) cos sin 0 sin cos 0 0 0 1 θ θ θ θ            (B) cos cos 0 cos sin 0 0 0 1 θ θ θ θ   _        (C) cos sin 0 sin cos 0 0 0 1 θ θ θ θ   _        (D) sin cos 0 cos sin 0 0 0 1 θ θ θ θ           _

(31)

www.targate.org

Page 31

(07) An

n n



array V is defined as follows V[i,j] =

i j for all i, j, 1i j, n then the sum of the elements of the array V is

(A) 0 (B) n – 1 (C) n23n2 (D) n n ( 1)

1.8 CALY- HAMILTON

Question Level – 01

eE1 / T1 / K8 / L1 / V1 / R11 / AC [GATE – CE – 2007] (01) If A = 3 2 1 0    _   

then A satisfies the relation

(A) A + 3I + 2A1 = O (B) A22A2IO (C) (A I A )( 2 )I O (D) eAO eE1 / T1 / K8 / L1 / V1 / R11 / AA [GATE – EE – 2007] (02) If A = 3 2 1 0    _    then 9 A equals (A) 511 A + 510 I (B) 309 A + 104 I (C) 154 A + 155 I (D) e9 A

Question Level – 02

(01) The characteristic equation of a 3x3 matrix P is

defined as

3 2

( ) | | 2 1 0.

α λ  λIP λ  λλ  

If I denotes identity matrix then the inverse of P will be

(A) P2P2I (B) P2P I

(C) (P2PI) (D) (P2 P2 )I