Questions without answers in Engineering Mathematics

For the complex function$$f(z)=f(x+i y)=x^{3}-4 x y^{2}+i\left(4 x^{2} y-2 y^{3}\right)$$consider the following two statements$P: f$ is not analytic at $(0,0)$.$Q: f$ doe...

Let $C$ be the circle $(x-1)^{2}+y^{2}=25$ oriented counterclockwise. Then, the value of the line integral$$\oint_{C}\left[\left(x^{5}-3 y\right) d x+\left(-2 x+e^{y^{2}}...

Let $W(x)$ denote the Wronskian of two linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$ of the differential equation$$(x-1) \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d...

Which of the following statements is/are true?$Y=1+1.8 X$ and $X=1-0.5 Y$ are estimated as lines of regression of $Y$ on $X$ and $X$ on $Y$, respectively.$Y=1+1.3 X$ and ...

Consider the matrix $A=\left[\begin{array}{ccc}1 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$. Then, which of the following statements is/are true?A...

Let $L$ be the lamina of the form $x^{2}+4 y^{2} \leq 64,0 \leq y \leq 4$, with density $\rho(x, y)=\mid x \mid y$. Then, the mass of $L$ (in integer) is $\_\_\_\_$

Let $X$ be a random variable having probability density function (pdf)$$f_X(x) =\left\{\begin{array}{ll}\dfrac{e}{2(e-1)} e^{-x}, & 0 < x < 1, \\\dfrac{1}{2}, & 1 < x < 2...

Match each entry of $\text{List-1}$ with a suitable entry in $\text{List-2}$ and choose the correct option.$\textbf{List-1}$$\textbf{List-2}$$\mathrm{P}$$\text{The sum of...

Consider the following partial differential equation (PDE)$$(y-1) \frac{\partial^{2} u}{\partial x^{2}}-(x-3)^{2} \frac{\partial^{2} u}{\partial y^{2}}+y^{2} \frac{\parti...

Let $\left[\begin{array}{ccc}l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & -10\end{array}\right]\left[\begin{array}{ccc}1 & u_{12} & u_{13} \\ 0 & 1 & u_{23}...

The initial value problem$$\begin{array}{l}\dfrac{d u}{d t}=u^{2}+t^{2}, \quad t \geq 0 \\\text { with } u(0)=1\end{array}$$is solved by using the explicit Euler method w...

​​Let $X$ and $Y$ be two random variables with mean $0$, variance $1$, and correlation coefficient $\frac{1}{3}$. Then the value of $\operatorname{Var}(X+3 Y)$ is equal t...

​​​​Consider the second order Partial Differential Equation $\textsf{(PDE)}$$$4 x^{2} \frac{\partial^{2} u}{\partial x^{2}}+4(x+y) \frac{\partial^{2} u}{\partial x \parti...

​Consider the infinite series$$ \begin{array}{l} \text { (P) }: \sum_{n=2}^{\infty} \frac{1}{(n \log n)^{1 / n}} \\\text { (Q): } \sum_{n=1}^{\infty} \frac{n^{n}}{(2 n)!}...

​​​​Suppose the polynomial $a+b x+c x^{2}+d x^{3}$ interpolates the data, $(-1,1),(0,3),(1,2)$ and $(2,4)$.Then which one of the following statements is correct?$a=-2 c, ...

​​Let $C$ be the positively oriented boundary of the domain bounded by the curves $y=2 x^{2}$ and $y^{2}=4 x$. Then the value of the line integral$$\oint_{C}\left(2 y^{2}...

Let $y(x)$ be the solution of the initial value problem,$$\begin{array}{c}x^{2} y^{\prime \prime}+x y^{\prime}-y=0, \quad x>0, \\y(1)=0, \quad y^{\prime}(1)=2 .\end{array...

Suppose that $2$ is an eigenvalue of the matrix$$\left[\begin{array}{ccc}0 & 3 & -\alpha \\0 & 1 & 0 \\1 & -1 & 3\end{array}\right]$$Then the value of $\alpha$ is equal t...

​​​​​Let $f(z)$ be an analytic function such that$$ {Re}\left(f^{\prime}(z)\right)=3 x^{2}-4 y-3 y^{2}, f(i)=0, f^{\prime}(0)=0, $$where $i=\sqrt{-1}$. Then the value of ...

​​Consider the function$$ f(x, y)=x^{2} y+2 x y^{2}-2 x^{2} y^{2}.$$Then which one of the following statements is correct?$\left(\frac{3}{2}, 0\right)$ is a point of loca...