Mathematics Tes...

What is the value of \(\lim _{{x} \rightarrow \infty} \frac{{x}^{3}+3 {x}^{2}+6 {x}+5}{{x}^{3}+2 {x}+6}\)?

A discrete random variable \(x\) has the probability functions as:

If \(y \frac{d y}{d x}=x\left[\frac{y^2}{x^2}+\frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi^{\prime}\left(\frac{y^2}{x^2}\right)}\right], x>0, \phi>0\), and \(y(1)=-1\), then \(\phi\left(\frac{y^2}{4}\right)\) is equal to:

If \(A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ -\frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{cc}1 & 0 \\ 1 & 1\end{array}\right), \mathrm{i}=\sqrt{-1}\), and \(Q=A^T B A\), then the inverse of the matrix \(\mathrm{AQ}^{2021} \mathrm{~A}^{\mathrm{T}}\) is equal to:

The nth term of the following sequence 25, -125, 625, -3125, …….. is:

What is the area of the region bounded by the curve \(f(x)=1-\frac{x^2}{4}, x \in[-2,2]\) and the \(x\)-axis?

Find the domain of the inverse trigonometric function \(\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)\) is:

If \(p\) and \(q\) are the lengths of the perpendiculars from the origin on the lines,

\(\mathrm{xcosec} \alpha-\mathrm{y} \sec \alpha=\mathrm{k} \cot 2 \alpha\) and \(\mathrm{x} \sin \alpha+\mathrm{y} \cos \alpha=\mathrm{k} \sin 2 \alpha\) respectively, then \(\mathrm{k}^2\) is equal to:

Find the area bounded by the curve between \(y=\sin x+\cos x\) in the interval \(0<\mathrm{x}<\frac{\pi}{2}\).

A balloon, which always remains spherical, has a variable diameter \(\frac{3}{2}(4 x +3)\). Find the rate of change of its volume with respect to \(x\).