Recent questions in Engineering Mathematics / GATE Overflow for GATE CSE

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151

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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152

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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153

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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154

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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155

GATE 2022 | MATHS | Q-65
Let M be a $3 × 3$ real matrix such that $M^2 = 2M + 3I$. If the determinant of $M$ is $−9$, then the trace of $M$ equals._______

Let M be a $3 × 3$ real matrix such that $M^2 = 2M + 3I$. If the determinant of $M$ is $−9$, then the trace of $M$ equals._______

rajveer43

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156

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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157

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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158

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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159

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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160

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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161

GATE 2022 | MATHS | LINEAR ALGEBRA
Suppose that the characteristic equation of $ M \in \mathbb{C}^{3 \times 3} $ is \[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0, \] where $ \alpha, \beta \in \mathbb{C} $ with $ \alpha + \beta \neq 0 $. Which of the following statements is TRUE? (A) $ M(I ... 1} (M^{-1} + \beta I) = M - \alpha I $ (D)$ M^{-1} (M^{-1} - \beta I) = M + \alpha I $

Suppose that the characteristic equation of $ M \in \mathbb{C}^{3 \times 3} $ is\[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0, \]where \( \alpha, \beta \in \m...

rajveer43

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162

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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rajveer43 asked Jan 10

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163

GATE 2023 | MATHS | Question 59
Let $A$ be a $3 \times 3$ real matrix with $\text{det}(A + iI) = 0$, where $i = \sqrt{-1}$ and $I$ is the $3 \times 3$ identity matrix. If $\text{det}(A) = 3$, then the trace of $A^2$ is $\underline{\hspace{1cm}}$.

Let $A$ be a $3 \times 3$ real matrix with $\text{det}(A + iI) = 0$, where $i = \sqrt{-1}$ and $I$ is the $3 \times 3$ identity matrix. If \(\text{det}(A) = 3...

rajveer43

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rajveer43 asked Jan 10

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164

GATE 2023 | Maths | Sample Ques for CS-IT
Let $G$ be an abelian group and $\Phi: G \rightarrow (\mathbb{Z}, +)$ be a surjective group homomorphism. Let $1 = \Phi(a)$ for some $a \in G$. Consider the following statements: $P$: For every $g \in G$, there exists an $n \in \ ... following statements is/are correct? (A) \(P$ is TRUE (B) $P$ is FALSE (C) $Q$ is TRUE (D) $Q$ is FALSE

Let $G$ be an abelian group and $\Phi: G \rightarrow (\mathbb{Z}, +)$ be a surjective group homomorphism. Let $1 = \Phi(a)$ for some $a \in G$.Consider the follow...

rajveer43

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rajveer43 asked Jan 10

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165

GATE 2023 | Maths | Problems for DA Paper
Consider the linear system $Mx = b$, where \[ M = \begin{bmatrix} 2 & -1 \\ -4 & 3 \end{bmatrix} \] and \[ b = \begin{bmatrix} -2 \\ 5 \end{bmatrix} \]. Suppose $M = LU$, where $L$ and $U$ are lower triangular and upper ... (A) both 𝑃 and 𝑄 are TRUE (B) 𝑃 is FALSE and 𝑄 is TRUE (C) 𝑃 is TRUE and 𝑄 is FALSE (D) both 𝑃 and 𝑄 are FALSE

Consider the linear system $Mx = b$, where \[ M = \begin{bmatrix} 2 & -1 \\ -4 & 3 \end{bmatrix} \]and\[ b = \begin{bmatrix} -2 \\ 5 \end{bmatrix} \].Suppose $M = LU$...

rajveer43

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166

GATE 2023 | Maths | Linear Alegbra
Let $ M = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{bmatrix} $ and $ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $. If $ 6M^{-1} = M^2 - 6M + \alpha I $ for some $ \alpha \in \mathbb{R} $, then the value of $ \alpha $ is equal to $\underline{\hspace{1cm}}$.

Let $ M = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{bmatrix} $ and $ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $. If \( 6...

rajveer43

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167

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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168

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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169

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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170

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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rajveer43 asked Jan 10

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