GATE QUESTION BANK
for
Computer Science
&
Information Technology
By
Contents
Subject Name
Topic Name
Page No.
#1. Mathematics
1-112
1 Linear Algebra 1 – 28
2 Probability & Distribution 29 – 57
3 Numerical Methods 58 – 73
4 Calculus 74 – 112
#2. Data Structure and Algorithm
113 – 177
5 Data Structure and Algorithm Analysis 113 – 136
6 Stacks and Queues 137 – 140
7 Trees 141 – 148
8 Height Balanced Trees (AVL Trees, B and ) 149 – 150
9 Priority Queues (Heaps) 151 – 156
10 Sorting Algorithms 157 – 160
11 Graph Algorithms 161 – 174
12 Hashing 175 – 177
#3. Operating System
178 – 210
13 Introduction to Operating System 178
14 Process Management 179 – 186
15 Threads 187
16 CPU Scheduling 188 – 193
17 Deadlocks 194 – 199
18 Memory Management & Virtual Memory 200 – 206
19 File System 207 – 208
20 I/O System 209 – 210
#4. Theory of Computation
211 – 235
21 Finite Automata 211 – 216
22 Regular Expression 217 – 223
23 Context Free grammar 224 – 229
24 Turing Machines 230 – 235
#5. Computer Organization & Architecture
236 – 263
25 Introduction to Computer Organization 236
26 Memory Hierarchy 237 – 246
27 Pipeline 247 – 252
28 Instruction Types 253 – 258
29 I/O Data Transfer 259 – 263
#6. Digital Logic
264 – 289
30 Number Systems & Code Conversions 264 – 268
32 Logic Gates 276 – 279
33 Logic Gate Families 280
34 Combinational and Sequential Digital Circuits 281 – 289
#7. Discrete Mathematics & Graph Theory
290 – 322
35 Mathematical Logic 290 – 297
36 Combinatorics 298 - 300
37 Sets and Relations 301 – 313
38 Graph Theory 314 – 322
#8. Database Management System
323 – 361
39 ER Diagrams 323 – 324
40 Functional Dependencies & Normalization 325 – 330
41 Relational Algebra & Relational Calculus 331 – 337
42 SQL 338 – 351
43 Transactions and Concurrency Control 352 – 357
44 File Structures (Sequential files, Indexing,
B and trees)
358 – 361
#9. Computer Networks
362 – 396
45 Introduction to Computer Networks 362 – 363
46 Medium Access Sublayer
(LAN Technologies: Ethernet, Token Ring)
364 - 367
47 The Data Link Layer (Flow and Error Control
Techniques)
368 – 373
48 Routing & Congestion Control 374 – 379
49 TCP/IP, UDP and Sockets, IP(V4) 380 – 390
50 Application Layer 391 – 393
51 Network Security 394 – 396
#10. Compiler Design
397 – 409
52 Introduction to Compilers 397
53 Syntax Analysis 398 – 403
54 Syntax Directed Translation 404 – 405
55 Intermediate Code Generation 406 – 409
#11. Software Engineering and Web Technology
410 – 412
56 Introduction to Software and Software
Engineering
410 – 411
57 Process Modeling 412 – 413
58 Project Management 414 – 415
59 Validation and Verification 416 – 418
60 HTML Structure 419 – 420
Linear Algebra
ME – 20051. Which one of the following is an
Eigenvector of the matrix[ ]? (A) [ ] (B) [ ] (C) [ ] (D) [ ]
2. A is a 3 4 real 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 ME – 2006
3. Multiplication of matrices E and F is G.
Matrices E and G are E [ os sin sin os ] and G [
]. What is the matrix F? (A) [ os sin sin os ] (B) [ sin os os sin ] (C) [ os sin sin os ] (D) [ sin os os sin ]
4. Eigen values of a matrix
S 0
1are 5 and 1. What are the
Eigenvalues of the matrix = SS?
(A) 1 and 25 (B) 6 and 4
(C) 5 and 1 (D) 2 and 10
5. Match the items in columns I and II.
Column I Column II P. Singular matrix 1. Determinant is not defined Q. Non-square matrix 2. Determinant is always one R. Real symmetric matrix 3. Determinant is zero S. Orthogonal matrix
4. Eigen values are always real 5. Eigen values are
not defined (A) P - 3 Q - 1 R - 4 S - 2 (B) P - 2 Q - 3 R - 4 S - 1 (C) P - 3 Q - 2 R - 5 S - 4 (D) P - 3 Q - 4 R - 2 S - 1 ME – 2007
6. The number of linearly independent
Eigenvectors of 0 1 is (A) 0 (B) 1 (C) 2 (D) Infinite
7. If a square matrix A is real and symmetric,
then the Eigenvalues (A) are always real
(B) are always real and positive (C) are always real and non-negative (D) occur in complex conjugate pairs
ME – 2008
8. The Eigenvectors of the matrix 0
1 are
written in the form 0 1 and 0 1. What is
a + b? (A) 0 (B) 1/2 (C) 1 (D) 2 9. The matrix [ p
] has one Eigenvalue equal to 3. The sum of the other two Eigenvalues is
(A) p (B) p – 1
(C) p – 2 (D) p – 3
10. For what value of a, if any, will the following system of equations in x, y and z have a solution
x y x y z x y z (A) Any real number (B) 0
(C) 1
(D) There is no such value
ME – 2009
11. For a matrix,M- *
x +, the transpose
of the matrix is equal to the inverse of the
matrix ,M- ,M- . The value of x is
given by (A) ( ) (B) ( ⁄ ) (C) ⁄ (D) ⁄ ME – 2010
12. One of the Eigenvectors of the matrix
0 1 is (A) 2 3 (B) 2 3 (C) 2 3 (D) 2 3 ME – 2011
13. Consider the following system of
equations:
x x x
x x
x x
The system has
(A) A unique solution (B) No solution
(C) Infinite number of solutions (D) Five solutions
14. Eigen values of a real symmetric matrix
are always (A) Positive (B) Real (C) Negative (D) Complex ME – 2012
15. For the matrix A=0
1 , one of the
normalized Eigenvectors is given as
(A) (√ ) (B) (√ √ ) (C) (√ √ ) (D) ( √ ) 16. x + 2y + z =4 2x + y + 2z =5 x – y + z = 1
The system of algebraic equations given above has
(A) a unique algebraic equation of x = 1, y = 1 and z = 1
(B) only the two solutions of ( x = 1, y = 1, z = 1) and ( x = 2, y = 1, z = 0) (C) infinite number of solutions.
(D) No feasible solution.
ME – 2013
17. The Eigenvalues of a symmetric matrix
are all
(A) Complex with non –zero positive imaginary part.
(B) Complex with non – zero negative imaginary part.
(C) Real
(D) Pure imaginary.
18. Choose correct set of functions, which are
linearly dependent.
