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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
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
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
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
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? 
- 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
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.
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
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.
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.
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.
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.
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 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
are nonzero, and one of its eigen values is zero. Which of the following statements is true?
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.
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 |
(where C is a closed curve given by |z| = 1) is 
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 = 4/3
- b)x = 0; y = 1/2 ; z = 2
- c)x = 1; y = 1/2 ; z = 2
- d)non-existent
Correct answer is option 'D'. Can you explain this answer?
Solution for the system defined by the set of equations
4y + 3z = 8;
2x – z = 2
and 3x + 2y =5 is
4y + 3z = 8;
2x – z = 2
and 3x + 2y =5 is
a)
x = 0; y = 1; z = 4/3
b)
x = 0; y = 1/2 ; z = 2
c)
x = 1; y = 1/2 ; z = 2
d)
non-existent
|
Dishani Desai answered |
Ans.(d)
Consider the matrix A =
,Now det (A) = 0
So byCramer's Rule the system has no solution
Consider the matrix A =
,Now det (A) = 0So byCramer's Rule the system has no solution
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.
is given by
The general solution of (x2 D2 – xD), y= 0 is :
- a) y = C1 + C2 ex
- b) y = C1 + C2 x2
- c)y = C1 x + C2 x2
- d)y = C1 + C2 x
Correct answer is option 'C'. Can you explain this answer?
The general solution of (x2 D2 – xD), y= 0 is :
a)
y = C1 + C2 ex
b)
y = C1 + C2 x2
c)
y = C1 x + C2 x2
d)
y = C1 + C2 x
|
Akshara Rane answered |
+ 4xD + 3y = 0) can be found by assuming a solution of the form y = e^(mx).
Substituting this into the differential equation gives:
x^2(m^2)e^(mx) + 4xme^(mx) + 3e^(mx) = 0
Dividing through by e^(mx) gives:
m^2x^2 + 4mx + 3 = 0
This is a quadratic equation in m, which can be solved using the quadratic formula:
m = (-4x ± √(16x^2 - 4(3)(x^2))) / (2x^2)
Simplifying this gives:
m = (-2 ± √1) / x
m1 = -1/x, m2 = -3/x
Therefore, the general solution is a linear combination of the two solutions:
y = c1e^(-x) + c2e^(-3x)
where c1 and c2 are arbitrary constants determined by initial or boundary conditions.
Substituting this into the differential equation gives:
x^2(m^2)e^(mx) + 4xme^(mx) + 3e^(mx) = 0
Dividing through by e^(mx) gives:
m^2x^2 + 4mx + 3 = 0
This is a quadratic equation in m, which can be solved using the quadratic formula:
m = (-4x ± √(16x^2 - 4(3)(x^2))) / (2x^2)
Simplifying this gives:
m = (-2 ± √1) / x
m1 = -1/x, m2 = -3/x
Therefore, the general solution is a linear combination of the two solutions:
y = c1e^(-x) + c2e^(-3x)
where c1 and c2 are arbitrary constants determined by initial or boundary conditions.
the eigen value corresponding to the eigenvector 
Using Cauchy’s integral theorem, the value of the integral (integration being taken in counter clockwise direction) 
- a)
- b)
- c)
- d)1
Correct answer is option 'A'. Can you explain this answer?
Using Cauchy’s integral theorem, the value of the integral (integration being taken in counter clockwise direction)
a)
b)
c)
d)
1
|
|
Yash Patel answered |
Here f (z) has a singularities at z i / 3
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 Fourier series of a real periodic function has only P. cosine terms if it is even Q. sine terms if it is even R. cosine terms if it is odd S. sine terms if it is odd Which of the above statements are correct? - a)P and S
- b)P and R
- c)Q and S
- d)Q and R
Correct answer is option 'A'. Can you explain this answer?
The Fourier series of a real periodic function has only
P. cosine terms if it is even
Q. sine terms if it is even
R. cosine terms if it is odd
S. sine terms if it is odd
Which of the above statements are correct?
a)
P and S
b)
P and R
c)
Q and S
d)
Q and R
|
|
Bijoy Menon answered |
Because sine function is odd and cosine is even function.
Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes is equally likely at both places. He can afford up to 13 minutes of total waiting time if he is to arrive at D on time. What is the probability that Manish will arrive late at D?
- a)9/128
- b)13/64
- c)119/128
- d)8/13
Correct answer is option 'A'. Can you explain this answer?
Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes is equally likely at both places. He can afford up to 13 minutes of total waiting time if he is to arrive at D on time. What is the probability that Manish will arrive late at D?
a)
9/128
b)
13/64
c)
119/128
d)
8/13
|
Machine Experts answered |
The Fourier series expansion of a symmetric and even function, f(x) where 
Will be - a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
The Fourier series expansion of a symmetric and even function, f(x) where
Will be
a)
b)
c)
d)
|
|
Rhea Reddy answered |
f(x) is symmetric and even, it’s Fourier series contain only cosine term. Now.
Find the Eigenvalues of matrix
- a)8, 2
- b)3, -3
- c)-3, -5
- d)5, 0
Correct answer is option 'A'. Can you explain this answer?
Find the Eigenvalues of matrix
a)
8, 2
b)
3, -3
c)
-3, -5
d)
5, 0
|
Sanya Agarwal answered |
A - λI = 0
Now, After taking the determinant:
(4 - λ)2 - 1 = 0
16 + λ2 - 8λ - 1 = 0
λ2 - 8λ + 15 = 0
(λ
- 3) (
λ - 5) = 0
λ
= 3, 5
s one of the eigen values is equal to -2. Which of the following is an eigen vector? 
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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