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Recent questions in Engineering Mathematics
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121
GATE 2018 | MATHS | Q-55
Let \( A \) be a \(3 \times 3\) matrix with real entries. If three solutions of the linear system of differential equations \(\dot{x}(t) = Ax(t)\) are given by \[ \begin{bmatrix} e^t - e^{2t} \\ -e^{t} + e^{2t} \\ e^t + e^{2t} \end{bmatrix}, \begin{bmatrix} ... \\ e^{-t} - 2e^t \\ -e^{-t} + 2e^t \end{bmatrix}, \] then the sum of the diagonal entries of \( A \) is equal to
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
86
views
linear-algebra
0
votes
2
answers
122
GATE 2018 | MATHS | Q-52
Consider the matrix \( A = I_9 - 2u^T u \) with \( u = \frac{1}{3}[1, 1, 1, 1, 1, 1, 1, 1, 1] \), where \( I_9 \) is the \(9 \times 9\) identity matrix and \( u^T \) is the transpose of \( u \). If \( \lambda \) and \( \mu \) are two distinct eigenvalues of \( A \), then \[ | \lambda - \mu | = \] _________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
97
views
linear-algebra
1
vote
1
answer
123
GATE 2018 | MATHS | Q-51
Consider \( \mathbb{R}^3 \) with the usual inner product. If \( d \) is the distance from \( (1, 1, 1) \) to the subspace ${(1, 1, 0), (0, 1, 1)}$ of \( \mathbb{R}^3 \), then \( 3d^2 \) is given by
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
132
views
linear-algebra
vector-space
0
votes
0
answers
124
GATE 2018 | MATHS | Q-50
Let \( M_2(\mathbb{R}) \) be the vector space of all \( 2 \times 2 \) real matrices over the field \( \mathbb{R} \). Define the linear transformation \( S : M_2(\mathbb{R}) \to M_2(\mathbb{R}) \) by \( S(X) = 2X + X^T \), where \( X^T \) denotes the transpose of the matrix \( X \). Then the trace of \( S \) equals________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
61
views
linear-algebra
vector-space
0
votes
1
answer
125
GATE 2018 | MATHS | Q-48
Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed random variables with probability density function given by \[ f_X(x; \theta) = \begin{cases} \theta e^{-\theta(x-1)}, & \text{if } x \geq 1 \\ 0, & \text{otherwise} \end{cases} \] Also, let \(X = \frac{1}{n ... {1}{X}\) (B) \(\frac{1}{X^{\frac{1}{\theta} - 1}}\) (C) \(\frac{1}{X - 1}\) (D) $X$
rajveer43
asked
in
Probability
Jan 11
by
rajveer43
97
views
probability
0
votes
1
answer
126
GATE 2018 | MATHS | QI47
Let \(\{X_i\}\) be a sequence of independent Poisson(\(\lambda\)) variables, and let \(W_n = \frac{1}{n} \sum_{i=1}^{n} X_i\). Then the limiting distribution of \(\sqrt{n}(W_n - \lambda)\) is the normal distribution with zero mean and variance given by (A) \(1\) (B) \(\sqrt{\lambda}\) (C) \(\lambda\) (D) \(\frac{\lambda}{2}\)
rajveer43
asked
in
Probability
Jan 11
by
rajveer43
115
views
probability
statistics
0
votes
1
answer
127
GATE 2018 | MATHS | Q-42 DA Practice Questions
Consider the following two statements: \(P\): The matrix \(\begin{bmatrix} 0 & 5 \\ 0 & 7 \end{bmatrix}\) has infinitely many LU factorizations, where \(L\) is lower triangular with each diagonal entry 1 and \(U\) is upper triangular. \(Q\): The matrix \( ... \(Q\) are TRUE (C) \(P\) is FALSE and \(Q\) is TRUE (D) Both \(P\) and \(Q\) are FALSE
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
166
views
linear-algebra
0
votes
1
answer
128
GATE 2018 | MATHS | Q-40
Which one of the following statements is true? (A) Every group of order 12 has a non-trivial proper normal subgroup (B) Some group of order 12 does not have a non-trivial proper normal subgroup (C) Every group of order 12 has a subgroup of order 6 (D) Every group of order 12 has an element of order 12
rajveer43
asked
in
Set Theory & Algebra
Jan 11
by
rajveer43
111
views
set-theory
0
votes
0
answers
129
GATE 2018 | MATHS | QUESTION 34
Let the cumulative distribution function of the random variable \(X\) be given by \[ F_X(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } 0 \leq x < \frac{1}{2} \\ \frac{1 + x}{2} & \text{if } \frac{1}{2} \leq x < 1 \\ 1 & \text{if } x \geq 1 \end{cases} \] Then, the probability \(P(X = \frac{1}{2})\) is given by
rajveer43
asked
in
Probability
Jan 11
by
rajveer43
63
views
probability
0
votes
1
answer
130
GATE 2018 | MATHS | Q-33