(A) sin x sin x n os x
(B) os x sin x n t n x
(C) os x sin x n os x
(D) os x sin x n os x
ME – 2014
19. Given that the determinant of the matrix
[ ] is 12 , the determinant of the matrix [ ] is (A) (B) (C) (D)
20. One of the Eigenvectors of the matrix 0 1 is (A) {– } (B) {– } (C) 2 3 (D) 2 3
21. Consider a 3×3 real symmetric matrix S
such that two of its Eigenvalues are with respective Eigenvectors [ x x x ] [ y y y ] If then x y + x y +x y equals (A) a (B) b (C) ab (D) 0
22. Which one of the following equations is a
correct identity for arbitrary 3×3 real matrices P, Q and R? (A) ( ) (B) ( ) (C) et ( ) et et (D) ( ) CE – 2005
1. Consider the system of equations ( )
( ) ( ) where is s l r Let
( ) e n Eigen -pair of an Eigenvalue and its corresponding Eigenvector for real matrix A. Let I be a (n × n) unit matrix. Which one of the following statement is NOT correct?
(A) For a homogeneous n × n system of
linear equations,(A ) X = 0 having
a nontrivial solution the rank of
(A ) is less than n.
(B) For matrix , m being a positive
integer, ( ) will be the Eigen
-pair for all i.
(C) If = then | | = 1 for all i.
(D) If = A then is real for all i.
2. Consider a non-homogeneous system of
linear equations representing 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
3. Consider the matrices , - , - and
, -. The order of , ( ) - will be
(A) (2 × 2) (B) (3 × 3
(C) (4 × 3) (D) (3 × 4
CE – 2006
4. Solution for the system defined by the set
of equations 4y + 3z = 8; 2x – z = 2 and 3x + 2y = 5 is (A) x = 0; y =1; z = ⁄ (B) x = 0; y = ⁄ ; z = 2 (C) x = 1; y = ⁄ ; z = 2 (D) non – existent
5. For the given matrix A = [
],
one of the Eigen values is 3. The other two Eigen values are
(A) (B)
(C) (D)
CE – 2007
6. The minimum and the maximum
Eigenvalue of the matrix [
]are 2 and 6, respectively. What is the other Eigenvalue?
(A) (B)
(C) (D)
7. For what values of and the following
simultaneous equations have an infinite of solutions? X + Y + Z = 5; X + 3Y + 3Z = 9; X + 2 Y + Z (A) 2, 7 (B) 3, 8 (C) 8, 3 (D) 7, 2
8. The inverse of the m trix 0 1 is (A) 0 1 (B) 0 1 (C) 0 1 (D) 0 1 CE – 2008
9. The product of matrices ( ) is
(A)
(B)
(C)
(D) PQ
10. The Eigenvalue of the matrix
[P] = 0 1 are (A) and 8 (B) and 5 (C) n (D) n
11. The following simultaneous equation
x + y + z = 3 x + 2y + 3z = 4 x + 4y + kz = 6
will NOT have a unique solution for k equal to (A) 0 (B) 5 (C) 6 (D) 7 CE – 2009
12. A square matrix B is skew-symmetric if
(A)
(B)
(C)
(D)
CE – 2011
13. [A] is square matrix which is neither
symmetric nor skewsymmetric and , -is its transpose. The sum and difference of these matrices are defined as
[S] = [A] + , - and [D] = [A] , - ,
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
14. The inverse of the matrix 0 i i
i i1 is ( ) 0 i i i i1 ( ) 0 i i i i1 ( ) 0 i i i i1 ( ) 0 i i i i1 CE – 2012
15. The Eigenvalues of matrix 0
1 are (A) 2.42 and 6.86 (B) 3.48 and 13.53 (C) 4.70 and 6.86 (D) 6.86 and 9.50 CE – 2013
16. There is no value of x that can
simultaneously satisfy both the given equations. Therefore, find the ‘le st squares error’ solution to the two equations, i.e., find the value of x that minimizes the sum of squares of the errors in the two equations.
2x = 3 and 4x = 1
17. What is the minimum number of
multiplications involved in computing the matrix product PQR? Matrix P has 4 rows and 2 columns, matrix Q has 2 rows and 4 columns, and matrix R has 4 rows and 1 column. __________
CE – 2014
18. Given the matrices J = [
] n K [ ], the product K JK is
19. The sum of Eigenvalues of the matrix, [M]
is, where [M] = [ ] (A) 915 (B) 1355 (C) 1640 (D) 2180
20. The determinant of matrix [ ] is ____________
21. The rank of the matrix
[
] is ________________
CS – 2005
1. Consider the following system of
equations in three real
variables x x n x
x x x
x x x
x x x
This system of equation has (A) no solution
(B) a unique solution
(C) more than one but a finite number of solutions
(D) an infinite number of solutions
2. What are the Eigenvalues of the following
2 2 matrix? 0 1 (A) n (B) n (C) n (D) n CS – 2006
3. F is an n x n real matrix. b is an n real
vector. Suppose there are two nx1
vectors, u and v such that u v , and
Fu=b, Fv=b. Which one of the following statement is false?
(A) Determinant of F is zero
(B) There are infinite number of
solutions to Fx=b
(C) There is an x 0 such that Fx=0
(D) F must have two identical rows
4. Let A be a 4x4 matrix with Eigenvalues
–5, –2, 1, 4. Which of the following is an
Eigenvalue of 0 I I 1, where I is the 4x4 identity matrix? (A) (B) (C) (D) CS – 2007
5. Consider the set of (column) vectors
defined by X={xR3 x1+x2+x3=0, where
XT =[x1, x2, x3]T }. Which of the following is
TRUE?
(A) {[1, 1, 0]T, [1, 0, 1]T} is a basis for
the subspace X.
(B) {[1, 1, 0]T, [1, 0, 1]T} is a linearly
independent set, but it does not span X and therefore, is not a basis of X.
(C) X is not the subspace for R3
(D) None of the above
CS – 2008
6. The following system of
x x x
x x x
x x x
Has unique solution. The only possible value (s) for is/ are
(A) 0
(B) either 0 or 1 (C) one of 0,1, 1
(D) any real number except 5
7. How many of the following matrices have
an Eigenvalue 1? 0 1 0 1 0 1 n 0 1 (A) One (B) two (C) three (D) four CS – 2010
8. Consider the following matrix
A = [
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
CS – 2011
9. Consider the matrix as given below
[
]
Which one of the following options provides the CORRECT values of the Eigenvalues of the matrix?
(A) 1, 4, 3
(B) 3, 7, 3
(C) 7, 3, 2
(D) 1, 2, 3
CS – 2012
10. Let A be the 2 2 matrix with elements
and .
Then the Eigenvalues of the matrix
are (A) 1024 and (B) 1024√ and √ (C) √ n √ (D) √ n √ CS – 2013
11. Which one of the following does NOT
equal [ x x y y z z ] (A) | x(x ) x y(y ) y z(z ) z | (B) | x x y y z z | (C) | x y x y y z y z z z | (D) | x y x y y z y z z z | CS – 2014
12. Consider the following system of
equations: x y x z x y z x y z
The number of solutions for this system is __________.
