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Given the matrix 
the eigenvector is- a)
- b)
- c)
- d)
Correct answer is 'C'. Can you explain this answer?
Given the matrix the eigenvector is
the eigenvector isa)
b)
c)
d)
|
Ritika Bose answered |
Characteristic equation
The number of linearly independent eigenvectors of 
- a)0
- b)1
- c)2
- d)Infinit e
Correct answer is 'B'. Can you explain this answer?
The number of linearly independent eigenvectors of
a)
0
b)
1
c)
2
d)
Infinit e
|
Pranjal Sen answered |
Number of linear independent vectors is equal to the sum of Geometric Multiplicity of eigen values. Here only eigen value is 2. To find Geometric multiplicity find n-r of (matrix-2I), where n is order and r is rank. Rank of obtained matrix is 1 and n=2 so n-r=1. Therefore the no of linearly independent eigen vectors is 1
A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads? - a)
- b)
- c)
- d)
Correct answer is option 'C'. Can you explain this answer?
A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads?
a)
b)
c)
d)
|
Rhea Reddy answered |
Let A be the event that first toss is head
And B be the event that second toss is head.
By the given condition rest all 8 tosses should be tail
∴ The probability of getting head in first two cases
The eigenvalues of
- a)-19,5,37
- b)19,-5,-37
- c)2,-3,7
- d)3,-5,37
Correct answer is option 'A'. Can you explain this answer?
The eigenvalues of
a)
-19,5,37
b)
19,-5,-37
c)
2,-3,7
d)
3,-5,37
|
|
Avinash Sharma answered |
The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix.
Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look atd
λ = 5, -19, 37
The three characteristic roots of the following matrix A 
- a)2,3
- b)1,2,2
- c)1,0,0
- d)0,2,3
Correct answer is option 'B'. Can you explain this answer?
The three characteristic roots of the following matrix A
a)
2,3
b)
1,2,2
c)
1,0,0
d)
0,2,3
|
Kabir Verma answered |
A is lower triangular matrix. So eigen values are only the diagonal elements.
the eigen value are
are –2 and 6, respectively. What is the other eigen value? 
Eigen values of a matrix 
are 5 and 1. What are the eigen values of the matrix S2 = SS?- a)1 and 25
- b)6 and 4
- c)5 and 1
- d)2 and 10
Correct answer is option 'A'. Can you explain this answer?
Eigen values of a matrix are 5 and 1. What are the eigen values of the matrix S2 = SS?
are 5 and 1. What are the eigen values of the matrix S2 = SS?a)
1 and 25
b)
6 and 4
c)
5 and 1
d)
2 and 10
|
EduRev GATE answered |
We know If λ be the eigen value of A ⇒λ2 is an eigen value of A2 .
For S matrix, if eigen values are λ1, λ2, λ3,... then for S² matrix, the eigen values will be λ²1 λ²2 λ²3......
For S matrix, if eigen values are 1 and 5 then for S² matrix, the eigen values are 1 and 25
For S matrix, if eigen values are 1 and 5 then for S² matrix, the eigen values are 1 and 25
are nonzero, and one of its eigen values is zero. Which of the following statements is true?
If the rank of a (5×6) 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 lineally independent rows and five lineally independent columns
- c)QQT will be invertible
- d)QTQ will be invertible
Correct answer is option 'A'. Can you explain this answer?
If the rank of a (5×6) 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 lineally independent rows and five lineally independent columns
c)
QQT will be invertible
d)
QTQ will be invertible
|
Kaavya Sengupta answered |
Given that rank Q is 4, so it will have 4 linearly independent columns as well as 4 linearly independent rows (∵ row rank of a matrix = column rank of a matrix)
So, A is correct and accordingly B is false.
C is false as order of QQ^T will be 5 x 5 and given that rank of Q is 4 i.e. < 5. So the matrix is QQ^T will be singular and hence not invertible.
Similarly D is false as order of Q^T Q will be 6 x 6 and the matrix is Q^T Q will be singular and hence not invertible.
