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\(n = \frac{sin i}{sin r}\) \(i = 90 - \alpha\) \(\therefore n = \frac{sin 90 - \alpha}{sin \beta}\) ⇒ \(n = \frac{cos \alpha}{sin \beta}\)

I = \(\frac{E}{R + r}\) For parallel of 5\Omega and 10\Omega \(\frac{1}{R}\) = \(\frac{1}{10}\) + \(\frac{1}{5}\) = \(\frac{1+2}{10}\) = \(\frac{3}{10}\) ∴R = \(\frac{10}{3}\) Total R = 3\(\frac{1}{3}\) + 3 x 1 = 7\(\frac{1}{3}\) ∴I = 6\(\div\) 7\(\frac{1}{3}\) = \(\frac{6}{1}\) x \(\frac{3}{22}\) = \(\frac{18}{22}\) = \(\frac{9}{11}\) ∴I = \(\frac{9}{11}\)

Energy = 1/2CV2 ∴Energy of P = 1/2 x 2 x V2 = V2 Energy of Q = 1/2 x 4 x V2 = 2V2 ∴ P/Q = V2/2V2 = 1/2 => P:Q = 1:2

C = \(\frac{\text{Q}}{\text{V}}\) => Q = CV : ∴ V = \(\frac{\text{Q}}{\text{C}}\) ∴V\(_1\) = \(\frac{\text{Q}}{C_1}\); V\(_2\) = \(\frac{\text{Q}}{C_2}\); V\(_3\) = \(\frac{\text{Q}}{C_3}\) for series \(\frac{1}{\text{C}}\) = \(\frac{1}{C_1}\) + \(\frac{1}{C_2}\) + \(\frac{1}{C_3}\) = \(\frac{1}{2} + \frac{1}{6} + \frac{1}{3}\) = \(\frac{3 +1 + 2}{6}\) = \(\frac{6}{6}\) = 1 ∴ C = 1\muF = 1.0 x 10\(^{-6}\)F ∴ Q = CV = 1.0 x 10\(^{-6}\) x 12 = 12.0 x 10\(^{-6}\)C ∴V\(_1\) = \(\frac{\text{Q}}{C_1}\) = \(\frac{12 \times 10^{-6}}{2 \times 10^{-6}}\) = 6V V\(_2\) = \(\frac{\text{Q}}{C_2}\) = \(\frac{12 \times 10^{-6}}{6 \times 10^{-6}}\) = 2V V\(_3\) = \(\frac{\text{Q}}{C_3}\) = \(\frac{12 \times 10^{-6}}{3 \times 10^{-6}}\) = 4V ∴ 6V, 2V and 4V

0.0 E since one of the electric fields points in the opposite direction (left)

for a parallel R and X, when R = 5\Omega, Effective resistance R = \(\frac{5X}{5 + X}\) and when R = 2\Omega Effective resistance R = \(\frac{2X}{2 + X}\) Thus since E = I(R+r), and if r = 0 then E = IR ∴(i) E = 3(\(\frac{5X}{(5+x)}\)); (ii) E = 6(\(\frac{2X}{(2+x)}\)) ∴3(\(\frac{5X}{(5+X)}\)) = 6(\(\frac{2X}{(2+X)}\)) => (\(\frac{5X}{(5+X)}\)) = 2(\(\frac{2x}{(2+X)}\)) ∴ (\(\frac{5X}{(5+X)}\)) = (\(\frac{4X}{(2+X)}\)) 5x(2+x) = 4x(5+x) 10x + 5x\(^2\) = 20x + 4x\(^2\) 5x\(^2\) - 4x\(^2\) = 20x - 10x => x\(^2\) = 10x ∴x\(^2\) - 10x = 0 x(x - 10) = 0 ∴ x = 0 or 10 ∴ x = 10\Omega

Ir.m.s = Vr.m.s/Z But Z = \sqrt R2 + XL2 = \sqrt 42 + 32 = \sqrt 16 + 9 = \sqrt 25 ∴Ir.m.s = 240/5 = 48.0A

In the Daniel cell, i. Depolarize = CuSO4 solution ii. Positive electrode = Copper container iii Negative electrode = zinc rod/plate

The passage describes how the rise of motor vehicles (cars, lorries, and motorcycles) led to changes in road transport, such as the adaptation of road surfaces through tar-spraying and efforts to widen and straighten roads. Although challenges remained, these adjustments indicate an overall improvement in road transport systems.

The passage highlights the time when motor cars and motorcycles were becoming more common, but before that, the railway was the primary means of transportation. The text mentions that cars and motorcycles "dashed along [the roads] at speeds which rivalled those of the express trains". This that before the motor car became widespread, express trains were the fastest means of transport.

The passage describes how the roads, which had been relatively quiet and underused after the introduction of the railway, were suddenly awakened by the rise of cars, motorcycles and lorries. These vehicles, especially with their increased speed and volume, put new demands on the road system, which was not initially designed to handle such traffic. This led to changes in the infrastructure (e.g., tar-spraying to reduce dust and grease) and posed challenges for those using the roads.

The expression "unless mechanically minded" refers to coachmen and grooms who would struggle to adapt to the new technology of motor vehicles. In other words, only those are can adapt to new technology would be able to continue making a living, as the rise of motor vehicles was making traditional roles like that of coachmen and grooms obsolete.

The passage mentions that by 1910, the motor car was "plainly conquering the highway," which means that motor cars had become widespread and were becoming a dominant presence on the roads. This means that by that time, motor cars were no longer rare or exceptional but had become common on the highways.

The first step when analysing data is to check the field record book and questionnaires for any issues that might affect the quality or reliability of the data. "Mistakes" fits best here because it refers to errors or inaccuracies that can occur during data collection or recording which need to be identified and corrected before analysis.

The sentence refers to the action of checking the field records and questionnaires for any issues, and it mentions that, in some cases, it might be possible to correct the discovered shortcomings. The word "corrections" fits perfectly as it refers to fixing errors or issues found in the data.