Recent questions in Mathematics Foundation

If $2 X+\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 8 \\ 7 & 2\end{array}\right]$, then $X$ is $\_\_\_\_$.$\left[\begin{array}{ll}2...

The correct integral of $\int \dfrac{\sin x}{\sqrt{1+\cos x}} d x$ (assuming integral constant as $c$ ) is $\_\_\_\_$ .$-2 \sqrt{2} \sin \dfrac{x}{2}+c$$-2 \sqrt{2} \cos ...

Given $f(x)=3 x^{4}+4 x^{3}-12 x^{2}+6$, the minimum value of $f(x)$ is $\_\_\_\_$.$-6$$-36$$-26$$-46$

A box contains $26$ cards with all English alphabets from $\text{A}$ to $\text{Z}$.The probability of randomly choosing the card with either '$\text{E}$' or '$\text{S}$' ...

Which ONE of the following options is the CORRECT formula for Chi-Square $\left(\chi^{2}\right)$ test?Given, $O_{1}, O_{2}, \ldots . O_{n}$ are the observed values, $E_{1...

A $6$-hour rainfall of $6$ cm at a station was found to have a return period of $40$ years.The probability that a $6$ -hour rainfall of this or larger magnitude will occu...

A first-order ordinary differential equation is given as follows:$$\dfrac{d y}{d x}+x^{2} y=0$$Which ONE of the following options CORRECTLY represents the characteristics...

Consider the matrix $A=\left[\begin{array}{ccc}2 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 10\end{array}\right]$.Which ONE of the following options CORRECTLY lists the eigenvalues ...

A system of linear equations is given as $\mathrm{A X}=\mathrm{B}$, where $\mathrm{A}=\left[\begin{array}{cc}2 & 5 \\ 5 & -2\end{array}\right], \quad \mathrm{B}=\left[\be...

Standard deviation for the particulate matter (PM) concentrations listed in the table is $\_\_\_\_$ $\mu \mathrm{g} / \mathrm{m}^{3}$ (rounded off to one decimal place).$...

The area in the first quadrant bounded by the function $y=(8-x)$ and the coordinate axes is $\_\_\_\_$ square units (answer in integer).

​​​​Assuming $s>\mid a \mid$; Laplace transform of $f(x)=\cosh a x$ is$\frac{s}{s^{2}+a^{2}}$$\frac{a}{s^{2}+a^{2}}$$\frac{s}{s^{2}-a^{2}}$$\frac{a}{s^{2}-a^{2}}$

​​​​For the ordinary differential equation $\frac{d^{2} y}{d x^{2}}+4 y=0$, the general solution is$y=c_{1} \cos 2 x+c_{2} \sin 2 x$$y=c_{1} \cosh 2 x+c_{2} \sinh 2 x$$y=...

​​​Consider the following two series$$\begin{array}{l} \mathrm{P}: \sum_{n=1}^{\infty} \frac{1}{n} \\ \mathrm{Q}: \sum_{n=1}^{\infty} \frac{1}{n^{2}} \end{array}$$Choose ...

​​​​​​Consider the following statementsEnvironmental pollutant concentration is generally modeled using lognormal distribution.Environmental pollutant concentration is ge...

$\lim _{x \rightarrow 0}\left(\frac{\ln (1+x)}{2 \sin x}\right)$ is $\_\_\_\_\_\_$ (rounded off to two decimal places).

A residential family is considering two cities for relocation. The data related to pollutant exposure and associated health cost per year are given in the following figur...

The value of $\int_{0}^{\infty} \frac{\sin 4 x}{\pi x} d x$ is $\_\_\_\_\_\_$ (rounded off to two decimal places)

The following is a system of linear equations$$\begin{array}{c} x-2 y+z=34 \\ 2 x+y+z=102 \\x+y-3 z=17\end{array}$$The value of $(x+y+z)$ is $\_\_\_\_\_\_$ (rounded off t...

A tank has inflow, outflow and stirring mechanism. Initially, the tank holds $500 \: L$ of a brine solution of concentration $200 \mathrm{~g} / \mathrm{L}$. At $t=0$, an ...