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211
GATE 2016 | MATHS | Q-12
Consider the following statements P and Q: (P) : If \( M = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{bmatrix} \), then M is singular. (Q) : Let S be a diagonalizable matrix. If T is a matrix such that \( ... ), then T is diagonalizable. Which of the above statements hold TRUE? (A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
87
views
linear-algebra
0
votes
0
answers
212
GATE 2016 | MATHS | Q-14
Consider a real vector space \( V \) of dimension \( n \) and a non-zero linear transformation \( T: \mathbb{V} \rightarrow \mathbb{V} \). If \( \text{dim}(T) < n \) and $T^2 = \lambda T$, for some \( \lambda \in \mathbb{R} \backslash \{0\} \), then which ... 0 \) for all \( X \in \mathbb{W} \) (C) \( T \) is invertible (D) \( \lambda\) is the only eigenvalue of \( T \)
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
74
views
linear-algebra
0
votes
0
answers
213
GATE 2018 | MATH | Q-64
Let \(X_1\) and \(X_2\) be independent geometric random variables with the same probability mass function given by \(P(X = k) = p(1 - p)^{k-1}\), \(k = 1, 2, \ldots\). Then the value of \(P(X_1 = 2 | X_1 + X_2 = 4)\) correct up to three decimal places is____________
rajveer43
asked
in
Probability
Jan 11
by
rajveer43
105
views
probability
statistics
0
votes
0
answers
214
GATE 2018 | MATHS | Q-63
Let $X$ be the number of heads in 4 tosses of a fair coin by Person 1 and let $Y$ be the number of heads in 4 tosses of a fair coin by Person 2. Assume that all the tosses are independent. Then the value of $P(X = Y )$ correct up to three decimal places is_________
rajveer43
asked
in
Probability
Jan 11
by
rajveer43
107
views
probability
0
votes
0
answers
215
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
87
views
linear-algebra
0
votes
0
answers
216
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
62
views
linear-algebra
vector-space
0
votes
0
answers
217
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
66
views
probability
0
votes
0
answers
218
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
51
views
linear-algebra
0
votes
0
answers
219
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
59
views
linear-algebra
0
votes
0
answers
220
Control Unit
Consider a microprogrammed control unit has to support 32 number of instructions. For each instruction execution control unit generate a sequence of 64 control words. Each micro instruction contains 3 fields: 118 control signals to support horizontal control unit, a MUX select field to select one of 8 inputs, and a next address field. The size of control memory needed is?
arnabjana09
asked
in
CO and Architecture
Jan 11
by
arnabjana09
121
views
co-and-architecture
control-unit
microprogramming
0
votes
0
answers
221
Diagonalization of Matrix
Debargha Mitra Roy
asked
in
Linear Algebra
Jan 11
by
Debargha Mitra Roy
54
views
matrix
linear-algebra
eigen-value
0
votes
0
answers
222
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
83
views
linear-algebra
eigen-value
matrix
0
votes
0
answers
223
Made Easy Mock Test 2
Rohit Chakraborty
asked
in
Mathematical Logic
Jan 11
by
Rohit Chakraborty
230
views
graph-theory
made-easy-test-series
engineering-mathematics
0
votes
0
answers
224
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
104
views
vector-space
linear-algebra
0
votes
0
answers
225
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
62
views
linear-algebra
1
vote
0
answers
226
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
69
views
linear-algebra
0
votes
0
answers
227
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
110
views
calculus
1
vote
0
answers
228
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
68
views
set-theory
0
votes
0
answers
229
GATE 2021 | MATHS | QUESTION
Let $ \mathbf{A} $ be a square matrix such that $ \det(\mathbf{xI} - \mathbf{A}) = \mathbf{x}^4 (\mathbf{x} - 1)^2 (\mathbf{x} - 2)^3 $, where $ \det(\mathbf{M}) $ denotes the determinant of a square matrix $ \mathbf{M} $ ... $ 0 $ of $ \mathbf{A} $ is __________
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
53
views
linear-algebra
0
votes
0
answers
230
GATE 2021 | MATHS | QUESTION
Let $ \mathbb{F} $ be a finite field, and $ \mathbb{F}^{\times} $ be the group of all nonzero elements of $ \mathbb{F} $ under multiplication. If $ \mathbb{F}^{\times} $ has a subgroup of order $ 17 $, then the smallest possible order of the field $ \mathbb{F} $ is ____________________________
rajveer43
asked
in
Mathematical Logic
Jan 10
by
rajveer43
106
views
discrete-mathematics
0
votes
0
answers
231
GATE 2021 | MATHS | PRACTICE PROBLEMS FOR DA PAPER
Let $ \langle \cdot, \cdot \rangle: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} $ be an inner product on the vector space $ \mathbb{R}^n $ over $ \mathbb{R} $. Consider the following statements: $P:$ ... P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
111
views
linear-algebra
vector-space
0
votes
0
answers
232
GATE 2021 | MATHS | Q-20
Let $ f: \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \to \mathbb{R} $ be given by $ f(x) = \frac{\pi}{2} + x - \tan^{-1}(x) $. Consider the following statements: $P:$ $ |f(x) - f(y)| < |x - y| $ ... Then the correct option is: (A) both P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE
rajveer43
asked
in
Set Theory & Algebra
Jan 10
by
rajveer43
57
views
functions
set-theory
0
votes
0
answers
233
GATE 2021 | MATHS | Q-14
Let $ \mathbf{R} $ be the row reduced echelon form of a $ 4 \times 4 $ real matrix $ \mathbf{A} $, and let the third column of $ \mathbf{R} $ be $ \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} $. Consider the following statements: $P:$ ... : (A) both P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
54
views
linear-algebra
0
votes
0
answers
234
GATE 2021 | MATHS | Q-11
Let $ \mathbf{A} $ be a $ 3 \times 4 $ matrix and $ \mathbf{B} $ be a $ 4 \times 3 $ matrix with real entries such that $ \mathbf{A}\mathbf{B} $ is non-singular. Consider the following statements: $P:$ Nullity of $ \mathbf{A} $ is $ 0 $. $Q:$ ... Then (A)]both P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
59
views
linear-algebra
1
vote
0
answers
235
Function of Matrix - Sylvester Theorem & Cayley-Hamilton Theorem
$Prove\ that,\ sin^2A+cos^2A=1,\ where \ A=\begin{bmatrix} 1&2 \\ -1&4 \end{bmatrix}.$
Debargha Mitra Roy
asked
in
Linear Algebra
Jan 10
by
Debargha Mitra Roy
58
views
eigen-value
0
votes
0
answers
236
Function of Matrix - Sylvester Theorem
$Given \ A=\begin{bmatrix} 1&20&0 \\ -1&7&1 \\ 3&0&-2 \end{bmatrix},\ find\ tan\ A\ .$
Debargha Mitra Roy
asked
in
Linear Algebra
Jan 10
by
Debargha Mitra Roy
40
views
eigen-value
0
votes
0
answers
237
GATE 2022 | MATHS | Q-56 SAMPLE QUESTION FOR DA
Consider \( \mathbb{R}^3 \) as a vector space with the usual operations of vector addition and scalar multiplication. Let \( x \in \mathbb{R}^3 \) be denoted by \( x = (x_1, x_2, x_3) \). Define subspaces \[ W1 := \{x \in \mathbb{R}^3 : x_1 + 2x_2 - x_3 = 0\} ... {R}^3) = 1 \) (C) \( \text{dim}(W1 + W2) = 2 \) (D) \( \text{dim}(W1 \cap W2) = 1 \)
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
64
views
linear-algebra
0
votes
0
answers
238
GATE 2022 | MATHS | Q-32
If the function \( f(x, y) = x^2 + xy + y^2 + \frac{1}{x} + \frac{1}{y} \), \( x \neq 0, y \neq 0 \), attains its local minimum value at the point \((a, b)\), then the value of \(a^3 + b^3\) is _________(rounded off to TWO decimal places).
rajveer43
asked
in
Calculus
Jan 10
by
rajveer43
77
views
calculus
0
votes
0
answers
239
GATE 2022 | MATHS | Q-27
The number of subgroups of a cyclic group of order 12 is ______________________
rajveer43
asked
in
Set Theory & Algebra
Jan 10
by
rajveer43
50
views
discrete-mathematics
0
votes
0
answers
240
GATE 2022 | MATHS | Q-25
Consider the linear system of equations \(Ax = b\) with \[ A = \begin{bmatrix} 3 & 1 & 1 \\ 1 & 4 & 1 \\ 2 & 0 & 3 \\ \end{bmatrix} \] and \[ b = \begin{bmatrix} 2 \\ 3 \\ 4 \\ \end ... for any initial vector. (C) The Gauss-Seidel iterative method converges for any initial vector. (D) The spectral radius of the Jacobi iterative matrix is less than 1.
rajveer43
asked
in
Linear Algebra
Jan 10
by
rajveer43
78
views
linear-algebra
matrix
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