In an M × N matrix such that all non-zero entries are covered in a rows and b column. Then the maximum number of non-zero entries, such that no two are on the same row or column, is - a)≤ a + b
- b)≤ max (a, b)
- c)≤ min[ M –a, N–b]
- d)≤ min {a, b}
Correct answer is option 'D'. Can you explain this answer?
In an M × N matrix such that all non-zero entries are covered in a rows and b column. Then the maximum number of non-zero entries, such that no two are on the same row or column, is
a)
≤ a + b
b)
≤ max (a, b)
c)
≤ min[ M –a, N–b]
d)
≤ min {a, b}
|
Naroj Boda answered |
Suppose a < b, for example let a = 3, b= 5, then we can put non-zero entries only in 3 rows and 5 columns. So suppose we put non-zero entries in any 3 rows in 3 different columns. Now we can’t put any other non-zero entry anywhere in matrix, because if we put it in some other row, then we will have 4 rows containing non-zeros, if we put it in one of those 3 rows, then we will have more than one non-zero entry in one row, which is not allowed.
So we can fill only “a” non-zero entries if a < b, similarly if b < a, we can put only “b” non-zero entries. So answer is ≤min(a,b), because whatever is less between a and b, we can put atmost that many non-zero entries.
Consider the function f(x) = |x|3, where x is real. Then the function f(x) at x = 0 isa)Continuous but not differentiable
b)Once differentiable but not twice
c)Twice differentiable but not thrice
d)Thrice differentiable
Correct answer is option 'C'. Can you explain this answer?
|
Diya Mukherjee answered |
The minimum point of the function f(x) = (x2/3) – x is at - a) x = 1
- b)x = -1
- c) x = 0
- d)x = 1/√3
Correct answer is option 'A'. Can you explain this answer?
The minimum point of the function f(x) = (x2/3) – x is at
a)
x = 1
b)
x = -1
c)
x = 0
d)
x = 1/√3
|
|
Avinash Sharma answered |
Correct Answer :- a
Explanation : f(x) = (x^2/3) - x
f'(x) = 2/3(x-1/2) - 1
f"(x) = -1/3(x-3/2)
For critical points. f′(x)=0
=> 2/3(x-1/2) - 1 = 0
f has minimum value of x = 1
- a)∞
- b)2
- c)0
- d)1
Correct answer is option 'C'. Can you explain this answer?
a)
∞
b)
2
c)
0
d)
1
|
Yash Patel answered |
the squeeze theorem for this. Recall that sinx is only defined on −1≤sinx≤1. Therefore
Consider the following two statements about the function f(x) = |x| P: f(x) is continuous for all real values of x Q: f(x) is differentiable for all real values of x Which of the f oll owi ng is TRU E? - a)P is true and Q is false
- b)P is false and Q is true
- c)Both P and Q are true
- d)Both P and Q are false
Correct answer is option 'A'. Can you explain this answer?
Consider the following two statements about the function f(x) = |x|
P: f(x) is continuous for all real values of x
Q: f(x) is differentiable for all real values of x
Which of the f oll owi ng is TRU E?
a)
P is true and Q is false
b)
P is false and Q is true
c)
Both P and Q are true
d)
Both P and Q are false
|
|
Avinash Sharma answered |
The graph of f(x) is
f(x) is continuous for all real values of x Lim |x| = Lim |x| = 0
as can be seen from graph of |x|.
and Lim f(x) = +1 as can be seen from graph of |x|
x → 0+
Left deriva tive ≠ Rig ht derivative
So |x| is continuous but not differentiable at x = 0.
For which value of x will the matrix given below become singular? 
- a)4
- b)6
- c)8
- d)12
Correct answer is option 'A'. Can you explain this answer?
For which value of x will the matrix given below become singular?
a)
4
b)
6
c)
8
d)
12
|
Gate Funda answered |
Let the given matrix be A. A is singular.
A fair dice is rolled twice. The probability that an odd number will follow an even number is - a)1/2
- b)1/6
- c)1/3
- d)1/4
Correct answer is option 'D'. Can you explain this answer?
