Recent questions in Mathematics Foundation

The probability of a boy or a girl being born is $1 / 2$. For a family having only three children, what is the probability of having two girls and one boy?$3 / 8$$1 / 8$$...

Ten cards in a pack are numbered as $1,2,3, \ldots 10$. The probability of drawing a card with an even number or a number which is a multiple of 5 from the pack is$4 / 10...

The eigen values of the matrix $\left[\begin{array}{ll}4 & 3 \\ 3 & 4\end{array}\right]$ are$1$$2$$7$$4$

If $\mathbf{X}$ is a vector, and $\mathbf{A}$ and $\mathbf{B}$ are linear operators; then the correct mathematical relationship(s) is/are$(\mathbf{A}+\mathbf{B}) \mathbf{...

Consider the equation $\frac{d y}{d x}-x^{2}+e^{x}=0$; with $y=1$ at $x=0$.The value of $y$ at $x=1$ is ________ $\text{(rounded off to 2 decimal places)}$.Take the value...

Which one is the solution $y(x)$ for the following ordinary differential equation and the specified boundary conditions?\[\frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+2 y=2 ...

A population $\text{(with mean $\mu$)}$ follows normal distribution. Ten samples $(N)$ are drawn at random with a mean value of " $x$ " and standard deviation of " $S$ "....

Consider the function $f(x)=\ln (\sin (x))$Expand $f(x+h)$ using Taylor's series. In this context, the correct statement(s) is/areSecond term in the Taylor's series i.e.,...

Consider the equation for a curve, $y=f(x)=x^{2}+x$.The area enclosed by the curve, the $x$-axis $\text{( $y=0$ line)}$; the vertical lines passing through $x=1$ and $x=2...

Given a fair six-faced dice where the faces are labelled $'1'$, $'2'$, $'3'$, $'4'$, $'5'$, and $'6'$, what is the probability of getting a $'1'$ on the first roll of the...

Given are two ordinary differential equations$\mathrm{P}: \frac{d y}{d x}+x=x \sin y$$\mathrm{Q}: \frac{d y}{d x}+x y=e^{x} y$The correct choice is$\mathrm{P}$ is linear;...

$\mathbf{P}$ and $\mathbf{Q}$ are square matrices. Consider the following$\mathrm{X}:\left(\mathbf{P}^{-\mathbf{1}}\right)^{-1}=\mathbf{P}$Y: Symmetric if $\mathbf{Q}=-\m...

Given are two infinite series$\qquad P: \sum \frac{n^{2}+1}{n^{2}}$ $\qquad Q: \sum\left(1+\frac{1}{n}\right)^{-n}$The correct choice is$\text{P}$ is convergent series; $...

A researcher compiled the following information about the performance of a kit in an outbreakInfection stateKit responseDisease (probability =0.002)Positive response (pro...

The value of $\frac{4}{\pi} \int_{0}^{\pi / 2} \sin ^{2} x d x$ is ___ ____ $\text{(rounded off to two decimal places)}$.

$\text{W1, W2, W3...W9}$ represent the holding times of $9$ water samples, which follow a normal distribution with mean $=8.33$ and standard deviation $=4.472$.$\text{M}$...

Second order ordinary differential equation $\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y=0$ has values $y=2$ and $\frac{d y}{d x}=1$ at $x=0$. The value of y at $x=1$ ...

Consider two matrices $\mathbf{P}=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]$ and $\mathbf{Q}=\left[\begin{array}{ll}5 & 4 \\ 0 & 2\end{array}\right]$. If $\...

For the function } f(x)=x \sqrt{4-x^{2}}, the maximum value in the range $-2 \leq x \leq 2$ is ____________ $\text { (rounded off to two decimal places). }$

A tangent is drawn on the curve of the function $y=x^{2}$ at the point $(x, y)=(3,9)$. The slope of the tangent is$3$$6$$9$$12$