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\((343)^{\frac{1}{3}} \times (0.14)^{-1} \times (25)^{-\frac{1}{2}}\) = \(\sqrt[3]{343} \times (\frac{14}{100})^{-1} \times (\sqrt{25})^{-1}\) = \(7 \times \frac{100}{14} \times \frac{1}{5}\) = 10.

Hint: Represent the question in a venn diagram, such that n(B ∩ M) = x n(B) only = 125-x n(M) only = 110-x => 125-x + 110-x + x = 220 => x = 15. n(B) only = 125 - 15 = 110

52.4 - 5.7 - 3.45 - 1.75 = 52.4 - (5.7 + 3.45 + 1.75) = 52.4 - 10.90 = 41.5

\((\sqrt{0.7} + \sqrt{70})^{2}\) = \((\sqrt{0.7} + \sqrt{70})(\sqrt{0.7} + \sqrt{70})\) = \(0.7 + 2\sqrt{0.7 \times 70} + 70\) = \(0.7 + 14 + 70 \) = 84.7

\(\frac{0.21 \times 0.072 \times 0.00054}{0.006 \times 1.68 \times 0.063}\) = \(\frac{9 \times 10^{-10}}{7 \times 10^{-8}}\) = \(1.2857 \times 10^{-2}\) = \(0.012857 \approxeq 0.01286\) (to 4 s.f)

Let the number of goats be X C.P of X goats = N4000X S.P of X goats = (75/100) x 4000X = N3000X 3000X = 180,000 => X = 180000/3000 = 60. X = 60 goats

dy/dx = 2x - 3 y = ∫2x - 3 dx => y = x2 - 3x + c 3 = (0)2 - 3(0) + c => c = 3 Thus y = x2 - 3x + 3

\(y = \sin^{2} (5x)\) Let u = sin 5x \(\frac{\mathrm d u}{\mathrm d x} = 5 \cos 5x\) \(\therefore y = u^{2}\) \(\frac{\mathrm d y}{\mathrm d u} = 2u\) \(\frac{\mathrm d y}{\mathrm d x} = 2u . 5 \cos 5x\) = \(10u \cos 5x\) = \(10 \sin 5x \cos 5x\)

y = 3x\(^2\) - 2x + 5 Slope = \(\frac{\mathrm d y}{\mathrm d x} = 6x - 2\) At x = 1, Slope : 6(1) - 2 = 4.

\(\int \sin 3x \mathrm d x = - \frac{\cos 3x}{3} + c\) = \(-\frac{1}{3} (\cos 3x) + c\)

Area of the circle (A) = \(\pi r^{2}\) \(\frac{\mathrm d A}{\mathrm d r} = 2\pi r\) \(r = 5cm\); \(\frac{\mathrm d r}{\mathrm d t} = 0.2\) \(\frac{\mathrm d A}{\mathrm d t} = \frac{\mathrm d A}{\mathrm d r} . \frac{\mathrm d r}{\mathrm d t}\) \(\frac{\mathrm d A}{\mathrm d t} = 2\pi \times 5 \times 0.2 = 2\pi\)

\(y = x^{2} - \frac{1}{x} = x^{2} - x^{-1}\) \(\frac{\mathrm d y}{\mathrm d x} = 2x - (- x^{-2})\) = \(2x + \frac{1}{x^{2}}\)

x\(^3\) - 5x\(^2\) - x + 5 = 0. \(x^{2}(x - 5) - 1(x - 5) = 0\) \((x^2 - 1)(x - 5) = 0 \implies (x - 1)(x + 1)(x - 5) = 0\) \(\therefore x = 1, -1, 5\)

The time (t) to do the work is inversely proportional to the number of workers (n). \(\implies t \propto \frac{1}{n}\) \(t = \frac{k}{n}\) \(5 = \frac{k}{45} \implies k = 45 \times 5 = 225\) \(\therefore t = \frac{225}{n}\) For 25 men, \(t = \frac{225}{25} = 9\) \(\therefore\) 25 men will do the work in 9 days.

\(\frac{x + 2}{4} - \frac{2x - 3}{3}\) < 4 \(\frac{3(x + 2) - 4(2x - 3)}{12}\) < 4 3x + 6 - 8x + 12 < 48 18 - 5x < 48 \(\implies\) - 5x < 48 - 18 -5x < 30 \(\implies \)x > - 6