13. The value of the dot product of the
Eigenvectors corresponding to any pair of
different Eigenvalues of a 4-by-4
symmetric positive definite matrix is __________.
14. If the matrix A is such that
[ ] , -
Then the determinant of A is equal to __________.
15. The product of the non – zero Eigenvalues
of the matrix [ ] is __________.
16. Which one of the following statements is
TRUE about every n n matrix with only real eigenvalues?
(A) If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
(B) If the trace of the matrix is positive, all its eigenvalues are positive. (C) If the determinant of the matrix is
positive, all its eigenvalues are positive.
(D) If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
ECE – 2005
1. Given an orthogonal matrix
A = [ ] . , - is (A) [ ⁄ ⁄ ⁄ ⁄ ]
(B) [ ⁄ ⁄ ⁄ ⁄ ] (C) [ ] (D) [ ⁄ ⁄ ⁄ ⁄ ]
2. Let, A=0 1 and = 0 ⁄ 1.
Then (a + b)=
(A) ⁄
(B) ⁄
(C) ⁄
(D) ⁄
3. Given the matrix 0 1 the
Eigenvector is (A) 0 1 (B) 0 1 (C) 0 1 (D) 0 1 ECE – 2006
4. For the matrix 0
1 , the Eigenvalue
corresponding to the Eigenvector
0 1 is (A) 2 (B) 4 (C) 6 (D) 8
5. The Eigenvalues and the corresponding
Eigenvectors of a 2 2 matrix are given by Eigenvalue Eigenvector = 8 v = 0 1 = 4 v = 0 1 The matrix is (A) 0 1 (B) 0 1 (C) 0 1 (D) 0 1
6. The rank of the matrix [
] (A) 0 (B) 1 (C) 2 (D) 3 ECE – 2007
7. It is given that X1 , X2 …… M are M
non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M
vector X1 , X2 … XM , X1 , X2 … XM is
(A) 2M (B) M+1 (C) M
(D) dependent on the choice of X1 , X2 …
XM. ECE – 2008
8. The system of linear equations
4x + 2y = 7, 2x + y = 6 has (A) a unique solution (B) no solution
(C) an infinite number of solutions (D) exactly two distinct solutions
9. All the four entries of the 2 x 2 matrix
P = 0pp p
p 1 are non-zero, and one of
its Eigenvalues is zero. Which of the following statements is true?
(A) p p p p
(B) p p p p
(C) p p p p
(D) p p p p
ECE – 2009
10. The Eigen values of the following matrix
are [ ] (A) 3, 3 + 5j, 6 j (B) 6 + 5j, 3 + j, 3 j (C) 3 + j, 3 j, 5 + j (D) 3, 1 + 3j, 1 3j
ECE – 2010
11. The Eigenvalues of a skew-symmetric
matrix are (A) Always zero
(B) Always pure imaginary (C) Either zero or pure imaginary (D) Always real
ECE – 2011
12. The system of equations x y z
x y z x y z
has NO solution for values of n given by (A) (B) (C) (D) ECE\EE\IN – 2012
13. Given that A = 0 1 and I = 0 1 ,
the value of A3 is (A) 15 A + 12 I (B) 19A + 30 (C) 17 A + 15 I (D) 17A +21 ECE – 2013
14. The minimum Eigenvalue of the following
matrix is [ ] (A) 0 (B) 1 (C) 2 (D) 3
15. Let A be a m n matrix and B be a n m
matrix. It is given that
Determinant(I ) determinant
(I ) where I is the k k identity
matrix. Using the above property, the determinant of the matrix given below is [ ] (A) 2 (B) 5 (C) 8 (D) 16 ECE – 2014
16. For matrices of same dimension M, N and
scalar c, which one of these properties DOES NOT ALWAYS hold?
(A) (M ) M
(B) ( M ) (M)
(C) (M N) M N
(D) MN NM
17. A real (4 × 4) matrix A satisfies the
equation I where 𝐼 is the (4 × 4)
identity matrix. The positive Eigenvalue of A is _____.
18. Consider the matrix
J [ ]
Which is obtained by reversing the order
of the columns of the identity matrix I .
Let I J where is a
non-negative real number. The value of for which det(P) = 0 is _____.
19. The determinant of matrix A is 5 and the
determinant of matrix B is 40. The determinant of matrix AB is ________.
20. The system of linear equations
( ) 4 5 ( ) h s (A) a unique solution
(B) infinitely many solutions (C) no solution
(D) exactly two solutions
21. Which one of the following statements is
NOT true for a square matrix A?
(A) If A is upper triangular, the Eigenvalues of A are the diagonal elements of it
(B) If A is real symmetric, the Eigenvalues of A are always real and positive
(C) If A is real, the Eigenvalues of A and are always the same
(D) If all the principal minors of A are positive, all the Eigenvalues of A are also positive
22. The maximum value of the determinant
among all 2×2 real symmetric matrices with trace 14 is ___. EE – 2005 1. If R = [
] , then top row of is
(A) , - (B) , -
(C) , -
(D) , -
2. For the matrix p = [
] , one of
the Eigenvalues is equal to 2 . Which of the following is an Eigenvector?
(A) [ ] (B) [ ] (C) [ ] (D) [ ]
3. In the matrix equation Px = q, which of
the following is necessary condition for the existence of at least 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
EE – 2006
Statement for Linked Answer Questions 4 and 5. P = [ ] , Q = [ ] , R = [ ] are three vectors
4. An orthogonal set of vectors having a
span that contains P,Q, R is
(A) [ ] [ ] (B) [ ] [ ] [ ] (C) [ ] [ ] [ ] (D) [ ] [ ] [ ]
5. The following vector is linearly
dependent upon the solution to the previous problem (A) [ ] (B) [ ] (C) [ ] (D) [ ] EE – 2007 6. X = [x , x . . . . x - is an n-tuple non-zero
vector. The n n matrix V = X (A) Has rank zero
(B) Has rank 1
(C) Is orthogonal (D) Has rank n
7. The linear operation L(x) is defined by
the cross product L(x) = b x, where
b =[0 1 0- and x =[x x x - are three
dimensional vectors. The matrix M of this operation satisfies
L(x) = M [ x x
x ]
Then the Eigenvalues of M are (A) 0, +1, 1
(B) 1, 1, 1
(C) i, i, 1 (D) i, i, 0
8. Let x and y be two vectors in a 3
dimensional space and <x, y> denote their dot product. Then the determinant det 0 x x y x y y 1 x 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
Statement for Linked Questions 9 and 10. Cayley-Hamilton Theorem states that a
square matrix satisfies its own
characteristic equation. Consider a
matrix.
