Match the chemical substances in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
| Column I | Column II | ||
|---|---|---|---|
| (A) | cellulose | (P) | natural polymer |
| (B) | nylon-6, 6 | (Q) | synthetic polymer |
| (C) | protein | (R) | amide linkage |
| (D) | sucrose | (S) | glycoside linkage |
Match gases under specified conditions listed in Column I with their properties/laws in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
| Column I | Column II | ||
|---|---|---|---|
| (A) | hydrogen gas (P = 200 atm, T = 273 K) | (P) | Compressibility factor $$\ne$$ 1 |
| (B) | hydrogen gas (P $$\sim$$ 0, T = 273 K) | (Q) | attractive forces are dominant |
| (C) | CO$$_2$$ (P = 1 atm, T = 273 K) | (R) | PV = nRT |
| (D) | real gas with very large molar volume | (S) | $$P(V-nb)=nRT$$ |
Let $$\alpha,\beta$$ be the roots of the equation $$x^2-px+r=0$$ and $$\frac{\alpha}{2},2\beta$$ be the roots of the equation $$x^2-qx+r=0$$. Then the value of r is
Let $$f(x)$$ be differentiable on the interval (0, $$\infty$$) such that $$f(1)=1$$, and $$\mathop {\lim }\limits_{t \to x} {{{t^2}f(x) - {x^2}f(t)} \over {t - x}} = 1$$ for each $$x > 0$$. Then $$f(x)$$ is
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