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Given \(\cos x = -\frac{5}{13}\) where \(180^\circ < x < 270^\circ\): Using: \(\sin^2 x + \cos^2 x = 1\) \(\sin^2 x = 1 - \left(-\frac{5}{13}\right)^2 = \frac{144}{169} \implies \sin x = -\frac{12}{13}\) \(\tan x = \frac{\sin x}{\cos x} = \frac{-\frac{12}{13}}{-\frac{5}{13}} = \frac{12}{5}\) Calculating \(\tan x - \sin x\): \(\tan x - \sin x = \frac{12}{5} - \left(-\frac{12}{13}\right) = \frac{12}{5} + \frac{12}{13}\) Finding a common denominator (65): \(\tan x - \sin x = \frac{12 \times 13}{65} + \frac{12 \times 5}{65} = \frac{156}{65} + \frac{60}{65} = \frac{216}{65}\) Thus, the value is: \(\tan x - \sin x = \frac{216}{65}\)

(\(\frac{1}{343}\))\(^{\frac{1}{3}}\) + (64)\(^{-\frac{1}{3}}\) - \(\frac{4}{9}\)\(^{-\frac{1}{2}}\) where \(343 = 7^3, 64 = 2^6, 4 = 2^2, 9 = 3^2\) \(7^{-3}\) \(^{\frac{1}{3}}\) + \(2^6\) \(^{\frac{-1}{3}}\) - \(\frac{2^2}{3^2}\) \(^{\frac{-1}{2}}\) \(7^{-1} + 2^{-2} - (\frac{2}{3})^{-1}\) \(\frac{1}{7} + \frac{1}{4} - \frac{3}{2}\) \(\frac{4+7-42}{28}\) -\(\frac{31}{28}\)

Given: \(\frac{3 - y}{2x} = \frac{6y + 7}{4x - 5}\) Substitute y = 2 \(\frac{3 - 2}{2x} = \frac{6(2) + 7}{4x - 5} \implies \frac{1}{2x} = \frac{19}{4x - 5}\) Cross-multiply: 1 \(\cdot\) (4x - 5) = 19 \(\cdot\) 2x 4x - 5 = 38x -5 = 38x - 4x -5 = 34x x = -\(\frac{5}{34}\)

first term (a) = \(\frac{5}{6}\) common difference = \(\frac{8}{6}\) - \(\frac{5}{6}\) → \(\frac{3}{6}\) or \(\frac{1}{2}\) A.P formula → T\(_n\) = a + (n - 1)d T\(_n\) = \(\frac{5}{6}\) + (19 - 1)\(\frac{1}{2}\) T\(_n\) = \(\frac{5}{6}\) + 9 → 9\(\frac{5}{6}\)

Total number of students = 20 + 16 + 10 + 6 = 52 students

No of women wearing both purple and white shoes = 21 + 9 = 30 women. No of women wearing neither purple nor white shoes = 51 - 30 = 21 women \(\therefore\) Probability of picking a woman wearing neither purple nor white shoes = \(\frac{21}{51}\) = \(\frac{7}{17}\).

The three main reasons why the weight of an object varies on the earth's surface is - The shape of the earth - Variation in Longitude and latitude -Rotation of the earth

W = P\DeltaV = 2 x 10\(^5\) * (12 x 10\(^{-6}\) - 2 x 10\(^{-6}\)) = 2 x 10\(^5\) 10 x 10\(^{-6}\) = 2.0J

Permanent magnets are made from "hard" ferromagnetic materials such as alnico and ferrite Permanent magnets are made from special alloys (ferromagnetic materials) such as iron, nickel and cobalt, several alloys of rare-earth metals and minerals such as lodestone. When a nonmagnetic piece of steel is applied to a magnet, the atoms within it rearrange themselves in a manner that creates a permanent magnet.

According to Faraday's Law, the mass of the substance deposited is directly proportional to the time for which the current flows, assuming a constant current. Therefore, the graph should show a linear increase in mass over time, which corresponds to option A

The reaction is not a natural radioactive decay. Natural decay occurs spontaneously without external influence; this reaction requires an external alpha particle to initiate it.

Element X is aluminum of valency 3+ Element Y is oxygen of valency 2- Their combination is \(Al_{2}O_{3}\)

Relative molecular mass of XO\(_2\) = 2 \times vapour density Relative molecular mass → (2 x 32) = 64 Atomic mass of X: X + [16 x 2] = 64 X = 64 - 32 = 32

Hygroscopy is the phenomenon of attracting and holding water molecules via either absorption or adsorption from the surrounding environment, which is usually at normal or room temperature. Deliquescence, the process by which a substance absorbs moisture from the atmosphere until it dissolves in the absorbed water and forms a solution.

In the modern periodic table, the elements are listed in order of increasing atomic number. The atomic number is the number of protons in the nucleus of an atom.