A = 0
1
9. A satisfies the relation
(A) A + 3 + 2 = 0 (B) A2 + 2A + 2 = 0 (C) (A+ ) (A 2) = 0 (D) exp (A) = 0 10. equals (A) 511 A + 510 (B) 309 A + 104 (C) 154 A + 155 (D) exp (9A) EE – 2008
11. If the rank of a ( ) 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) Q will be invertible
(D) Q will be invertible
12. The characteristic equation of a ( )
matrix P is defined as
() = | P| = = 0
If I denotes identity matrix, then the inverse of matrix P will be
(A) ( I)
(B) ( I)
(C) ( I)
(D) ( I)
13. A is m n full rank matrix with m > n and
is an identity matrix. Let matrix
A+ = ( ) , then, which one of the
following statements is FALSE?
(A) A A+ A = A
(B) (AA+ ) = A A+
(C) A+ A =
(D) A A+ A = A+
14. Let P be a real orthogonal matrix. x⃗
is a real vector [x x - with length
x⃗ (x x ) . Then, which one of the
following statements is correct?
(A) x⃗ x⃗ where at least one vector satisfies x⃗ x⃗
(B) x⃗ x⃗ for all vectors x⃗
(C) x⃗ x⃗ where at least one vector satisfies x⃗ x⃗
(D) No relationship can be established between x⃗ and x⃗
EE – 2009
15. The trace and determinant of a
matrix are known to be –2 and –35 respe tively It’s Eigenv lues re
(A) –30 and –5 (B) –37 and –1
(C) –7 and 5 (D) 17.5 and –2
EE – 2010
16. For the set of equations
x x x x = 2
x x x x = 6
The following statement is true (A) Only the trivial solution
x x x x = 0 exists
(B) There are no solutions
(C) A unique non-trivial solution exists (D) Multiple non-trivial solutions exist
17. An Eigenvector of [ ] is (A) , (B) , (C) , (D) , -EE – 2011 18. The matrix[A] = 0 1 is decomposed
into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are
(A) 0
(B) 0 1 and 0 1 (C) 0 1 and 0 1 (D) 0 1 and 0 1 EE – 2013 19. The equation 0 1 0 x x 1 0 1 has (A) No solution
(B) Only one solution 0xx 1 0 1.
(C) Non – zero unique solution (D) Multiple solution
20. A matrix has Eigenvalues – 1 and – 2. The
corresponding Eigenvectors are 0
1 and
0
1 respectively. The matrix is
(A) 0 1 (B) 0 1 (C) 0 1 (D) 0 1 EE – 2014
21. Given a system of equations:
x y z x y z
Which of the following is true regarding its solutions?
(A) The system has a unique solution for
any given and
(B) The system will have infinitely many
solutions for any given and
(C) Whether or not a solution exists
depends on the given and
(D) The system would have no solution
for any values of and
22. Which one of the following statements is
true for all real symmetric matrices? (A) All the eigenvalues are real. (B) All the eigenvalues are positive. (C) All the eigenvalues are distinct. (D) Sum of all the eigenvalues is zero.
23. Two matrices A and B are given below:
0pr qs1 [p q pr qs
pr qs r s ]
If the rank of matrix A is N, then the rank of matrix B is (A) N (B) N (C) N (D) N IN – 2005
1. Identify which one of the following is an
Eigenvector of the matrix A = 0
1?
(A) [ 1 1]T
(B) [3 1]T
(C) [1 1]T
(D) [ 2 1]T
2. Let A be a 3 3 matrix with rank 2. Then
AX = 0 has
(A) only the trivial solution X = 0
(B) one independent solution
(C) two independent solutions
(D) three independent solutions
IN – 2006
Statement for Linked Answer Questions 3 and 4
A system of linear simultaneous equations is given as Ax=B where
[ ] n [ ]
3. The rank of matrix A is
(A) 1 (B) 2
(C) 3 (D) 4
4. Which of the following statements is true?
(A) x is a null vector (B) x is unique (C) x does not exist
(D) x has infinitely many values
5. For a given matrix A, it is observed
that 0
1 0 1 n 0 1 0 1
(A)
A
2
1
1 0
1
1
1
1
0
2
1
2
(B)A
1
1 1 0
2
1
1
2 0 2
1
1
(C)A
1
1
1 0
2
1
1
2
0
2
1
1
(D)A
0
2
1
3
IN – 2007 6. Let A = [ ] i j n with n n= i. j. Then the rank of A is
(A) (B)
(C) n (D) n
7. Let A be an n×n real matrix such that
= I and y be an n- dimensional vector. Then the linear system of equations Ax=Y has
(A) no solution (B) a unique solution
(C) more than one but finitely many independent solutions
(D) Infinitely many independent solutions IN – 2009 8. The matrix P =[ ] rotates a vector
about the axis[ ] by an angle of
(A)
(B)
(C) (D)
9. The Eigenvalues of a (2 2) matrix X are
2 and 3. The Eigenvalues of matrix
( I) ( I) are
(A) (B)
(C) (D)
10. Let P 0 be a 3 3 real matrix. There exist
linearly independent vectors x and y such that Px = 0 and Py = 0. The dimension of the range space of P is
(A) 0 (B) 1
(C) 2 (D) 3
IN – 2010
11. X and Y are non-zero square matrices of
size n n. If then
(A) |X| = 0 and |Y| 0 (B) |X| 0 and |Y| = 0 (C) |X| = 0 and |Y| = 0 (D) |X| 0 and |Y| 0
12. A real n × n matrix A = [ ] is defined as
follows: { i i j
otherwise
The summation of all n Eigenvalues of A is (A) n(n ) (B) n(n ) (C) ( )( ) (D) n IN – 2011 13. The matrix M = [ ] has Eigenvalues . An Eigenvector corresponding to the Eigenvalue 5 is
, - . One of the Eigenvectors of
the matrix M is (A) , (B) , -(C) , √ (D) , -IN – 2013
14. The dimension of the null space of the
matrix [ ] is (A) 0 (B) 1 (C) 2 (D) 3
15. One of Eigenvectors corresponding to the
two Eigenvalues of the matrix 0 1 is
(A) [ j] 0 j
1
(B) 0 1 0 1
(C) [j] 0 1
IN – 2014
16. For the matrix A satisfying the equation
given below, the eigenvalues are , - [ ] [ ] (A) ( 𝑗,𝑗) (B) (1,1,0) (C) ( ) (D) (1,0,0)
Answer Keys and Explanations
ME 1. [Ans. A] [ ]h r teristi equ tions is | I|
( )( )( )
∴ Real eigenvalues are 5, 5 other two are complex
Eigenvector corresponding to is ( I)
(or) →( )
Verify the options which satisfies relation (1)
Option (A) satisfies.
2. [Ans. B] Given n in onsistent ( ) n ( ⁄ ) ( ( ) minimum of m n) For inconsistence ( ) ( ⁄ ) ∴ he highest possi le r nk of is 3. [Ans. C] Given , E = [ os sin sin os ] and G = [ ] Now E × F = G
∴ ,E- [ sin os os sin ]
4. [Ans. A]
For S matrix, if Eigenvalues are
… … … then for matrix,
the Eigenvalues will be , , ………
For S matrix, if Eigenvalues are 1 and 5
then for matrix, the Eigenvalues are 1
and 25. 5. [Ans. A] 6. [Ans. B] 0 1 Eigenv lues re 2, 2 No ( I) ( I) . /
No. of L.I Eigenvectors
(no of v ri les) ( I)
7. [Ans. A]
( I) .
olving for , Let the symmetric and real
matrix be A = 0 1 Now | | Which gives ( ) ⟹ ( ⁄ )
⟹
Hence real Eigen value.