A fair dice is rolled twice. The probability that an odd number will follow an even number is
a)
1/2
b)
1/6
c)
1/3
d)
1/4
|
Raghavendra Sharma answered |
Probability of even number =3/6 =1/2
Probability of odd number =3/6 =1/2
Both are independent so probability=1/2.1/2 =1/4
Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix
A satisfies the relation
- a)A +3I + 2A -2 =0
- b)A2+2A +2I=0
- c)(A+1I)( A+2I)=0
- d)exp(A) =0
Correct answer is option 'C'. Can you explain this answer?
Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix
A satisfies the relation
a)
A +3I + 2A -2 =0
b)
A2+2A +2I=0
c)
(A+1I)( A+2I)=0
d)
exp(A) =0
|
Telecom Tuners answered |
Characteristic equation of A is
over the triangle is 
The state variable description of a linear autonomous system is, X= AX, Where X is the two dimensional state vector and A is the system matrix given by 
The roots of the characteristic equation are - a)-2 and +2
- b)-j2 and +j2
- c)-2 and -2
- d)+2 and +2
Correct answer is option 'A'. Can you explain this answer?
The state variable description of a linear autonomous system is, X= AX,
Where X is the two dimensional state vector and A is the system matrix given by
The roots of the characteristic equation are
a)
-2 and +2
b)
-j2 and +j2
c)
-2 and -2
d)
+2 and +2
|
|
Janhavi Sarkar answered |
Characteristic equation will be :λ2 -4 =0 thus root of characteristic equation will be +2 and - 2.
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
Correct answer is option 'D'. Can you explain this answer?
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
|
|
Janhavi Singh answered |
Let the matrix be A. We know, Trace (A)=sum of eigen values.
- a)0
- b)π/2
- c)1/5
- d)π/3
Correct answer is option 'B'. Can you explain this answer?
a)
0
b)
π/2
c)
1/5
d)
π/3
|
Akash Kumar answered |
Explain the questions
The distance between the origin and the point nearest to it on the surface z2 = 1 + xy is - a)1
- b)√3/2
- c)√3
- d)-2
Correct answer is option 'A'. Can you explain this answer?
The distance between the origin and the point nearest to it on the surface z2 = 1 + xy is
a)
1
b)
√3/2
c)
√3
d)
-2
|
Baishali Bajaj answered |
or pr – q^2 = 4 – 1 = 3 > 0 and r = +ve
so f(xy) is minimum at (0,0)
Hence, minimum value of d^2 at (0,0)
d2 = x^2 + y^2 + xy + 1 = (0)^2 + (0)^2 + (0)(0) + 1 = 1
Then the nearest point is
z^2 = 1 + xy = 1+ (0)(0) = 1
or z = 1
The Fourier series for the function f(x)=sin2x is - a)sinx+sin2x
- b)1-cos2x
- c)sin2x+cos2x
- d)0.5-0.5cos2x
Correct answer is option 'D'. Can you explain this answer?
The Fourier series for the function f(x)=sin2x is
a)
sinx+sin2x
b)
1-cos2x
c)
sin2x+cos2x
d)
0.5-0.5cos2x
|
Ashish Chakraborty answered |
Here f(x ) = sin2 x is even function, hence f( x ) has no sine term.
The following differential equation has 
a)degree = 2, order = 1b)degree = 1, order = 2 c)degree = 4, order = 3 d)degree = 2, order = 3Correct answer is option 'B'. Can you explain this answer?
|
Anmol Basu answered |
Order of highest derivative = 2
Hence, most appropriate answer is (b)
Hence, most appropriate answer is (b)
The following system of equations 
has a unique solution. The only possible value(s) for a is/are - a)0
- b)either 0 or 1
- c)one of 0, 1 or –1
- d)any real number other than 5
Correct answer is option 'D'. Can you explain this answer?
The following system of equations
has a unique solution. The only possible value(s) for a is/are
a)
0
b)
either 0 or 1
c)
one of 0, 1 or –1
d)
any real number other than 5
|
|
Gate Funda answered |
System has unique Soln if rank (A) = rank ( A ) = 3 . It is possible if a ≠ 5.
An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is- a)0.5
- b)0.18
- c) 0.12
- d) 0.06
Correct answer is option 'C'. Can you explain this answer?
