Questions without answers in Engineering Mathematics

Consider the two functions $f_{1}(x)=\dfrac{x^{2}-4}{x-2}$ and $f_{2}(x)=x^{2}-2 x+2$. Which of the following is the value of $\left(f_{1}(x)+f_{2}(x)\right)$ as $x \righ...

Which of the following is one of the eigenvalues for the matrix given below?\[\left[\begin{array}{cc}3 & 4 \\4 & -3\end{array}\right]\]$1$$3$$5$$7$

Which of the following functions has the highest area under the curve between $x=0$ and $x={10}?$$y=x+8$$y=3 x$$y=2 x+1$$y=x+2$

The equation $\dfrac{d^{2} y}{d x^{2}}-y=0$ has a solution of the form $y=e^{A x}$. The value(s) of $A$ satisfying this is/are:$0$$1$$-1$$-\infty$

For the given matrix, which of the statements given below is/are true?\[\left[\begin{array}{ccc}3 & 1 & 5 \\2 & -1 & 0 \\5 & 2 & 9\end{array}\right]\]The matrix is full r...

Consider the equation $\dfrac{d y}{d x}=\dfrac{1}{x}$The value of the integral $\int_{1}^{2} d y$ using trapezoidal method and interval $h=0.25$ is $\_\_\_\_$. (rounded o...

If the straight lines given by the following two equations are parallel to each other, the value of $\boldsymbol{a}$ is $\_\_\_\_$. (answer in integer)\[4 x+7 y=6 ; \quad...

A heat exchanger during operation in a bioprocess has a steady temperature of $90^{\circ} \mathrm{C}$. After completion of its operation, it was shut down and it was obse...

From the following plot of $\dfrac{d y}{d x}$ versus $x$ and if $y(2)=5$, the value of $y(3)$ is $\_\_\_\_$. (rounded off to one decimal place) 

​​​​The minimum value of the function$$ f(x)=x+\frac{4}{x} $$for $x>0$ is$1$$2$$3$$4$

Consider a nonlinear algebraic equation, $e^{x}-2=0$. Using the Newton-Raphson method, with the initial guess of $x_{0}=1$, the approximated value of the root of the equa...

The value of $k$, for which the linear equations $2 x+3 y=6$ and $4 x+6 y=3 k$ have at least one solution, is $\_\_\_\_\_$ (Answer in integer)

Two fair six-sided dice, with sides numbered $1$ to $6$ , are thrown once. The probability of getting $7$ as the sum of the numbers on the top side of the dice is $\_\_\_...

​​Let $m$ and $n$ be fixed real numbers. If the function $y(t)=C_{1} e^{t}+C_{2} e^{-t}$ is a solution of$$ \frac{d^{2} y}{d t^{2}}+m \frac{d y}{d t}+n y=0$$for any const...

​​​​If the function$$ f(x)=\left\{\begin{array}{ll}\sin 2 x, & \text { for } x>0 \\a+b x, & \text { for } x \leq 0\end{array}\right.$$where $a$ and $b$ are constants, is ...

Let $a_{0}=0$ and define $a_{n}=\frac{1}{2}\left(1+a_{n-1}\right)$ for all positive integers $n \geq 1$. The least value of $n$ for which $\left|1-a_{n}\right|<\frac{1}{2...

Let $A=\left(\begin{array}{ccc}1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1\end{array}\right)$. If the eigenvalues of $A$ are $-2,1$, and $2$ , then the value of $k$ is $\_\_\_\_...

A value of $k$ for which the linear equations $(k-1) x+3 y=0$ and $2 x+k y=0$ have a non-zero solution is $\_\_\_\_\_\_\_$.$1$$2$$3$$4$

The value of the series $1+\sin x+\cos ^{2} x+\sin ^{3} x+\cdots$ at $x=\frac{\pi}{4}$ is $\_\_\_\_\_\_\_$.$\frac{1}{\sqrt{2}+1}$$\frac{\sqrt{2}}{\sqrt{2}+1}$$\frac{1}{\s...