8. [Ans. B]
Let 0
1 eigenv lues re n
Eigen vector corresponding to is ( I) . / . x y/ . / By simplifying .K / . / y t king K
Eigen vector corresponding to =2 is ( I) . / .xy/ . / By simplifying ( K K ) 4 ⁄ 5 by taking K ⁄ ⁄ 9. [Ans. C]
Sum of the diagonal elements = Sum of the Eigenvalues ⟹ 1 + 0 + p = 3+S ⟹ S= p 2 10. [Ans. B] ( ⁄ ) [ ] [ ] → → [ ] → [ ] If system will h ve solution
11. [Ans. A] iven M M → MM I [ x ] [ x ] 0 1
Equating the elements x ⁄
12. [Ans. A]
0
1 → Eigenv lues re
Eigenve tor is x x verify the options
13. [Ans. C] [ ] [ ] → [ ] → [ ] ( ) infinite m ny solutions 14. [Ans. B]
Eigenvalues of a real symmetric matrix are always real
15. [Ans. B]
0
1 eigenv lues v lue
Eigen vector will be .
/
Norm lize ve tor
[ √( ) ( ) √( ) ( ) ] * √ ⁄ √ ⁄ + 16. [Ans. C]
The given system is x y z x y z x y z
Use Gauss elimination method as follows Augmented matrix is
, | - [ | ] → [ | ] → [ | ] nk ( ) nk ( | )
So, Rank (A) = Rank (A|B) = 2 < n (no. of variables)
So, we have infinite number of solutions
17. [Ans. C]
Suppose the Eigenvalue of matrix A is
( i )(s y) and the Eigenvector is
‘x’ where s the onjug te p ir of Eigenvalue and Eigenvector is ̅ n x̅. So Ax = x … ①
and x̅ ̅x̅……②
king tr nspose of equ tion ②
x̅ x̅ ̅… ③ [( ) n ̅ is s l r ] x̅ x x̅ ̅x x̅ x x̅ ̅x … , - x̅ x x̅ ̅x (x̅ x) ̅(x̅ x) ( ̅ re s l r ) ̅
( x x̅ re Eigenve tors they nnot e zero ) i i
i 0
Hence Eigenvalue of a symmetric matrix are real
18. [Ans. C]
We know that
os x os x sin x
( ) os x sin x ( ) os x
Hence 1, 1 and 1 are coefficients. They are linearly dependent.
19. [Ans. A] | | So, | | | |
(Taking 2 common from each row) ( )
20. [Ans. D]
0
1 eigen v lues
Eigenve tor is verify for oth n
21. [Ans. D]
We know that the Eigenvectors
corresponding to distinct Eigenvalues of real symmetric matrix are orthogonal. [ x x x ] [ y y y ] x y x y x y 22. [Ans. D] ( )
In case of matrix PQ QP (generally)
CE
1. [Ans. C]
If = i.e. A is orthogonal, we can
only s y th t if is n Eigenv lue of
then also will be an Eigenvalue of A,
which does not necessarily imply that | | = 1 for all i.
2. [Ans. A]
In an over determined system having more equations than variables, it is necessary to have consistent unique solution, by definition
3. [Ans. A]
With the given order we can say that order of matrices are as follows:
3×4
Y 4×3
( ) 3×3 P 2×3 3×2 P( ) (2×3) (3×3) (3×2) 2 × 2 ( ( ) ) 2×2 4. [Ans. D]
The augmented matrix for given system is [ | ]→ [ | ]
Then by Gauss elimination procedure [ | ]→ [ | ] → [ | ] ( ⁄ ) ( ) ( ) ( ⁄ )
∴ olution is non – existent for above system. 5. [Ans. B] ∑ = Trace (A) + + = Trace (A) = 2 + ( 1) + 0 = 1 Now = 3 ∴ 3 + + = 1
Only choice (B) satisfies this condition.
6. [Ans. B] ∑ = Trace (A) + + = 1 + 5 + 1 = 7 Now = 2, = 6 ∴ 2 + 6 + = 7 = 3 7. [Ans. A]
The augmented matrix for given system is [
| ]
Using Gauss elimination we reduce this to an upper triangular matrix to find its rank [ | ] → [ | ] → [ | ]
Now for infinite solution last row must be completely zero i e – 2 = 0 n – 7 = 0 n 8. [Ans. A] Inverse of 0 1 is 0 1 ( )0 1 ∴ 0 1 ( )0 1 0 1 9. [Ans. B] ( ) P = ( ) P ( ) ( ) = ( ) (I) = 10. [Ans. B] A = 0 1 Characteristic equation of A is | | = 0 (4 ) ( 5 ) 2 × 5 =0 + 30 = 0 6, 5 11. [Ans. D]
The augmented matrix for given system is [ k | ] 6 x y z 7 [ ]
Using Gauss elimination we reduce this to an upper triangular matrix to find its rank
[ k | ] → [ | ] → [ | ] Now if k
Rank (A) = rank (A|B) = 3 ∴ Unique solution
If k = 7, rank (A) = rank (A|B) = 2 which is less than number of variables ∴ When K = 7, unique solution is not possible and only infinite solution is possible
12. [Ans. A]
A square matrix B is defined as
skew-symmetric if and only if = B
13. [Ans. D]
By definition A + is always symmetric
is symmetri is lw ys skew symmetri is skew symmetri 14. [Ans. B] 0 1 = ( )0 1 ∴ 0 i i i i1 ,( i)( i) i -0 i i i i1 = 0 i i i i1 15. [Ans. B] 0 1
Sum of the Eigenvalues = 17
Product of the Eigenvalues = From options, 3.48 + 13.53 = 17 (3.48)(13.53) = 47 16. [Ans. 0.5] 0.5 17. [Ans. 16] M trix , - , - , -
The product of matrix PQR is
, - , - , -
The minimum number of multiplications involves in computing the matrix product PQR is 16 18. [Ans. 23] [ ] [ ] [ ] [ ] K JK , - [ ] , , - 19. [Ans. A] Sum of Eigenvalues
= Sum of trace/main diagonal elements = 215 + 150 + 550
= 915
20. [Ans. 88]
The determinant of matrix is [ ] → [ ] → [ ] → [ ]
Interchanging Column 1& Column 2 and taking transpose [ ] | |
* ( ) ( )+ = ( ) 21. [Ans. 2] [ ] → [ ( ) ( ) ( ) ( ) ( ) ] [ ]
( ) no. of non zero rows = 2
CS
1. [Ans. B]
The augmented matrix for the given system is [
| ]
Using elementary transformation on above matrix we get, [ | ] → [ ⁄ ⁄ | ⁄ ] → [ | ] Rank ([A B]) = 3 Rank ([A]) = 3 Since
Rank ([A B]) = Rank ([A]) = number of
variables, the system has unique solution.
