Match List-I with List-II

List-(I) Function	Lis-(II) Derivative
(A) \(y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^2},|x|<1\)	(I) \(\frac{d y}{d x}=\frac{1}{2 y-1}\)
(B) \(y=\sqrt{x+y}, x+y>0, y \neq \frac{1}{2}\)	(II) \(\frac{d y}{d x}=10^x \log _e 10\)
(C) \(y=\log _{10} x, x>0\)	(III) \(\frac{d y}{d x}=0\)
(D) \(y=10^*\)	(IV) \(\frac{d y}{d x}=\frac{1}{x \log _e 10}\)

Choose the correct answer from the options given below:

The angle between the line \(\vec{r}=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})\) and the plane \(\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=4\) is :

A parallelogram is constructed on the vectors \(\vec{a}=3 \vec{\alpha}-\vec{\beta}, \vec{b}=\vec{\alpha}+3 \vec{\beta}\). If \(|\vec{\alpha}|=|\vec{\beta}|=2\) and the angle between \(\vec{\alpha}\) and \[\vec{\beta}\] is \(\frac{\pi}{3}\), then length of the diagonal of the parallelogram is:

If the length of a simple pendulum is decreased by \(3 \backslash \%\), the percentage change in its period T is
[Note : \(T=2 \pi \sqrt{\frac{L}{g}}\) where \(L\) is the length of pendulum and \(g\) is constant]

If \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ -4 & 2 & 4\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & -2 \\ 3 & -4 \\ 2 & 4\end{array}\right]\) then product \(A B\) is :

Corner points of the feasible region for an LPP are: \((0,2),(3,0),(6,0),(6,8)\) and \((0,5)\). Let \(z=4 x+6 y\) be the objective function. Then, Max \(z\) - Min \(z\) is equal to:

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\), then the value of \(\cos ^{-1} x+\cos ^{-1} y\) is:

Cloning of alien DNA by the host involves -

The correct expression and unit of molar conductivity (\(\Lambda_{m}\)) are:

If the magnetic field \(\vec{B} = \hat{i} - 0.3\hat{j}\) and the area \(\vec{A} = \hat{i} + 6\hat{j} - 3\hat{k}\) then the magnetic flux through this area is:

Question based on the following passage :
P-Block elements are placed in group 13 to 18 in the periodic table. Group 15 is nitrogen family.
Nitrogen can react with hydrogen, oxygen, halogens etc. It shows multiple oxidation states in its various oxides i.e. \(N_2O, N_2O_3, N_2O_5\) etc. Important allotropic forms of phosphorus are white, red and black phosporus. Various compounds of phosphorus are phosphine, \(PCl_3, PCl_5\) and oxoacids of phosphorus. Oxidation states \(+1\) to \(+5\) are seen in various oxoacids of phosphorus. Group 17 elements are collectively known as halogens. Bond dissociation enthalpy of \(F_2\) is smaller than \(Cl_2\).
The gases present in group 18 are chemically unreactive, they form only few compounds. Neil Bartlett (in 1962) observed the reaction of a noble gas. Since then, many noble gas compounds have been synthesised.
The process used for the preparation of Oleum is :

A line through the points \(A(3,4,1)\) and \(B(5,1,6)\) is drawn.
The coordinates of the points where the line through the points \(A \backslash \& B\) crosses the \(X Y\) plane is:

Einstein photo-electric equation states that \(E_k = h\nu - \phi\). In the equation, \(E_k\) refers to