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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
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{bm...
rajveer43
84
views
rajveer43
asked
Jan 11
Linear Algebra
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 \[ | \lambda - \mu | = \] _________
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 t...
rajveer43
93
views
rajveer43
asked
Jan 11
Linear Algebra
linear-algebra
+
–
1
votes
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
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 \( \mathbb{R}^3 \), ...
rajveer43
128
views
rajveer43
asked
Jan 11
Linear Algebra
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________
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...
rajveer43
61
views
rajveer43
asked
Jan 11
Linear Algebra
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) $X$
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} \the...
rajveer43
91
views
rajveer43
asked
Jan 11
Probability
probability
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–
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) \(\frac{\lambda}{2}\)
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...
rajveer43
109
views
rajveer43
asked
Jan 11
Probability
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
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 trian...
rajveer43
171
views
rajveer43
asked
Jan 11
Linear Algebra
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
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 p...
rajveer43
104
views
rajveer43
asked
Jan 11
Set Theory & Algebra
set-theory
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–
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
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} \\...
rajveer43
61
views
rajveer43
asked
Jan 11
Probability
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
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...
rajveer43
74
views
rajveer43
asked
Jan 11
Probability
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_________
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 \( \math...
rajveer43
47
views
rajveer43
asked
Jan 11
Linear Algebra
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 ._______
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...
rajveer43
54
views
rajveer43
asked
Jan 11
Linear Algebra
linear-algebra
+
–
0
votes
0
answers
133
Diagonalization of Matrix
Debargha Mitra Roy
54
views
Debargha Mitra Roy
asked
Jan 11
Linear Algebra
matrix
linear-algebra
eigen-value
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–
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 = $ ______
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 di...
Debargha Mitra Roy
83
views
Debargha Mitra Roy
asked
Jan 11
Linear Algebra
linear-algebra
eigen-value
matrix
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–
0
votes
0
answers
135
Made Easy Mock Test 2
Rohit Chakraborty
221
views
Rohit Chakraborty
asked
Jan 11
Mathematical Logic
graph-theory
made-easy-test-series
engineering-mathematics
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–
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 $A$.)
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 : ...
rajveer43
101
views
rajveer43
asked
Jan 10
Linear Algebra
vector-space
linear-algebra
+
–
2
votes
1
answer
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
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...
rajveer43
90
views
rajveer43
asked
Jan 10
Linear Algebra
linear-algebra
+
–
1
votes
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________________
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 + ...
rajveer43
68
views
rajveer43
asked
Jan 10
Linear Algebra
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 ______________
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
106
views
rajveer43
asked
Jan 10
Calculus
calculus
+
–
1
votes
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
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 uncou...
rajveer43
64
views
rajveer43
asked
Jan 10
Set Theory & Algebra
set-theory
+
–
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