Let \(X\) and \(Y\) have a joint probability density function given by \[ f_{X,Y}(x, y) = \begin{cases} 2 & \text{if } 0 \leq x \leq 1 - y \text{ and } 0 \leq y \leq 1 \\ 0 & \text{otherwise} \end{cases} \] If \(f_Y\) denotes the marginal probability density function of \(Y\), then \(f_Y(1/2)\) is given by
rajveer43
asked
in
Probability
Jan 11
by
rajveer43
78
views
probability
0
votes
0
answers
131
GATE 2018 | MATHS | Q-24
Consider the subspaces \[ W_1 = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1 = x_2 + 2x_3 \} \] \[ W_2 = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1 = 3x_2 + 2x_3 \} \] of \( \mathbb{R}^3 \). Then the dimension of \(W_1 + W_2\) equals_________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
49
views
linear-algebra
0
votes
0
answers
132
GATE 2018 | MATHS | Q-23
Let A = A = \begin{bmatrix} a & 2f & 0 \\ 2f & b & 3f \\ 0 & 3f & c \\ \end{bmatrix} , where $a, b, c, f$ are real numbers and $f not equalto0$. The geometric multiplicity of the largest eigenvalue of A equals ._______
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
55
views
linear-algebra
0
votes
0
answers
133
Diagonalization of Matrix
Debargha Mitra Roy
asked
in
Linear Algebra
Jan 11
by
Debargha Mitra Roy
53
views
matrix
linear-algebra
eigen-value
0
votes
0
answers
134
Diagonalization of Matrix - Orthogonal Transformation
Consider a symmetric matrix $M=\begin{bmatrix} \frac{1}{3} & 0 & \frac{2}{3}\\ 0&1 &0 \\ \frac{2}{3}&0 & \frac{1}{3} \end{bmatrix}$. An orthogonal matrix $O$ which can diagonalize this matrix by an orthogonal transformation $O^TMO$ is given by $O = $ ______
Debargha Mitra Roy
asked
in
Linear Algebra
Jan 11
by
Debargha Mitra Roy
79
views
linear-algebra
eigen-value
matrix
0
votes
0
answers
135
Made Easy Mock Test 2
Rohit Chakraborty
asked
in
Mathematical Logic
Jan 11
by
Rohit Chakraborty
225
views
graph-theory
made-easy-test-series
engineering-mathematics
0
votes
0
answers
136
GATE 2019 | Maths | DA Sample questions
Let $V$ be the vector space of all $3 \times 3$ matrices with complex entries over the real field. If $W_1 = \{A \in V : A = \bar{\mathbf{A}}^T \}$ and $W_2 = \{A \in V : trace(A)=0\}$, then the dimension of $W_1 + W_2$ is equal to ______________ ($\bar{\mathbf{A}}^T $ denotes the conjugate transpose of $A$.)
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
99
views
vector-space
linear-algebra
0
votes
0
answers
137
GATE 2019 | MATHS | LINEAR ALGEBRA
Let $ \mathbf{M} $ be a $3 \times 3$ real symmetric matrix with eigenvalues $0, 2$ and $a$ with the respective eigenvectors $\mathbf{u} = \begin{bmatrix} 4 \\ b \\ c \end{bmatrix}$, $\mathbf{v} = \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}$, and ... of the above statements are TRUE? (A) I, II and III only (B) I and II only (C) II and IV only (D) III and IV only
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
59
views
linear-algebra
1
vote
0
answers
138
GATE 2019 | MATHS | QUESTION 40
If the characteristic polynomial and minimal polynomial of a square matrix $ \mathbf{A} $ are $(\lambda - 1)(\lambda + 1)^4 (\lambda - 2)^5$ and $(\lambda - 1)(\lambda + 1)(\lambda - 2)$ respectively, then the rank of the matrix $ \mathbf{A} + \mathbf{I} $, where $ \mathbf{I} $ is the identity matrix of the appropriate order, is________________
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
68
views
linear-algebra
0
votes
0
answers
139
GATE 2019 | MATHS | LIMIT
Let $ u_n = \frac{(n!)!}{1 \cdot 3 \cdot 5 \cdots (2n - 1)} $ (the set of all natural numbers). Then $ \lim\limits_{n \to \infty} \frac{n}{u_n} $ is equal to ______________
rajveer43
asked
in
Calculus
Jan 10
by
rajveer43
109
views
calculus
1
vote
0
answers
140
GATE 2019 | maths | set theory
Consider the following statements: I.The set $ \mathbb{R} $ is uncountable. II.The set $ \{ f : f \text{ is a function from } \mathbb{N} \text{ to } \{0, 1\} \} $ is uncountable. III.The set $ \{ p : p \text{ is a prime number} \} $ is uncountable. ... of the above statements are TRUE? (A)] I and IV only (B) II and IV only (C) II and III only (D) I, II, and IV only
rajveer43
asked
in
Set Theory & Algebra
Jan 10
by
rajveer43
67
views
set-theory
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