Given that \(p_0\) and \(p_1\) represent base and current prices of a commodity, \(q_0\) and \(q_1\) refer to the base and current quantities, then the formula used to calculate price relatives

\(\tan ^{-1} 3+\tan ^{-1} \lambda=\tan ^{-1}\left(\frac{3+\lambda}{1-3 \lambda}\right)\) is valid for which values of \(\lambda\) ?

Match List-l with List-ll

List-I	List-II
(A) Integrating factor of \(x d y-\left(y+x^2\right) d x=0\)	(I) \(x^2\)
(B) Integrating factor of \(x d y+\left(2 y+x^2\right) d x=0\)	(II) \(x^3\)
(C) Integrating factor of \(\left(3 y-x^2\right) d x+x d y=0\)	(III) \(x\)
(D) Integrating factor of \(\left(y+3 x^2\right) d x+x d y=0\)	(IV) \(\frac{1}{x}\)

Choose the correct answer from the options given below :

In an electromagnetic wave, the ratio of energy densities of electric and magnetic fields is :

Let the corner points of the bounded feasible region of the linear programming problem \(( LPP ) Z=a x+b y\) be \((0,0),(2,0),\left(\frac{20}{19}, \frac{45}{19}\right)\) and \((0,3)\).
If the optimal value of \(Z\) occurs at both points \((2,0)\) and \(\left(\frac{20}{19}, \frac{45}{19}\right)\), then the relation between \(a\) and \(b\) is:

The value of \(\tan ^{-1}(2)+\tan ^{-1}(3)\) is equal to

The value of \(x\) for which the matrix \(A = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{bmatrix}\) is singular is:

\(\text{R-Br} \xrightarrow{\text{Mg, dry Ether}} \text{P} \xrightarrow{\text{EtOH}} \text{Q}\)
What is Q?

The degree of the differential equation \(\left(2+\left(\frac{d y}{d x}\right)^2\right)^{\frac{3}{2}}=a^2 \frac{d^2 y}{d x^2}\) is :

The general solution of the differential equation \(\frac{x d y}{d x}+4 y=x^3,(x \neq 0)\) is :

Answer the question on the basis of passage given below:
In the periodic table, the d-block contains the elements of group \(3\) to \(12\). The d-orbitals are progressively filled in each of the four long periods. The elements of d block referred as transition metals have partly filled d orbitals and exhibit certain characteristic properties such as variety of oxidation states, formation of coloured ions, act as catalyst and show paramagnetic behaviour.
The two inner transition metals series \(4f\) and \(5f\) are known as Lanthanoids and Actinoids respectively. The lanthanoids resemble one another more closely as compared to ordinary transition elements in any series.
In context of the Lanthanoids, which of the following statements is/are NOT correct?
A. There is a gradual decrease in the radii of the members with increasing atomic number in the series
B. Availability of \(4f\) electrons results in the formation of compounds in \(+4\) oxidation state for all the members of the series
C. Because of similar properties, the separation of lanthanoids is not easy
D. \(La(OH)_3\) is least basic among hydroxides of lanthanoids
E. Ce can act as an oxidising agent
Choose the correct answer from the options given below:

Which of the following country has the highest biodiversity?

Two point charges of 2 C each are separated from each other by a distance of 1 m in water (K = 80). The force between the charges is: