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The solution of the differential equation $\dfrac{dy}{dx} = (x-y)^2$, subject to $y(1)=1$, is:
Let $v = x - y$, so $\frac{dv}{dx} = 1 - \frac{dy}{dx} = 1 - v^2$.
Separate: $\frac{dv}{1-v^2} = dx \Rightarrow \frac{1}{2}\ln\left|\frac{1+v}{1-v}\right| = x + C$
i.e. $\ln\left|\frac{1+x-y}{1-x+y}\right| = 2x + C'$
At $(1,1)$: $v=0$, $\ln(1)=2+C' \Rightarrow C'=-2$
Final: $-\ln\left|\dfrac{1-x+y}{1+x-y}\right| = 2(x-1)$ which matches option B.
- \(408 = 2^3 \times 3 \times 17\)
- \(462 = 2 \times 3 \times 7 \times 11\)
- \(516 = 2^2 \times 3 \times 43\)
Na⁺, F⁻, O²⁻ are isoelectronic (10 electrons). Increasing radius means decreasing nuclear charge:
- Na⁺: Z=11 → smallest
- F⁻: Z=9
- O²⁻: Z=8 → largest
Option (1) is wrong: H has 1 proton; H⁻ is larger (more electron repulsion) and H⁺ has no electron at all (just a proton, ~10⁻¹⁵ m). Option (4) is wrong as N³⁻ >> Al³⁺ in size.
- $P_{1} = 35 \text{ psi}$, $T_{1} = 27 + 273 = 300 \text{ K}$
- $P_{2} = 40 \text{ psi}$, $T_{2} = ?$
- $35/300 = 40/T_{2}$
- $T_{2} = (40 \times 300) / 35 \approx 342.8 \text{ K}$
- Converting to Celsius: $342.8 - 273 = 69.8^{\circ}\text{C} \approx 70^{\circ}\text{C}$
Sharks belong to which class of vertebrates?
Class Chondrichthyes includes cartilaginous fish (sharks, rays). Osteichthyes are bony fish. Echinodermata are starfish etc.; Urochordata are tunicates.
The plasma membrane is selectively permeable mainly due to the presence of:
Integral membrane proteins (e.g., channels, carriers, pumps) control selective transport. Lipids form the barrier but without proteins, permeability is limited to small nonpolar molecules.
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