If \(f(2)=4\) and \(f^{\prime}(2)=1\), then
\[
\lim _{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}
\]
is equal to

Choose the correct option about the matrices given below
\(\begin{aligned} & A=\left[\begin{array}{ccc}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} & 0 \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1\end{array}\right] \\ & B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \frac{\pi}{3} & \sin \frac{\pi}{3} \\ 0 & -\sin \frac{\pi}{3} & \cos \frac{\pi}{3}\end{array}\right] \\ & C=\left[\begin{array}{ccc}\cos \frac{\pi}{6} & 0 & \sin \frac{\pi}{6} \\ 0 & 1 & 0 \\ -\sin \frac{\pi}{6} & \cos \frac{\pi}{6} & 0\end{array}\right]\end{aligned}\)
\(D=\left[\begin{array}{ccc}
\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\
-\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\)

If \(x^2+y^2=t+\frac{1}{t}\) and \(x^4+y^4=t^2+\frac{1}{t^2}\), then \(\frac{d y}{d x}\) is equal to

\[
\int e^x\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right) d x=
\]

\(f(x)\) is a quadratic polynomial satisfying the condition \(f(x)+f\left(\frac{1}{x}\right)=f(x) f\left(\frac{1}{x}\right)\). If \(f(-1)=0\), then the range of \(f\) is

The equation of the hyperbola which passes through the point \((2,3)\) and has the asymptotes \(4 x+3 y-7=0\) and \(x-2 y-1=0\) is

The general solution of the differential equation \(\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x\) is

\(\int_0^{400 \pi} \sqrt{1-\cos 2 x} d x=\)

The equation of a tangent to the circle \(x^2+y^2+2 x-12 y\) \(-132=0\) which is perpendicular to the line \(12 x+5 y+k=0\) is

\(f(x)\) is a quadratic expression such that \(f(x)\) is negative when \(x \in\left(-\infty,-\frac{5}{3}\right) \cup(3, \infty)\) and positive when \(x \in\left(-\frac{5}{3}, 3\right) \cdot g(x)\) is another quadratic expression such that \(g(x)\) is negative when \(x \in\left(3, \frac{9}{2}\right)\) and positive when \(x \in R-\left[3, \frac{9}{2}\right]\). Then, the sign of \(f(x) g(x)\) in \([0,5]\) is

In a Freundlich adsorption isotherm, if the slope is unity and k is 0.1, the extent of adsorption at 2 atm is \((\log 2=0.30)\)

At $291 \mathrm{K}$, saturated solution of ${\mathrm{BaSO}}_4$ was found to have a specific conductivity of $3.648\times {10}^{-6}{\mathrm{ohm}}^{-1}{ \mathrm{cm}}^{-1}$ and that of water being used is $1.25\times {10}^{-6}{\mathrm{ohm}}^{-1}{ \mathrm{cm}}^{-1}$. If the ionic conductance's of ${\mathrm{Ba}}^{2+}$ and ${\mathrm{SO}}_4^{2-}$ are $110$ and $136.6{\mathrm{ohm}}^{-1}{ \mathrm{cm}}^2{ \mathrm{mol}}^{-1}$ respectively, the solubility of ${\mathrm{BaSO}}_4$ at $291 \mathrm{K}$ will be [Atomic masses of$\left.\mathrm{Ba}-137,S-32,0-16\right]$

Suppose that the sides passing through the vertex \((\alpha, \beta)\) of a triangle are bisected at right angles by the lines \(y^2-8 x y-9 x^2=0\). Then, the centoid of the triangle is