2. [Ans. B]
0
1
The characteristic equation of this matrix is given by | I| | | ( )( ) = 1, 6
∴ The Eigenvalues of A are 1 and 6
3. [Ans. D]
Given that Fu =b and Fv =b
If F is non singular, then it has a unique inverse.
Now, u = b and v= b
Since is unique, u = v but it is given
th t u v his is contradiction. So F must be singular. This means that
(A) Determinant of F is zero is true. Also (B) There are infinite number of
solution to Fx= b is true since |F| = 0 (C) here is n su h the is also true, since X has infinite number of solutions., including the X = 0 solution
(D) F must have 2 identical rows is false, since a determinant may become zero, even if two identical columns are present. It is not necessary that 2 identical rows must be present for |F| to become zero.
4. [Ans. C]
It is given that Eigenvalues of A is 5, 2, 1, 4 Let P = 0 I I 1 Eigenvalues of P : | I| | I I | ( ) I I I Eigenvalue of P is ( 5 +1 ), ( 2+ 1), (1+ 1), (4+1 ), ( 5 1 ), ( 2 1 ),(1 1), (4 1) = 4, 1, 2, 5, 6, 3,0,3 5. [Ans. B] X= {x |x x x + = ,x x x - then,
{ [1, 1, 0]T , [1,0, 1 ]T } is a linearly
independent set because one cannot be
obtained from another by scalar
multiplication. However (1, 1, 0) and (1,0, 1) do not span X, since all such
combinations (x1, x2, x3) such that
x1+ x2+ x3 =0 cannot be expressed as
linear combination of (1, 1,0) and (1,0, 1)
6. [Ans. D]
The augmented matrix for above system is [ | ] → [ | ] → [ | ] Now as long as – 5 0, rank (A) =rank (A|B) =3
∴ can be any real value except 5. Closest correct answer is (D). 7. [Ans. A] Eigenvalues of 0 1 | | = 0 = 0 , 1 Eigenvalues of 0 1 | | = 0 = 0 = 0, 0 Eigenvalues of 0 1 | |= 0 ( ) = 0 ( ) = i or 1 = 1 –i or 1 + i Eigenvalues of 0 1 | | = 0 ( )( ) = 0 ( ) = 0 = –1, 1
Only one matrix has an Eigenvalue of 1
which is 0
1
Correct choice is (A)
8. [Ans. D] | x y| ( ) ( y) x When ( y) x y x ( ) When ( y) x y x ( ) x y Solving (1) & (2) x y 9. [Ans. A]
The Eigenvalues of a upper triangular matrix are given by its diagonal entries. ∴ Eigenvalues are 1, 4, 3 only
10. [Ans. D]
0
1
Eigenvalues of the matrix (A) are the roots of the characteristic polynomial given below. | | ( ) ( ) ( )( ) √ Eigenvalues of A are √ n √ respectively So Eigenvalues of (√ ) n ( √ ) n n √ n √
11. [Ans. A]
→ p q
Since 2nd & 3rd columns have been
swapped which introduces a –ve sign Hence (A) is not equal to the problem
12. [Ans. 1] x y x z x y z x y z ugmente m trix is [ ] [ ] → → → [ ] → → [ ] → [ ] ( ) ( ) no of v ri les ∴ nique solution exists
13. [Ans. 0]
The Eigenvectors corresponding to distinct Eigenvalues of real symmetric matrix are orthogonal 14. [Ans. 0] [ ] | | ( ) 15. [Ans. 6] Let A = [ ] Let X = [ x x x x x ] e eigen ve tor
By the definition of eigenvector, AX =
[ ][ x x x x x ] [ x x x x x ] x x x x x x x x x x x x x x x x x x x x x x n x x x x x x (I) If s y x x x x x x x x x x (2) If Eigenv lue
∴ Three distinct eigenvalues are 0, 2, 3 Product of non zero eigenvalues = 2 × 3 = 6
16. [Ans. A]
If the trace or determinant of matrix is positive then it is not necessary that all eigenvalues are positive. So, option (B), (C), (D) are not correct
ECE 1. [Ans. C] Since, , - 2. [Ans. A] We know, =I 0 1 6 7 = 0 1 0 1 0 1 Or 2a 0.1b=0,
2a
b
10
, 1
60
a
a + b =1 1 21 7
3 60 60 20
3. [Ans. C] 0 1 (A I)=0 ( 4 ) (3 ) 2 4=0 2 + 20=0 = 5, 4 Putting = 5, 0 1 1 2
x
x
= 0 x + 2x = 0 x = 2x 12
x
= 21
x
Hence, 0 1 is Eigenvector. 4. [Ans. C] 0 1 We know th t it is Eigenvalue Then Eigenvector is x x Verify the options (C)5. [Ans. A] or m trix 0 1 We know |I A|=0 | | 2 –I2 +32 =0 = 4, 8 (Eigenvalues) For = 4, ( I ) = 0 1 v = 0 1 For = 8, ( I ) = 0 1 v = 0 1 6. [Ans. C] [ ] [ ] ( ) 7. [Ans. C]
There are M non-zero, orthogonal vectors, so there is required M dimension to represent them ’ 8. [Ans. B] Approach 1: Given 4x + 2y =7 and 2x + y =6
4 2 x
7
2 1 y
6
0 0 x
5
2 1 y
6
On comparing LHS and RHS0= 5, which is irrelevant and so no solution. Approach 2: 4x + 2y =7
7
or 2x y=
2
2x+y=6Since both the linear equation represent parallel set of straight lines, therefore no solution exists.
Approach 3:
Rank (A)=1; rank (C)=2,
As Rank (A) rank (C) therefore no solution exists.
9. [Ans. C]
Matrix will be singular if any of the Eigenvalues are zero.
| |= 0 For = 0, P = 0 |pp p p | = 0 p p p p 10. [Ans. D]
Approach1: Eigenvalues exists as complex conjugate or real
Approach 2: Eigenvalues are given by | | = 0 ( )(( ) ) = 0 , j j 11. [Ans. C]
Eigenvalue of skew – symmetric matrix is either zero or pure imaginary.
12. [Ans. B]
Given equations are x y z x y z and x y z If and ,
then x y z have Infinite solution If and , then
x y z ( ) no solution x y z
If n
x y z will have solution x y z
and will also give solution
13. [Ans. B]
0 1
Characteristic Equations is
By Cayley Hamilton theorem
I I ∴ ( I) I 14. [Ans. A] [ ] → ( ) [ ] | | | | Product of Eigenvalues = 0 ∴ Minimum Eigenv lue h s to e ‘ ’
15. [Ans. B] Let , - [ ] I I [ ] Then AB = [4]; BA [ ] Here m = 1, n = 4
And et(I ) et(I )
et of , - et of [ ] 16. [Ans. D]
Matrix multiplication is not commutative in general.
