If the coefficients of \(x^2\) and \(x^3\) in the expansion of \((3 + kx)^9\) are equal, then the value of '\(k\)' is

The equation of a plane passing through the line of intersection of the planes \(x+2 y+3 z=2\), \(x-y+z=3\) and at a distance \(\dfrac{2}{\sqrt{3}}\) from the point \((3,1,-1)\) is

Find the general solution of \(\sin 2 x+\cos x=0\).

- The converse of the contrapositive of the conditional \(p \rightarrow \sim q\) is

\(\text { The modulus of the following complex number } \dfrac{1+i}{1-i}-\dfrac{1-i}{1+i} \text { is }\)

Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are

If \(\mathbf{p}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{q}=4 \hat{\mathbf{k}}-\hat{\mathbf{j}}\) and \(\mathbf{r}=\hat{\mathbf{i}}+\hat{\mathbf{k}}\), then the unit vector in tile direction of \(3 p+q-2 r\) is

Match the IUPAC names in Column II with the correct structures given in Column I.

	Structures of compounds.		IUPAC name
A		P	1-Bromomethyl-3-(2,2-dimethylpropyl)benzene
B		Q	1-Bromo-4-(1-methylethyl)benzene
C		R	1-Bromo-2-(1-methylpropyl)benzene
D		S	1-Bromo-4-(2-methylpropyl)benzene

The electric field versus distance graph is shown as given. Select the correct statement from the following.
E- electric field
R-radius
r - distance from the centre

\(\text { Value of } \cos 105^{\circ} \text { is }\)

Choose the correct metal/ ion from the brackets which ..........
A. has chemical reactivity similar to that of the first few members of the Lanthanoids \((\mathrm{Zn}, \mathrm{Ca}, \mathrm{Fe}, \mathrm{Cu})\).
B. has stable \(4 \mathrm{f}^7\) electronic configuration, but acts as a strong reducing agent and converts to \(\mathrm{M}^{3+}\) state. \(\left(\mathrm{Eu}^{2+}, \mathrm{Ce}^{2+}, \mathrm{Pr}^{2+}, \mathrm{Dy}^{2+}\right)\)
C. is a colorless ion \(\left(\mathrm{Tm}^{3+}, \mathrm{Lu}^{3+}, \mathrm{Gd}^{3+}, \mathrm{Sm}^{3+}\right)\).
D. shows stable +2 oxidation state and is diamagnetic \((\mathrm{Ce}, \mathrm{Sm}, \mathrm{Ho}, \mathrm{Yb})\)

Excitation energy of a hydrogen like ion in its first excitation state is \(40.8 \mathrm{eV}\). Energy needed to remove the electron from the ion in ground state is

How much faster would a reaction proceed at \(25^{\circ} \mathrm{C}\) than at \(0^{\circ} \mathrm{C}\) if the activation energy is \(65 \mathrm{~kJ}\) ?