Recent questions without answers / GATE Overflow for GATE CSE

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241

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

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rajveer43 asked Jan 11

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242

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

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rajveer43 asked Jan 11

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243

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

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rajveer43 asked Jan 11

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244

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

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rajveer43 asked Jan 11

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245

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

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246

Diagonalization of Matrix

Debargha Mitra Roy

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Debargha Mitra Roy asked Jan 11

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247

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

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Debargha Mitra Roy asked Jan 11

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248

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

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rajveer43 asked Jan 10

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249

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

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250

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

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rajveer43 asked Jan 10

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251

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

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252

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 __________

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}) $ denote...

rajveer43

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253

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 ____________________________

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} $ ...

rajveer43

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254

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

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} $. Consid...

rajveer43

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rajveer43 asked Jan 10

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255

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

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: $P:$...

rajveer43

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rajveer43 asked Jan 10

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256

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

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 ...

rajveer43

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257

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

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...

rajveer43

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rajveer43 asked Jan 10

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258

Function of Matrix - Sylvester Theorem & Cayley-Hamilton Theorem
$Prove\ that,\ sin^2A+cos^2A=1,\ where \ A=\begin{bmatrix} 1&2 \\ -1&4 \end{bmatrix}.$

$Prove\ that,\ sin^2A+cos^2A=1,\ where \ A=\begin{bmatrix} 1&2 \\ -1&4 \end{bmatrix}.$

Debargha Mitra Roy

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Debargha Mitra Roy asked Jan 10

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259

Function of Matrix - Sylvester Theorem
$Given \ A=\begin{bmatrix} 1&20&0 \\ -1&7&1 \\ 3&0&-2 \end{bmatrix},\ find\ tan\ A\ .$

$Given \ A=\begin{bmatrix} 1&20&0 \\ -1&7&1 \\ 3&0&-2 \end{bmatrix},\ find\ tan\ A\ .$

Debargha Mitra Roy

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Debargha Mitra Roy asked Jan 10

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260

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 $

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...

rajveer43

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261

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).

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 va...

rajveer43

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262

GATE 2022 | MATHS | Q-27
The number of subgroups of a cyclic group of order 12 is ______________________

The number of subgroups of a cyclic group of order 12 is ______________________

rajveer43

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263

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.

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{bm...

rajveer43

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rajveer43 asked Jan 10

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264

GATE 2022 | Maths | Q-12
Consider $P :$ Let $ M \in \mathbb{R}^{m \times n} $ with $ m > n \geq 2 $. If $ \text{rank}(M) = n $, then the system of linear equations $ Mx = 0 $ has $ x = 0 $ as the only solution. $Q:$ Let \( E \in \mathbb{R}^{n \ ... following statements is TRUE? (A) Both P and Q are TRUE (B) Both P and Q are FALSE (C)P is TRUE and Q is FALSE (D) P is FALSE and Q is TRUE

Consider$P :$ Let $ M \in \mathbb{R}^{m \times n} $ with $ m n \geq 2 $. If $ \text{rank}(M) = n $, then the system of linear equations $ Mx = 0 $ has \( x = 0 \...

rajveer43

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265

GATE 2023 | Maths | Question 60
Let $ A = [a_{ij}]$ be a $3 \times 3$ real matrix such that \[ A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 0 \\ 1 ... is the degree of the minimal polynomial of $ A $, then $ a_{11} + a_{21} + a_{31} + m $ equals $\underline{\hspace{1cm}}$.

Let $ A = [a_{ij}]$ be a $3 \times 3$ real matrix such that \[ A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A \beg...

rajveer43

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266

GATE 2023 | Maths | Practice Problem for DA Paper
Let $ T: \mathbb{R}^4 \rightarrow \mathbb{R}^4 $ be a linear transformation, and the null space of $ T $ be the subspace of $ \mathbb{R}^4 $ given by \[ \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : 4x_1 + 3x_2 + 2x_3 + x_4 = 0 \}. \] If $ \text{Rank}(T - 3I) = ... \( x(x - 3) $ (B) $ x(x - 3)^3 $ (C) $ x^3(x - 3) $ (D) $ x^2(x - 3)^2 $

Let $ T: \mathbb{R}^4 \rightarrow \mathbb{R}^4 $ be a linear transformation, and the null space of $ T $ be the subspace of $ \mathbb{R}^4 $ given by\[ \{ (x_1, x_2...

rajveer43

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267

GATE 2023 | MATHS | Practice Problem for DA Paper
Let $M = \begin{bmatrix} 4 & -3 \\ 1 & 0 \end{bmatrix}$. Consider the following statements: $P:\ M^8 + M^{12}$ is diagonalizable. $Q:\ M^7 + M^9$ is diagonalizable. Which of the following statements is correct? (A) $𝑃$ is $TRUE$ ... $𝑃$ is $FALSE$ and $𝑄$ is $TRUE$ (C) Both $𝑃$ and $𝑄$ are $FALSE$ (D) Both $𝑃$ and $𝑄$ are $TRUE$

Let $M = \begin{bmatrix} 4 & -3 \\ 1 & 0 \end{bmatrix}$. Consider the following statements:$P:\ M^8 + M^{12}$ is diagonalizable.$Q:\ M^7 + M^9$ is diagonalizable.Wh...

rajveer43

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268

GATE 2022 | Statistics | GATE-DA Sample Question
A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is $0.7$. If the company stops the quarterly dividend, the probability that the next ... (rounded off to three decimal places) that the company will not pay quarterly dividend in the long run is ______.

A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is $0.7$. If the com...

rajveer43

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269

GATE DA - 2024 | Linear Algebra
Let $𝐴$ be an $𝑛 𝑛$ real matrix. Consider the following statements. $(I)$ If $𝐴$ is symmetric, then there exists $𝑐 ≥ 0$ such that $𝐴 + 𝑐𝐼_𝑛$ is symmetric and positive definite, where $𝐼_𝑛$ is the $𝑛 𝑛$ identity matrix $(II)$ If $𝐴$ is symmetric and ... above statements is/are true? (A) Only (I) (B) Only (II) (C) Both (I) and (II) (D) Neither (I) nor (II)

Let $𝐴$ be an $𝑛 × 𝑛$ real matrix. Consider the following statements.$(I)$ If $𝐴$ is symmetric, then there exists $𝑐 ≥ 0$ such that $𝐴 + 𝑐𝐼_�...

rajveer43

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270

GATE Practice Problem | Probability
Two defective bulbs are present in a set of five bulbs. To remove the two defective bulbs, the bulbs are chosen randomly one by one and tested. If $𝑋$ denotes the minimum number of bulbs that must be tested to find out the two defective bulbs, then $𝑃(𝑋 = 3)$ (rounded off to two decimal places) equals _______________

Two defective bulbs are present in a set of five bulbs. To remove the two defective bulbs, the bulbs are chosen randomly one by one and tested. If $𝑋$ denotes the mini...

rajveer43

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rajveer43 asked Jan 10

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