The solution of \(x \frac{d y}{d x}=y(\log y-\log x+1)\) is

\(\begin{aligned}
& \frac{4 x^2+5}{(x-2)^4}=\frac{A}{(x-2)}+\frac{B}{(X-2)^2}+\frac{C}{(X-2)^3}+\frac{D}{(X-2)^4} \text { then } \\
& \sqrt{\frac{A}{C}+\frac{B}{C}+\frac{D}{C}}=
\end{aligned}\)

The area (in sq. units) of the triangle formed by the tangent and normal to the ellipse \(9 x^2+4 y^2=72\) at the point \((2,3)\) with the \(X\)-axis is

If \((\sqrt{3}-\mathrm{i})^{\mathrm{n}}=2^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}\), then the least possible value of n is

If the angle \(2 \theta\) is acute, then the acute angle between the pair of straight lines
\[
\begin{aligned}
& x^2(\cos \theta-\sin \theta)+2 x y \cos \theta \\
& \quad+y^2(\cos \theta+\sin \theta)=0, \text { is }
\end{aligned}
\]

For all \(\mathrm{x} \in[0,2024]\) assume that \(f(x)\) is differentiable, \(f(0)=-2\) and \(f^{\prime}(x) \geq 5\). Then the least possible value of \(f\) (2024) is

The equation of the circle which cuts all the three circles \(4(x-1)^2+4(y-1)^2=1,4(x+1)^2+4(y-1)^2=1\) and \(4(x+1)^2+4(y+1)^2=1\) orthogonally is

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

Which of the following is true in the case of an adiabatic process, where \(\gamma=C_p / C_V\) ?

The increasing order of the atomic radius of \(\mathrm{Si}, \mathrm{S}, \mathrm{Na}, \mathrm{Mg}, \mathrm{Al}\) is

In a first order reaction, the concentration of the reactant is reduced from \(0.6 \mathrm{~mol} \mathrm{~L}^{-1}\) to \(0.2 \mathrm{~mol} \mathrm{~L}^{-1}\) in \(5 \mathrm{~min}\). What is the rate constant of the reaction? \((\log 3=0.4771)\)

The following graph is obtained for a gas at different temperatures \(\left(\mathrm{T}_1, \mathrm{~T}_2, \mathrm{~T}_3\right)\). What is the correct order of temperature \(（x\)-axis \(=\) velocity; \(y\)-axis \(=\) number of molecules)

The Cartesian form of the curve given by \(x=\frac{a}{2}\left(t+\frac{1}{t}\right)\), \(y=\frac{a}{2}\left(t-\frac{1}{t}\right), t\) is a parameter, is