An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is
a)
0.5
b)
0.18
c)
0.12
d)
0.06
|
Tanu Shindekar answered |
The probability of a student failing in both papers is 0.12
X(t) is a real valued function of a real variable with period T. Its trigonometric Fourier Series expansion contains no terms of frequency ω = 2π (2k ) /T ; k = 1, 2,.... Also, no sine terms are present. Then x(t) satisfies the equation - a)x ( t ) =−x(t − T)
- b)x (t) = −x(T − t)= −x (−t)
- c)x (t) = x(T− t) = −x (t −T / 2)
- d)x (t ) = x(t−T) = x (t −T / 2)
Correct answer is option 'D'. Can you explain this answer?
X(t) is a real valued function of a real variable with period T. Its trigonometric Fourier Series expansion contains no terms of frequency ω = 2π (2k ) /T ; k = 1, 2,.... Also, no sine terms are present. Then x(t) satisfies the equation
a)
x ( t ) =−x(t − T)
b)
x (t) = −x(T − t)= −x (−t)
c)
x (t) = x(T− t) = −x (t −T / 2)
d)
x (t ) = x(t−T) = x (t −T / 2)
|
Nandini Banerjee answered |
No sine terms are present.
∴x(t ) is even function.
The eigen values of a skew-symmetric matrix are- a)Always zero
- b)always pure imaginary
- c)Either zero or pure imaginary
- d)always real
Correct answer is option 'C'. Can you explain this answer?
The eigen values of a skew-symmetric matrix are
a)
Always zero
b)
always pure imaginary
c)
Either zero or pure imaginary
d)
always real
|
Byomkesh Kamila answered |
Here,(a_ij)=-(a_ji)#for all diagonal elements,a_ij=a_ii =a_jj.
a_ii=-a_ii#2a_ii=0#a_ii=0/2=0 for all values of i≤n.
a_ii=-a_ii#2a_ii=0#a_ii=0/2=0 for all values of i≤n.
with the condition that y = 1 at x = 1, is
A function n(x) satisfies the differential equation 
where L is a constant. The boundary conditions are: n(0)=K and n(∞)= 0. The solution to this equation is - a)n(x) = K exp(x/L)
- b)n(x) = K exp( x / √L )
- c)n(x) = K2 exp(-x/L)
- d)n(x) = K exp( -x/L)
Correct answer is option 'D'. Can you explain this answer?
A function n(x) satisfies the differential equation where L is a constant. The boundary conditions are: n(0)=K and n(∞)= 0. The solution to this equation is
where L is a constant. The boundary conditions are: n(0)=K and n(∞)= 0. The solution to this equation is a)
n(x) = K exp(x/L)
b)
n(x) = K exp( x / √L )
c)
n(x) = K2 exp(-x/L)
d)
n(x) = K exp( -x/L)
|
|
Sanvi Kapoor answered |
For finite solution c1 = 0
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
Correct answer is option 'C'. Can you explain this answer?
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
|
|
Jaideep Dasgupta answered |
Here sample space = 9
The required probability of drawing 2 washers, 3 nuts and 4 bolts respectively without replac ement
For the following set of simultaneous equations:
1.5x – 0.5y = 2
4x + 2y + 3z = 9
7x + y + 5z = 10 - a)The solution is unique
- b)Infinitely many solutions exist
- c)The equations are incompatible
- d)Finite number of multiple solutions exist
Correct answer is option 'A'. Can you explain this answer?
For the following set of simultaneous equations:
1.5x – 0.5y = 2
4x + 2y + 3z = 9
7x + y + 5z = 10
1.5x – 0.5y = 2
4x + 2y + 3z = 9
7x + y + 5z = 10
a)
The solution is unique
b)
Infinitely many solutions exist
c)
The equations are incompatible
d)
Finite number of multiple solutions exist
|
Dipika Nambiar answered |
(a)
∴ rank of(
) = rank of(A) = 3
) = rank of(A) = 3∴ The system has unique solution.
Chapter doubts & questions for Engineering Mathematics - RRB JE for Civil Engineering 2025 is part of Civil Engineering (CE) exam preparation. The chapters have been prepared according to the Civil Engineering (CE) exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
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