Recent activity in Engineering Mathematics

For the matrix$[A]=\left[\begin{array}{ccc}1 & -1 & 0 \\-1 & 2 & -1 \\0 & -1 & 1\end{array}\right]$which of the following statements is/are TRUE?$[A]\{x\}=\{b\}$ has a un...

Two vectors $\left[\begin{array}{llll}2 & 1 & 0 & 3\end{array}\right]^{T} $ and $\left[\begin{array}{cccc}1 & 0 & 1 & 2\end{array}\right]^{T}$ belong to the null space of...

The eigenvalues of $[A]=\left[\begin{array}{ccc}2 & -3.5 & 6 \\ 3.5 & 5 & 2 \\ 8 & 1 & 8.5\end{array}\right]$ are $\lambda_{1}=-1.547, \lambda_{2}=12.330$, and $\lambda_{...

Matrix $\text{A}$ has the eigenvalues $1,2$, and $3$. The Trace of $\mathrm{A}^{2}$ is$6$$14$$20$$8$

The rank of the matrix $\begin{bmatrix} 5 & 0 & -5 & 0\\ 0 & 2 & 0 & 1\\ -5 & 0 & 5 & 0\\ 0 & 1 & 0 & 2 \end{bmatrix}$ is$1$$2$$3$$4$

The rank of matrix $\begin{bmatrix} 1 & 2 & 2 & 3\\ 3 & 4 & 2 & 5\\ 5 & 6 & 2 & 7\\ 7 & 8 & 2 & 9 \end{bmatrix}$ is$1$$2$$3$$4$

What is the minimum number of multiplications involved in computing the matrix product $PQR?$ Matrix P has $4$ rows an $2$ columns, matrix $Q$ has $2$ rows and $4$ column...

Let $y$ be a non-zero vector of size $2022 \times 1.$ Which of the following statement(s) is/are $\text{TRUE}?$$yy^{T}$ is a symmetric matrix.$y^{T}y$ is an eigenvalue of...

$\text{P}$ and $\text{Q}$ are two square matrices of the same order. Which of the following statement(s) is/are correct?If $\text{P}$ and $\text{Q}$ are invertible, then ...

For the matrix$$[A]=\left[\begin{array}{lll}1 & 2 & 3 \\3 & 2 & 1 \\3 & 1 & 2\end{array}\right]$$which of the following statements is/are TRUE?The eigenvalues of $[A]^{T}...

Consider two matrices $\mathbf{A}=\left[\begin{array}{lll}2 & 1 & 4 \\ 1 & 0 & 3\end{array}\right]$ and $\mathbf{B}=\left[\begin{array}{cc}-1 & 0 \\ 2 & 3 \\ 1 & 4\end{ar...

​​​​​Suppose $\lambda$ is an eigenvalue of matrix $A$ and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $B=A-2 I$, where $I$ is the i...

Pick the CORRECT eigen value(s) of the matrix $[A]$ from the following choices.$$[A]=\left[\begin{array}{ll} 6 & 8 \\ 4 & 2 \end{array}\right]$$$10$$4$$-2$$-10$

For the matrix $[A]$ given below, the transpose is $\_\_\_\_\_\_\_\_\_\_\_$$$[A]=\left[\begin{array}{lll} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{array}\right]$$$\left[\...

The matrix $M$ is defined as $$M = \begin{bmatrix} 1 & 3\\ 4 & 2 \end{bmatrix}$$ and has eigenvalues $5$ and $-2$. The matrix $Q$ is formed as $$Q = M^{3} - 4M^{2} - 2M$$...

​​​​​The smallest positive root of the equation\[x^{5}-5 x^{4}-10 x^{3}+50 x^{2}+9 x-45=0\]lies in the range$0 < x \leq 2$$2 < x \leq 4$$6 \leq x \leq 8$$10 \leq x \leq 1...

​​​​​The second-order differential equation in an unknown function $u: u(x, y)$ is defined as\[\frac{\partial^{2} u}{\partial x^{2}}=2\]Assuming $g: g(x), f: f(y)$, and $...

Consider the function given below and pick one or more CORRECT statement(s) from the following choices.$$f(x)=x^{3}-\frac{15}{2} x^{2}+18 x+20$$$f(x)$ has a local minimum...

Numerically integrate, $f(x)=10x-20x^2$ from lower limit $a=0$ to upper limit $b=0.5$. Use Trapezoidal rule with five equal subdivisions. The value (in $\text{units}, \te...

​​​​​​For the following partial differential equation,\[x \frac{\partial^{2} f}{\partial x^{2}}+y \frac{\partial^{2} f}{\partial y^{2}}=\frac{x^{2}+y^{2}}{2}\]which of th...

Consider two Ordinary Differential Equations $\text{(ODEs)}$:$\text{P}$: $\frac{d y}{d x}=\frac{x^{4}+3 x^{2} y^{2}+2 y^{4}}{x^{3} y}$$\mathrm{Q}: \frac{d y}{d x}=\frac{-...

The maximum value of the function $h(x)=-x^{3}+2 x^{2}$ in the interval $[-1,1.5]$ is equal to $\_\_\_\_$ (rounded off to $1$ decimal place).

Which of the following probability distribution functions (PDFs) has the mean greater than the median?Function $1$Function $2$Function $3$Function $4$

A remote village has exactly $1000$ vehicles with sequential registration numbers starting from $ 1000.$ Out of the total vehicles, $ 30 \% $ are without pollution clear...

​​​​​​Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:$\frac{\partial^{2} u}{\partial t^{...

A water resources project with an expected life of $25$ years has to be designed for an acceptable risk of $5 \%$ against a design flood. The return period for the design...

​​​​The value of $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+x}\right)$ is equal to$-1$$-0.5$$-2$$0$

​​​​Let $A=\left[\begin{array}{cc}1 & 1 \\ 1 & 3 \\ -2 & -3\end{array}\right]$ and $b=\left[\begin{array}{l}b_{1} \\ b_{2} \\ b_{3}\end{array}\right]$. For $A x=b$ to be ...

Pick the CORRECT solution for the following differential equation$$\frac{d y}{d x}=e^{x-y}$$$y=\ln \left(e^{x}+\right.$ Constant $)$$\ln (y)=x+$ Constant$\ln (y)=\ln \lef...

The "order" of the following ordinary differential equation is $\_\_\_\_\_\_\_\_\_\_$$$\frac{d^{3} y}{d x^{3}}+\left(\frac{d^{2} y}{d x^{2}}\right)^{6}+\left(\frac{d y}{d...

The eigenvalues of matrix $\begin{bmatrix} 9 & 5 \\ 5 & 8 \end{bmatrix}$ are$-2.42$ and $6.86$$3.48$ and $13.53$$4.70$ and $6.86$$6.86$ and $9.50$

Integration of $\ln (x)$ with $x$ i.e.,$$ \int \ln (x) d x= \_\_\_\_\_\_\_\_\_\_\_\_\_$$$x \cdot \ln (x)-x+$ Constant$x-\ln (x)+$ Constant$x \cdot \ln (x)+x+$ Constant$\l...

​​​​​​The probability that a student passes only in Mathematics is $\frac{1}{3}$. The probability that the student passes only in English is $\frac{4}{9}$. The probabilit...

In the differential equation $\frac{d y}{d x}+\alpha x y=0, \alpha$ is a positive constant. If $y=1.0$ at $x=0.0$, and $y=0.8$ at $x=1.0$, the value of $\alpha$ is_______...