17. [Ans. *] Range 0.99 to 1.01
Let ‘ ’ e Eigenv lue of ‘ ’ hen ‘ ’ will
e Eigenv lue of ‘ ’
A. = I =
Using Cauchey Hamilton Theorem,
18. [Ans. *] Range 0.99 to 1.01 I J I J [ ] | | 19. [Ans. *] Range 199 to 201
From matrix properties we know that the determinant of the product is equal to the product of the determinants.
That is if A and B are two matrix with determinant | | n | | respectively, then | | | | | | ∴ | | | | | | 20. [Ans. B] [ ] → → [ ] → [ ] ( ) ( | ) no of v r les Infinitely many solutions
21. [Ans. B]
onsi er 0
whi h is re l symmetri m trix h r teristi equ tion is | I|
( )
∴ (not positive) ( ) is not true
(A), (C), (D) are true using properties of Eigenvalues
22. [Ans. *] Range 48.9 to 49.1
Real symmetric matrices are diagnosable Let the matrix be
0x x1 s tr e is
So determinant is product of diagonal entries
So | | x x
∴ M ximum v lue of etermin nt x x ∴ | | EE 1. [Ans. B] R = [ ] j( ) | | , of tor( )-| )-| | | = | | = 1(2 + 3) – 0(4 + 2) – 1 (6 – 2) = 1
Since we need only the top row of , we
need to find only first column of (R) which after transpose will become first row adj(A). cof. (1, 1) = + | | = 2 + 3 = 5 cof. (2, 1) = | | = 3 cof. (2, 1) = + | | = + 1 ∴ cof. (A) = [ ] Adj (A) =, of ( )- = [ ] Dividing by |R| = 1 gives = [ ] ∴ Top row of = , - 2. [Ans. D]
Since matrix is triangular, the Eigenvalues are the diagonal elements themselves
namely = 3, 2 & 1.
Corresponding to Eigenvalue = 2, let
us find the Eigenvector
[A - ] x̂ = 0 [ ] [ x x x ] [ ] Putting in above equation we get, [ ] [ x x x ] [ ] Which gives the equations,
5x x x = 0 . . . (i)
x = 0 . . . (ii) 3x = 0 . . . (iii) Since eqa (ii) and (iii) are same we have
5x x x = 0 . . . (i)
x = 0 . . . (ii)
Putting x = k, we get
x = 0, x = k and 5x k = 0
x = k
∴ Eigenvectorss are of the form [ x x x ] * k k + i.e. x x x = k : k : 0 = : 1 : 0 = 2 : 5 : 0 ∴ [ x x x ]=[ ] is an Eigenvector of matrix p. 3. [Ans. A]
Rank [P|Q] = Rank [P] is necessary for existence of at least one solution to x q.
4. [Ans. A]
We need to find orthogonal vectors, verify the options.
Option (A) is orthogonal vectors
( ) ( )
Option (B), (C), (D) are not orthogonal
5. [Ans. B]
The vector ( ) is linearly dependent upon the solution obtained in
Q. No. 4 namely , - and , -
We can easily verify the linearly dependence as | | 6. [Ans. B] hen n n m trix x x * x x x x x x x x x x x x x x x x x x +
Take x common from 1st row,
x common from 2nd row ……
x common from nth row.
It h s r nk ‘ ’ 7. [Ans. D] L(x) = | k⃗ x x x | = (x ) ( ) k⃗ ( x ) = x x k⃗ = [ x x ] L(x) = M [ x x x ]
Comparing both , we get, M = [ ] Hence Eigenvalue of M : | M | | | ( ) ( ) ( ) i i 8. [Ans. B] x y x y x x x n x y y x | x x x y y x y y | |x x y y x y | x y (x y)
= Positive when x and y are linearly independent. 9. [Ans. A] A = 0 1 |A – | = 0 | | = 0
A will satisfy this equation according to Cayley Hamilton theorem
i.e. I = 0
Multiplying by on oth si es we get
I = 0
I = 0
10. [Ans. A]
To calculate
Start from I = 0 which has
derived above I ( I)( I) I ( I) I I ( I)( I) I ( I) I I ( I) ( I) I
11. [Ans. A]
If rank of (5 6 ) matrix is 4,then surely
it must have exactly 4 linearly
independent rows as well as 4 linearly independent columns.
12. [Ans. D]
If characteristic equation is
= 0
Then by Cayley – Hamilton theorem,
I = 0
=
Multiplying by on both sides,
= I = ( I)
13. [Ans. D]
Choice (A) = A is correct
Since = A [ ( ) - A
= A [ ( ) -
Put = P
Then A [ ] = A. = A
Choice (C) = is also correct since
= ( )
= I
14. [Ans. B]
Let orthogonal matrix be
P = 0 os in
in os 1
By Property of orthogonal matrix A
I So, x⃗ = [ x os x in x in x os ] || x⃗ || = √(x os x in ) (x in x os ) || x⃗ || = √x x
|| x⃗ || = || x ̅|| for any vector x ̅
15. [Ans. C]
Trace = Sum of Principle diagonal elements.
16. [Ans. D]
On writing the equation in the form of AX =B * + * x x x x + * + Argument matrix C =* + → , * + nk ( ) nk( ) Number of variables = 4
Since, Rank (A) = Rank(C) < Number of variables
Hence, system of equations are consistent and there is multiple non-trivial solution exists. 17. [Ans. B] Characteristic equation | I| | | (1 ) ( ) ( )
Eigenve tors orrespon ing to is ( I) [ ] [ x x x ] [ ] 2x x x x At x x x x x x At x , x
Eigenvectors = c[ ]{Here c is a constant}
18. [Ans. D]
, - ,L-, - ⟹ Options D is correct
19. [Ans. D]
x x … (i)
x x … (ii)} (i) n (ii) re s me
∴ x x
20. [Ans. D] Eigen value Eigenvectors 0 1 n 0 1 Let matrix 0 1 x x 0 1 0 1 0 1 0 1 0 1 0 1 Solving 0 1 0 1 21. [Ans. B]
Since there are 2 equations and 3 variables (unknowns), there will be infinitely many
solutions. If if then
x y z x y z x z y For any x and z, there will be a value of y. ∴ Infinitely many solutions
22. [Ans. A]
For all real symmetric matrices, the Eigenvalues are real (property), they may be either ve or ve and also may be same. The sum of Eigenvalues necessarily not be zero.
23. [Ans. C]
0p qr s1
( pplying → p q
→ r s element ry tr nsform tions)
[p q pr qs pr qs r s ] ∴ hey h ve s me r nk N IN 1. [Ans. B] Given: 0 1
Characteristic equation is,
|A I|= |
|
i.e., (1 ) (2 ) 2
Thus the Eigenvalue are 1, 2.
