The first order rate constant for the decomposition of \(\mathrm{CaCO}_{3}\) at \(700\, \mathrm{~K}\) is \(6.36 \times 10^{-3}\, \mathrm{~s}^{-1}\) and activation energy is \(209\, \mathrm{~kJ} \,\mathrm{~mol}^{-1}\). Its rate constant (in \(\mathrm{s}^{-1}\) ) at \(500\, \mathrm{~K}\) is \(\mathrm{x} \times 10^{-6}\). The value of \(\mathrm{x}\) is ..... . Nearest integer) Given \(\mathrm{R}=8.31\, \mathrm{~J} \,\mathrm{~K}^{-1} \,\mathrm{~mol}^{-1} ; \log 6.36 \times 10^{-3}=-2.19\) \(\left.10^{-4.79}=1.62 \times 10^{-5}\right]\)

Number of carbocations from the following that are not stabilized by hyperconjugation is _______.

Number of oxygen atoms present in chemical formula of fuming sulphuric acid is _______.

\({image}\) The above reaction requires which of the following reaction conditions ?

Correct statements among \(i\) to \(iv\) regarding silicones are \((i)\) they are polymers with hydrophobic character \((ii)\) they are biocompatible \((iii)\) in general, they have high thermal stability and low dielectric strength \((iv)\) usually they are resistant to oxidation and used as greases.

Both acetaldehyde and acetone (individually) undergo which of the following reactions?
A. Iodoform Reaction
B. Cannizaro Reaction
C. Aldol Condensation
D. Tollen's Test
E. Clemmensen Reduction
Choose the correct answer from the options given below:

The wavelength of photon 'A' is 400 nm. The frequency of photon 'B' is \(10^{16} s^{-1}\). The wave number of photon 'C' is \(10^{4} cm^{-1}\). The correct order of energy of these photons is:

Let \(\mathrm{L}_1: \frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}\) and \(\mathrm{L}_2: \frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}, \alpha \in \mathbf{R}\), be two lines, which intersect at the point \(B\). If \(P\) is the foot of perpendicular from the point \(A(1,1,-1)\) on \(L_2\), then the value of \(26 \alpha(\mathrm{~PB})^2\) is _________

The area (in sq. units) of the parallelogram whose diagonals are along the vectors \(8\hat i - 6\hat j\) and \(3\hat i + 4\hat j - 12\hat k\) , is

Refer to the logic circuit given below. For two inputs \((A=1, B=1)\) and \((A=0, B=1)\), output \((Y)\) will be __________.

Two cells of emf \(2\, E\) and \(E\) with internal resistance \(r _{1}\) and \(r _{2}\) respectively are connected in series to an external resistor \(R\) (see \(figure\)). The value of \(R ,\) at which the potential difference across the terminals of the first cell becomes zero is

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

Let the plane containing the line of intersection of the planes \(P 1: x+(\lambda+4) y+z=1\) and \(P2:\) \(2 x+y+z=2\) pass through the points \((0,1,0)\) and \((1,0,1)\). Then the distance of the point \((2 \lambda, \lambda,-\lambda)\) from the plane \(P 2\) is