GATE QUESTION BANK
for
Civil Engineering
By
Contents
Subject Name
Topic Name
Page No.
#1. Mathematics
1-148
1 Linear Algebra 1 – 28
2 Probability & Distribution 29 – 57
3 Numerical Methods 58 – 73 4 Calculus 74 – 112 5 Differential Equations 113 – 131 6 Complex Variables 132 – 143 7 Laplace Transform 144 – 148
#2. Fluid Mechanics
149 - 169
1 Pressure and its Measurement 149
2 Hydrostatic Forces on Plane Surfaces 150 – 151
3 Kinematics of Flow 152 – 153
4 Fluid Dynamics 154 – 156
5 Flow Through Pipes 157 – 158
6 Impulse Momentum Equation and Its Application 159
7 Flow through Orifices and Mouth Pieces 160
8 Boundary Layer Flow 161
9 Viscous Flow 162 – 166
10 Dimensional Analysis 167
11 Impacts of jets and Turbines 168
12 Open Channel Flow 169
#3. Hydrology & Irrigation
170 - 191
1 Hydraulics & Hydraulic Machinery 170 – 177
2 Irrigation 178
3 Water Requirements of Crops 179 – 181
4 Sediment, Transport and Design of Irrigation
Channels 182
5 Hydrology 183 – 191
#4. 4.Environmental Engineering
192 - 214
1 Quality Standards of Water 192 – 193
2 Water Supply and its Treatment 194 – 198
3 Waste Water Treatment 199 – 207
4 Sludge Disposal 208 – 209
5 Domestic Waste Water Treatment 210 – 211
#5. Structural Analysis
215 - 235
1 Trusses and Arches 215 – 221
2 Influence Line Diagram and Rolling Loads 222 – 224
3 Slope and Deflection Method 225 – 231
4 Degree of Static Indeterminacy 232
5 Force Energy Method 233
6 Matrix Method of Structural Analysis 234 – 235
#6. Mechanics
236 - 257
1 Simple Stress and Strain Relationship 236 – 241
2 Bending Moment and Shear Force Diagram 242 – 246
3 Thin Walled Pressure Vessel 247
4 Simple Bending Theory 248 – 251
5 Torsion 252 – 253
6 Column and Struts 254 – 256
7 Analysis of Statically Determinate Structures 257
#7. RCC
258 - 277
1 Concrete Technology 258 – 259
2 Basic of Mix Design 260 – 263
3 Design of RCC structures 264
4 Analysis of Ultimate Load Capacity 265 – 272
5 Basic Elements of Pre-stressed Concrete 273 – 274
6 Design of Pre-Stressed Concrete Beams 275 – 276
7 Concrete Design 277
#8. Steel
278 - 290
1 Introduction 278 – 279
2 Plastic Analysis 280 – 283
3 Welded Connections 284 – 286
4 Design of Tension Member 287
5 Compression member 288
6 Beams 289 – 290
#9. Geotechnical Engineering
291 – 325
1 Three Phase System, Fundamental
Definitions and Relationship 291
2 Index Properties and Soil Classification 292 – 295
3 Permeability and Seepage 296 – 300
4 Consolidation 301 – 304
5 Compaction 305
6 Stress Analysis 306 – 309
7 Stress Analysis 310 – 311
9 Earth Pressure 314 – 315
10 Stability of Slopes 316 – 317
11 Bearing Capacity 318 – 322
12 Pile Foundation 323 – 325
#10. Transportation & Surveying
326 - 358
1 Introduction to Transportation 326
2 Geometric design of highway 327 – 332
3 Traffic Characteristics 333 – 336
4 Traffic Signs and Signal Design 337 – 339
5 Intersection Design 340 – 341
6 Testing and Specifications of Paving Materials 342 – 344
7 Design of Rigid Flexible Pavements 345 – 348
8 Introduction 349 – 350
9 Measurement of Distance & Direction 351
10 Theodolite & Traversing 352 – 355
Linear Algebra
ME – 2005
1. 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) [ os sin sin os ]
(C) [ sin os os sin ]
(D) [ os sin 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 ii 1 is ( ) 0 i i ii 1 ( ) 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 II 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 x y y x y y 1
(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 1 and 0 1 (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 0xx 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: 0p qr s1 [ppr qs r q pr qs 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
5. For a given matrix A, it is observed that 0 1 0 1 n 0 1 0 1 Then matrix A is (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 (D) [j ] 0j1
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 ( KK ) 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
Taking transpose 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 ( ) 3×3
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 z7 [ ]
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 ii 1 ,( i)( i) i -0 i i i i1 = 0 ii i i 1 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 ontr i tion o 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 II 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 1 2 x = 2 1 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 RHS
0= 5, which is irrelevant and so no solution. Approach 2: 4x + 2y =7 7 or 2x y= 2 2x+y=6
Since 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
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 1
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 os in 1
By Property of orthogonal matrix A I
So, x⃗ = [ xx os x in x os ] in || 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]
19. [Ans. D]
x x … (i)
x x … (ii)} (i) n (ii) re s me ∴ x x
So it has multiple solutions.
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 orrespon ing to the Eigenv lues we have
[A- I]X=0 1 0xy1=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
X X1 2
1 1 1 2 & D = 1 2 0 1 0 0 0 2 Hence 1 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 = | |
| | j j [ j j ] 0xx 1 0 1 x x j j [j j ] 0xx 1 0 1 x j x 16. [Ans. C] A [ ] = [ ] →| | | | | | → | | ( | | |
| two rows ounter lose thus | |
| |)
=Product of eigenvalues Verify options
Probability and Distribution
ME - 2005
1. 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
(B) In a symmetric distribution the value of mean, mode and median are the same
(C) In a positively skewed distribution mean > median > mode
(D) In a negatively skewed distribution mode > mean > median
CE - 2006
2. A class of first years B. Tech students is composed of four batches A, B, C and D each consisting of 30 students. It is found that the sessional marks of students in Engineering Drawing in batch C have a mean of 6.6 and standard deviation of 2.3. The mean and standard deviation of the marks for the entire class are 5.5 and 4.2 respectively. It is decided by the course instruction to normalize the marks of the students of all batches to have the same mean and standard deviation as that of the entire class. Due to this, the marks of a student in batch C are changed from 8.5 to
(A) 6.0
(B) 7.0 (C) 8.0 (D) 9.0
3. There are 25 calculators in a box. Two of them are defective. Suppose 5 calculators are randomly picked for inspection (i.e. each has the same chance of being selected). What is the probability that only one of the defective calculators will be included in the inspection?
(A) ⁄
(B) ⁄ (C) (D) ⁄ ⁄
CE - 2007
4. If the standard deviation of the spot speed of vehicles in a highway is 8.8 kmph and the mean speed of the vehicles is 33 kmph, the coefficient of variation in speed is (A) 0.1517 (B) 0.1867 (C) 0.2666 (D) 0.3646 CE - 2008
5. If probability density function of a random variable x is
f(x) { x for x nd for ny other v lue of x Then, the percentage probability P . x / is
(A) 0.247 (B) 2.47
(C) 24.7 (D) 247
6. A person on a trip has a choice between private car and public transport. The probability of using a private car is 0.45. While using the public transport, further choices available are bus and metro out of which the probability of commuting by a bus is 0.55. In such a situation, the probability, (rounded upto two decimals) of using a car, bus and metro, respectively would be (A) 0.45, 0.30 and 0.25 (B) 0.45, 0.25 and 0.30 (C) 0.45, 0.55 and 0.00 (D) 0.45, 0.35 and 0.20 CE - 2009
7. The standard normal probability function can be approximated as
(x )
exp( | x | )
Where x = standard normal deviate. If mean and standard deviation of annual precipitation are 102 cm and 27 cm respectively, the probability that the annual precipitation will be between 90 cm and 102 cm is
(A) 66.7%
(B) 50.0% (C) 33.3% (D) 16.7%
CE - 2010
8. Two coins are simultaneously tossed. The probability of two heads simultaneously appearing is
(A) 1/8
