Mathematics Questions and Answers

\(\frac{dm}{dt}\) = 4 - 0.6t

\(\int\)dm = \(\int\)(4 - 0.6t)dt

m = \(4t - 0.3t^2 + c\), when t = 0, m = 2g

∴ c = 2

m = \(4t - 0.3t^2 + 2\), when t = 5 minutes

m = \(4(5) - 0.3(5)^2 + 2 = 20 - 7.5 + 2\)

= 14.5

\(y = x \sin x\)

\(\frac{\mathrm d y}{\mathrm d x} = x \cos x + \sin x\)

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

= \(2 \cos x - x \sin x\)

\(\lim \limits_{x \to 2} \frac{(x - 2)(x^2 + 3x - 2)}{x^2 - 4}\)

\(\frac{(x - 2)(x^{2} + 3x - 2)}{x^{2} - 4} = \frac{(x - 2)(x^{2} + 3x - 2)}{(x - 2)(x + 2)}\)

= \(\frac{(x^{2} + 3x - 2)}{x + 2}\)

\(\therefore \lim \limits_{x \to 2} \frac{(x - 2)(x^2 + 3x - 2)}{x^2 - 4} = \lim \limits_{x \to 2} \frac{x^{2} + 3x - 2}{x + 2}\)

= \(\frac{2^{2} + 3(2) - 2}{2 + 2}\)

= \(\frac{4 + 6 - 2}{4} = 2\)

sin \(\theta\) = \(\frac{m}{n}\) 

Opp = m; Hyp = n

Adj = \(\sqrt{n^{2} - m^{2}}\)

\(\cos \theta = \frac{\sqrt{n^{2} - m^{2}}}{n}\)

= \(\frac{\sqrt{(n + m)(n - m)}}{n}\)

Equation (5, 7) parallel to the line 7x + 5y = 12

5Y = -7x + 12

y = \(\frac{-7x}{5}\) + \(\frac{12}{5}\)

Gradient = \(\frac{-7}{5}\)

∴ Required equation = \(\frac{y - 7}{x - 5}\) = \(\frac{-7}{5}\) i.e. 5y - 35 = -7x + 35

5y + 7x = 70