If x, y, be the component of Eigenvectors corresponding to the Eigenv lues we have
[A- I- 0
1 0
x y1=0 For =1, we get the Eigenvector as 0
1
Hence, the answer will be ,
-2. [Ans. B]
AX=0 and (A) = 2 n = 3
No. of linearly independent solutions = n r
= 3 = 1
3. [Ans. C]
There are 3 non-zero rows and hence rank (A) = 3
4. [Ans. C]
Rank (A) = 3 (This is Co-efficient matrix) Rank (A:b) =4(This is Augmented matrix) s r nk( ) r nk ( ) olution oes not exist.
5. [Ans. C]
We know Hen e from the given problem, Eigenvalue & Eigenvector is known. 1 2 1 2
1
1
X
, X
,
1,
2
1
2
We also know that , where
P
1 2
1 1
X X
1 2
& D = 1 20
1 0
0
0 2
Hence1
1
1 0
2
1
A
1
2
0
2
1
1
6. [Ans. B] A= [ ] = [ ]Using elementary transformation
[
] Hence, rank (A) =1
7. [Ans. B]
Given I
Hence rank (A) = n
Hence AX= Y will have unique solution
8. [Ans. C] 9. [Ans. C] Approach 1: Assume, 0 1 I 0 1 ∴ A ( I) ( I) 0 1 0 1 0 1 0 1 0 1 Now | I | | | ( )( ) = 0 Approach 2: Eigenvalues of ( I) is = 1, 1/2 Eigenvalues of (X+5I) is = 3, 2 Eigenvalues of ( I) (X+5I) is = , 10. [Ans. D] 11. [Ans. C]
A null matrix can be obtained by multiplying either with one null matrix or two singular matrices.
12. [Ans. A] A = [ ] i if i j = 0 otherwise. For n n matrix A = [ n ]
For diagonal matrix Eigenvalues are diagonal elements itself.
∴ n n(n )
13. [Ans. B]
If AX = →
From this result [1, 2, - is also vector
for M
14. [Ans. B]
Dim of null space [A]= nullity of A.
For given A = [
] Apply row operations
[ ] → [ ] → [ ] ∴ ( )
By rank – nullity theorem
Rank [A]+ nullity [A]= no. of columns[A] Nullity [A]= 3 ∴ Nullity , - 15. [Ans. A] A = | | Characteristics equation | I| | | j j
[ j j] 0 x x 1 0 1 x x j j [j j ] 0 x x 1 0 1 x j x 16. [Ans. C] A [ ] = [ ] →| | | | | | → | | ( | | |
| two rows ounter lose thus | | | |)
=Product of eigenvalues Verify options
Probability and Distribution
ME - 20051. A single die is thrown twice. What is the
probability that the sum is neither 8 nor 9?
(A) ⁄
(B) ⁄
(C) ⁄
(D) ⁄
2. A lot has 10% defective items. Ten items
are chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is
(A) 0.0036 (B) 0.1937
(C) 0.2234 (D) 0.3874
ME - 2006
3. Consider a continuous random variable
with probability density function
f(t) = 1 + t for 1 t 0
= 1 t for 0 t 1
The standard deviation of the random variable is:
(A) √ ⁄
(B) √ ⁄
(C) ⁄
(D) ⁄
4. A box contains 20 defective items and 80
non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective?
(A) ⁄
(B) ⁄
(C) ⁄
(D) ⁄
ME - 2007
5. Let X and Y be two independent random
variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE?
(A) E (XY) = E (X) E (Y) (B) Cov (X, Y) = 0
(C) Var (X + Y) = Var (X) + Var (Y)
(D) (X Y ) ( (X)) ( (Y))
ME - 2008
6. A coin is tossed 4 times. What is the
probability of getting heads exactly 3 times? (A) ⁄ (B) ⁄ (C) ⁄ (D) ⁄ ME - 2009
7. The standard deviation of a uniformly
distributed random variable between 0 and 1 is
(A) √ (B) √
(C) √ ⁄
(D) √
8. If three coins are tossed simultaneously,
the probability of getting at least one head is (A) 1/8 (B) 3/8 (C) 1/2 (D) 7/8 ME - 2010
9. A box contains 2 washers, 3 nuts and 4
bolts. Items are drawn from the box at
random one at a time without
replacement. The probability of drawing 2 washers first followed by 3 nuts and subsequently the 4 bolts is
(A) 2/315 (B) 1/630
(C) 1/1260 (D) 1/2520
ME - 2011
10. An unbiased coin is tossed five times. The
outcome of each toss is either a head or a tail. The probability of getting at least one head is________ (A) ⁄ (B) ⁄ (C) ⁄ (D) ⁄ ME - 2012
11. A box contains 4 red balls and 6 black
balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set has one red ball and two black balls is
(A) 1/20 (B) 1/12
(C) 3/10 (D) 1/2
ME - 2013
12. Let X be a normal random variable with
mean 1 and variance 4. The probability (X ) is
(A) 0.5
(B) Greater than zero and less than 0.5 (C) Greater than 0.5 and less than 1.0 (D) 1.0
13. The probability that a student knows the
correct answer to a multiple choice
question is . If the student dose not know
the answer, then the student guesses the answer. The probability of the guessed
answer being correct is . Given that the
student has answered the questions correctly, the conditional probability that the student knows the correct answer is
(A) ⁄
(B) ⁄
(C) ⁄
(D) ⁄
ME - 2014
14. In the following table x is a discrete
random variable and P(x) is the probability density. The standard deviation of x is x 1 2 3 P(x) 0.3 0.6 0.1 (A) 0.18 (B) 0.3 (C) 0.54 (D) 0.6
15. Box contains 25 parts of which 10 are
defective. Two parts are being drawn simultaneously in a random manner from the box. The probability of both the parts being good is ( ) ( ) ( ) ( )
16. Consider an unbiased cubic dice with
opposite faces coloured identically and each face coloured red, blue or green such that each colour appears only two times on the dice. If the dice is thrown thrice,
the probability of obtaining red colour on top face of the dice at least twice is _______
17. A group consists of equal number of men
and women. Of this group 20% of the men and 50% of the women are unemployed. If a person is selected at random from this group, the probability of the selected person being employed is _______
18. A machine produces 0, 1 or 2 defective
pieces in a day with associated probability of 1/6, 2/3 and 1/6, respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are
(A) 1 and 1/3 (B) 1/3 and 1
(C) 1 and 4/3 (D) 1/3 and 4/3
19. A nationalized bank has found that the
daily balance available in its savings accounts follows a normal distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings account holders, who maintain an average daily balance more than Rs. 500 is _______
20. The number of accidents occurring in a
plant in a month follows Poisson distribution with mean as 5.2. The probability of occurrence of less than 2 accidents in the plant during a randomly selected month is (A) 0.029 (B) 0.034 (C) 0.039 (D) 0.044 CE - 2005
1. Which one of the following statements is
NOT true?
(A) The measure of skewness is
dependent upon the amount of dispersion
