Under the same reaction conditions, initial concentration of \(1.386 \mathrm{~mol} \mathrm{dm}^{-3}\) of a substance becomes half in 40 s and 20 s through first-order and zero-order kinetics, respectively. Ratio \(\left(\frac{k_1}{k_0}\right)\) of the rate constants for first order \(\left(k_1\right)\) and zero \(\operatorname{order}\left(k_0\right)\) of the reaction is

The work function ( \(\Phi\) ) of some metals is listed below. The number of metals which will show photoelectric effect when light of \(300 \mathrm{~nm}\) wavelength falls on the metal is

The number of stereoisomers obtained by bromination of trans-2-butene is

With reference to aqua regia, choose the correct option(s).

Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Paragraph I: An organic compound \(\mathbf{P}\) with molecular formula \(\mathrm{C}_9 \mathrm{H}_{18} \mathrm{O}_2\) decolorizes bromine water and also shows positive iodoform test. \(\mathbf{P}\) on ozonolysis followed by treatment with \(\mathrm{H}_2 \mathrm{O}_2\) gives \(\mathbf{Q}\) and \(\mathbf{R}\). While compound \(\mathbf{Q}\) shows positive iodoform test, compound \(\mathbf{R}\) does not give positive iodoform test. \(\mathbf{Q}\) and \(\mathbf{R}\) on oxidation with pyridinium chlorochromate (PCC) followed by heating give \(\mathbf{S}\) and \(\mathbf{T}\), respectively. Both \(\mathbf{S}\) and \(\mathbf{T}\) show positive iodoform test.
Complete copolymerization of 500 moles of \(\mathbf{Q}\) and 500 moles of \(\mathbf{R}\) gives one mole of a single acyclic copolymer \(\mathbf{U}\).
[Given, atomic mass: \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16\)]
Question: Sum of number of oxygen atoms in \(\mathbf{S}\) and \(\mathbf{T}\) is

Among \(\left[\mathrm{Co}(\mathrm{CN})_4\right]^{4-},\left[\mathrm{Co}(\mathrm{CO})_3(\mathrm{NO})\right], \mathrm{XeF}_4,\left[\mathrm{PCl}_4\right]^{+},\)\(\left[\mathrm{PdCl}_4\right]^{2-},\left[\mathrm{ICl}_4\right]^{-},\left[\mathrm{Cu}(\mathrm{CN})_4\right]^{3-}\) and \(\mathrm{P}_4\) the total number of species with tetrahedral geometry is _______

Paragraph :
A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The maximum value of \(v_0\) for which the disk will roll without slipping is

Statement I p-hydroxybenzoic acid has a lower boiling point than o-hydroxybenzoic acid.
Statement II o-hydroxybenzoic acid has intramolecular hydrogen bonding.

A biconvex lens of focal length \(f\) forms a circular image of radius \(r\) of sun in focal plane. Then, which option is correct?

Among the following complexes, the total number of diamagnetic species is_____
\(\left[\mathrm{Mn}\left(\mathrm{NH}_3\right)_6\right]^{3+},\left[\mathrm{MnCl}_6\right]^{3-},\left[\mathrm{FeF}_6\right]^{3-},\left[\mathrm{CoF}_6\right]^{3-},\)\(\left[\mathrm{Fe}\left(\mathrm{NH}_3\right)_6\right]^{3+}\), and \(\left[\mathrm{Co}(\mathrm{en})_3\right]^{3+}\)
[Given, atomic number: \(\mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27\);
\(\text { en } \left.=\mathrm{H}_2 \mathrm{NCH}_2 \mathrm{CH}_2 \mathrm{NH}_2\right]\)

Paragraph:

A box \(B_{1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \(B_{2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \(B_{3}\) contains \(3\) white balls, \(4\) red balls and \(5\) black balls.


Question:

If \(1\) ball is drawn from each of the boxes \(B_{1}, B_{2}\) and \(B_{3}\), the probability that all \(3\) drawn balls are of the same colour is

The number of values of \(\theta\) in the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) such that \(\theta \neq \frac{n \pi}{5}\) for \(n=0, \pm 1, \pm 2\) and \(\tan \theta=\cot 5 \theta\) as well as \(\sin 2 \theta=\cos 4 \theta\) is