The solution of the differential equation $\frac{d y}{d x}=y+e^{-x}$ that satisfies $y(0)=-\frac{1}{2}$ is $\_\_\_\_\_\_\_$.$-\frac{1}{2} e^{-\frac{x}{2}}$$-\frac{1}{2} e...

The six faces of a cube (die) are numbered as $1,2,3,4,5$ and $6$, and it is rolled once. An outcome is the observed number on the top face. If the probability of getting...

The value of the limit $\lim _{x \rightarrow \infty} \frac{x}{2} \ln \left(1+\frac{2024}{x}\right)$ is $\_\_\_\_\_\_\_\_$.

Let $y(x)=x^2 \ln x$ for $x>0$, be a solution of $x^2 \frac{d^2 y}{d x^2}+4 y=\alpha x \frac{d y}{d x}$. Then the value of $\alpha$ is $\_\_\_\_\_\_\_\_\_$.

The absolute relative error in evaluating the integral $\int_0^1 x^2 d x$ by the trapezoidal rule with the step size $0.25$ is $\%$ $\_\_\_\_\_\_\_\_$ (rounded off to $2$...

The value of $\lim _{x \rightarrow 0}\left[\frac{\cos 2 x-\cos 4 x}{x^{2}}\right]$ is ___________.

Two fair six-sided dice are thrown. The probability of getting $12$ as the product of the numbers on the dice (rounded off to two decimal places) is __________.

The number of different possible ways of forming five intramolecular disulfide bonds with ten cysteine residues of a protein is ______________.

If $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right)$, the value of $\left|A^{4}+3 A^{2}-5 A+6 I\right|$ is _____________.

If $f(x)=\frac{\sin x+\cos x}{\sin x-\cos x}$, the value of $f^{\prime}(x)$ at $x=0$ is ___________.

If $f(2)=5$ and $(f(x))(f(x+1))=3$ for all real values of $x$, the value of $f(10)$ is ____________.

Ten playing cards numbered $1,2,3, \ldots, 10$ are placed face down on a table. One card is drawn at random, its number recorded, and then replaced face down. A card is d...

What is the order of the differential equation given below?$\dfrac{d^{2}y}{dx^{2}} – 6x = 3x^{4} – 2x^{3} + 2$$1$$2$$3$$4$

If the eigenvalues of a $2 \times 2$ matrix $P$ are $4$ and $2,$ then the eigenvalues of the matrix $P^{-1}$ are$0, 0$$0.0625, 0.25$$0.25, 0.5$$2, 4$

Consider a nonlinear algebraic equation, $x\ln x + x – 1 = 0.$ Using the Newton-Raphson method, with the initial guess of $x_{0} = 3,$ the value of $x$ after one iteratio...

The probability density function of a random variable $X$ is $p(x) = 2e^{-2x}.$ The probability $P(1 \leq X \leq 2)$ (rounded off to two decimal places) is _________.

The maximum value of the function $f(x) = 3x^{2} – 2x^{3}$ for $x>0$ is ________.

A $2\times 2$ matrix $\textbf{P}$ has an eigen value $\lambda_{1}=2$ with eigenvector $x_{1}=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and another eigenvalue $\lambda_{2}=5...

Consider the ordinary differential equation $\dfrac{dy}{dx} = f(x,y) = 2x^{2} – y^{2}.$ If $y(1)=1,$ the value(s) of $y(1.5),$ using the Euler’s implicit method $\left[y_...

Consider a random variable $X$ with mean $\mu_{x}=0.1$ and variance $\sigma_{x}^{2}=0.2.$ A new random variable $Y=2X+1$ is defined. The variance of the random variable $...

For $x_{1}>0$ and $x_{2}>0,$ the value of $\displaystyle{}\lim_{x_{1} \to x_{2}}\frac{x_{1}-x_{2}}{x_{2}\ln\left(\frac{x_{1}}{x_{2}}\right)}$ is __________.