If \(y=\sin \left(\log _e x\right)\), then \(x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}\) is equal to

If the acute angle between the circles \(S \equiv x^2+y^2+2 k x+4 y-3=0\) and \(S^1 \equiv x^2+y^2-4 x+2 k y+9=0\) is \(\operatorname{Cos}^{-1}\left(\frac{3}{8}\right)\) and the centre of \(S^1=0\) lies in the first quadrant, then the radical axis of \(S=0\) and \(S^1=0\) is

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

If the function \(f(x)=a x^3+b x^2+26 x-24\) satisfies the conditions of Rolle's theorem in \([2,4]\) and \(f^{\prime}\left(3+\frac{1}{\sqrt{3}}\right)=0\), then the value of \(a b\) is equal to

If \(\mathrm{P}\) is the set of all real numbers \(\alpha\) such that the product of the lengths of perpendiculars from \((\alpha, 1)\) to the pair of straight lines \(3 x^2+7 x y+2 y^2=0\) is \(\frac{\sqrt{2}}{5}\) then the sum of the elements in \(\mathrm{P}\) is

For any real number \(\lambda \neq 1\), the centre of the circle that passes through \(\mathrm{A}(1, \lambda), \mathrm{B}(\lambda, 1)\) and \((\lambda, \lambda)\) is

If \(y=\sin ^{-1} x\) then \(\left(1-x^2\right) y_2-x y_1=\)

Statement (I) The elimination of arbitrary constants from \(\alpha, \beta\) and \(\gamma\) from \(y=(\alpha+\beta+\gamma) x\) results in a differential equation of order three.
Statement (II) The elimination of arbitrary constants \(\alpha, \beta\) and \(\gamma\) from \(y=\alpha x+\beta \sin x+\gamma e^x\) results in a differential equation of order three.

${}_{90}{}^{232}\mathrm{Th}$ emits $6\alpha$ and $4\beta$ particles and gets converted into a lead. The mass number and atomic number of lead is _____

In \(\triangle A B C, \frac{16 \mathrm{R} s \Delta \sin \frac{A}{2} \sin \frac{B}{2} \cos \frac{C}{2}}{s-c}\) is equal to

Among the given group 14 elements, the one with highest density is

Consider the following.
Statement-I \(\quad: \mathrm{CCl}_4\) does not undergo hydrolysis. But \(\mathrm{SiCl}_4\) undergoes hydrolysis.
\(\begin{array}{ll}\text { Statement-II } \quad: & \text { Thermal and chemical stability of } \mathrm{GeX}_4 \text { is more than } \\ & \mathrm{GeX}_2\end{array}\)
The correct answer is

\(\int_0^1 \sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right) d x\) is equal to :