Among the following, the number of elements showing only one non-zero oxidation state is O, Cl, F, N, P, Sn, Tl, \(\mathrm{Na}, \mathrm{Ti}\)

If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride-ammonia complex (which behaves as a strong electrolyte) is $-{0.0558}^{\mathrm{o}}\mathrm{C}$ , the number of chloride(s) in the coordination sphere of the complex is
$[{\mathrm{K}}_{\mathrm{f}}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}=1.86\mathrm{ }\mathrm{K}\mathrm{ }\mathrm{k}\mathrm{g}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}]$

The carboxyl functional group \((-\mathrm{COOH})\) is present in

The substituents \(R_{1}\) and \(R_{2}\) for nine peptides are listed in the table given below. How many of these peptides are positively charged at \(\mathrm{pH}=7.0\) ?

\(\begin{array}{|c|c|c|}\hline Peptide & \mathbf{R}_{\mathbf{1}} & \mathbf{R}_{\mathbf{2}} \\\hline I & \mathrm{H} & \mathrm{H} \\\hline II & \mathrm{H} & \mathrm{CH}_{3} \\\hline III & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{H} \\\hline IV & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline V & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VI & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline VII & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VIII & \mathrm{CH}_{2} \mathrm{OH} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline IX & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \mathrm{CH}_{3} \\\hline\end{array}\)

The order of the oxidation state of the phosphorus atom in $H_{3}P{O}_2, H_{3}P{O}_4, H_{3}P{O}_3$ and $H_4{P}_2{O}_6$ is

$50 \mathrm{mL}$ of $0.2$ molal urea solution (density $=1.012 \mathrm{g} {\mathrm{mL}}^{-1}$ at $300 \mathrm{K}$ ) is mixed with $250 \mathrm{mL}$ of a solution containing $0.06 \mathrm{g}$ of urea. Both the solutions were prepared in the same solvent. The osmotic pressure (in torr) of the resulting at $300 \mathrm{K}$ is
[Use : Molar mass of urea $=60 \mathrm{g} {\mathrm{mol}}^{-1}$; gas constant, $\mathrm{R}=62 \mathrm{L}-\mathrm{torr} {\mathrm{K}}^{-1} {\mathrm{mol}}^{-1}$;
Assume, $\left.{\mathrm{\Delta }}_{mix}\mathrm{H}=0,{\mathrm{\Delta }}_{\text{mix }}\mathrm{V}=0\right]$

Match each of the reactions given in Column I with the corresponding products(s) given in Column II.

Consider a disc rotating in the horizontal plane with a constant angular speed \(\omega\) about its centre \(\mathrm{O}\). The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles \(P\) and \(Q\) are simultaneously projected at an angle towards \(R\). The velocity of projection is in the \(y\) - \(z\) plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed \(1 / 8\) rotation, (ii) their range is less than half the disc radius, and (iii) \(\omega\) remains constant throughout. Then

Words of length$10$ are formed using the letters A, B, C, D, E, F, G, H, I, J. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, $\frac{y}{9x}=$

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Let \(p, q\) be integers and let \(\alpha, \beta\) be the roots of the equation, \(x^{2}-x-1=0\), where \(\alpha \neq \beta\). For \(n=0,1,2, \ldots\), let \(a_{n}=p \alpha^{n}+q \beta^{n}\)
FACT: If \(a\) and \(b\) are rational numbers and \(a+b \sqrt{5}=0\), then \(a=0=b\).

Question:
\(a_{12}=\)

Let a, b, c be positive integers such that $\frac{b}{a}$ is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of $\frac{a^2+a-14}{a+1}$ is ________

Let \(f\) be a real-valued function defined on the interval \((0, \infty)\), by \(f(x)=\ln x+\int_0^x \sqrt{1+\sin t} d t\). Then which of the following statement(s) is (are) true ?

The entropy versus temperature plot for phases $\alpha$ and $\beta$ at $1$ bar pressure is given. ${\mathrm{S}}_{\mathrm{T}}$ and ${\mathrm{S}}_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{K}$, respectively. The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{K}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}=1 \mathrm{J} {\mathrm{mol}}^{-1} {\mathrm{K}}^{-1}$. Assume $\left({\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}\right)$ is independent of temperature in the range of $200$ to $700 \mathrm{K}.{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively. The value of enthalpy change, ${\mathrm{H}}_{\beta }-{\mathrm{H}}_{\alpha }$ (in ${\mathrm{Jmol}}^{-1}$ ), at $300 \mathrm{K}$ is