Further Mathematics Questions and Answers

Given lines \(OA\) and \(OB\) inclined at angle \(\theta\), the line \(AB\) is gotten using cosine rule.

\(|AB|^{2} = |OA|^{2} + |OB|^{2} - 2|OA||OB|\cos \theta\)

\(|AB|^{2} = 4^{2} + 5^{2} - 2(4)(5)\cos 60\)

= \(16 + 25 - 20\)

\(|AB|^{2} = 21 \implies |AB| = \sqrt{21}\)

\(\implies \text{The two particles are} \sqrt{21} m \text{apart in 1 sec}\)

In two seconds, the particles will be \(2\sqrt{21} m\) apart.

\(F = F_{1} + F_{2}\)

\((2i - 5j) + (-3i + 4j) = (-i - j)\)

\(F = ma \implies (-1, -1) = 5a\)

\(a = (-\frac{1}{5}, -\frac{1}{5})\)

\(|a| = \sqrt{(\frac{-1}{5})^2 + (\frac{-1}{5})^2} = \sqrt{2}{25}\)

\(|a| = \frac{\sqrt{2}}{5} ms^{-2}\)

The number of arrangements for the 4 letters = \(^{4}P_{4} = \frac{4!}{(4 - 4)!}\)

\(4! = 24\)

Alphabetical order is just 1 of the arrangements for the letters 

= \(\frac{1}{24}\)

Given \(V = xi +yj\), the direction cosines are \(\frac{x}{|V|}, \frac{y}{|V|}\).

\(|4i - 3j| = \sqrt{4^{2} + (-3)^{2}} = \sqrt{25} = 5\)

Direction cosines = \(\frac{4}{5}, \frac{-3}{5}\).

\(\overrightarrow{OA} = (3, 4)\)

\(\overrightarrow{OB} = (5, -6)\)

\(\overrightarrow{OM} = (\frac{3 + 5}{2}, \frac{4 + (-6)}{2})\)

= \((4, -1) = 4i - j\)