2020 H2 Math Paper 1 Answers

(a)

{\begin{pmatrix} - 1 \\ 1 \\ - 3 \end{pmatrix}.}

(b)

{49.2^{\circ}.}

{5 x + 9 y = 14.}

(a)

{k = -9.}

(b)

{f(x) = 3 x - \frac{9}{2} x^2 + \frac{9}{2} x^3 + \ldots}

(a)

{\frac{1}{2} \sqrt{2} \left( \cos \frac{5}{12} \pi + \mathrm{i} \sin \frac{5}{12} \pi \right).}

(b)

z_4 = \sqrt{2} \left( \cos \frac{1}{12} \pi + \mathrm{i} \sin \frac{1}{12} \pi \right) \allowbreak \textrm{ or } {z_4 = \sqrt{2} \left( \cos - \frac{11}{12} \pi + \mathrm{i} \sin - \frac{11}{12} \pi \right).}

(a)

{\mathbf{a}}

is parallel to

{\mathbf{b}.}

{\mathbf{a} = k \mathbf{b}, k \in \mathbb{R}, k \neq 0.}

(b)

(i)

The set of all possible positions of the point

{R}

forms a line passing through the point

{P}

and parallel to the direction vector

{\mathbf{q}.}

(ii)

{3 x - 5 y + 2 z = - 5.}

The set of all possible positions of the point

{R}

forms a plane passing through the point

{P}

and perpendicular to the normal vector

{\mathbf{q}.}

{k = 2.}

{t = - 8.}

{\textrm{Other root} = - \frac{2}{5} + \frac{6}{5} \mathrm{i}.}

(a)

{2 x + \frac{1}{4} \cos ( 4 x ) + C.}

(b)

{\frac{1}{4} \pi^2 + \frac{1}{8} \pi.}

(c)

{\frac{9}{4} \pi.}

(a)

(i)

{u_{30} = \frac{95}{2}.}

(ii)

{\frac{3345}{2}.}

(b)

(i)

{S_\infty = 20.}

(ii)

{\textrm{Smallest } n = 18.}

(a)

{y=2x}

and

{y=-\frac{1}{2}x}

are perpendicular since their gradients are

{2}

and

{-\frac{1}{2}}

respectively and

{m_1m_2 = 2(-\frac{1}{2}) = -1.}

{y=2x}

makes an angle

{\tan^{-1}(2)}

clockwise to the positive

{x\textrm{-axis}}

and

{y=-\frac{1}{2}x}

makes an angle

{-\tan^{-1}(-\frac{1}{2})}

anticlockwise to the positive

{x\textrm{-axis}.}

Hence

\tan^{-1}(2)-\tan^{-1}(-\frac{1}{2}) = \frac{1}{2} \pi.

(b)

{0 < k \leq 4.5.}

(c)

(d)

{\left( \frac{\pi}{2} - \frac{4}{3} \ln 2 \right) \textrm{ units}^2.}

(a)

{\frac{\mathrm{d}P}{\mathrm{d}t} = - \frac{3}{100} P.}

(b)

{P = A \mathrm{e}^{- \frac{3}{100} t}.}

{\textrm{As } t \to \infty, P \to 0.}

Hence over many years the number of sheep will approach 0 and there will be no more sheep.

(c)

\frac{\mathrm{d}P}{\mathrm{d}t} = - \frac{3}{100} P + n

(d)

{P = A \mathrm{e}^{- \frac{3}{100} t} + \frac{100n}{3}.}

(e)

{n = 15.}

(b)

{x = \sqrt{4a+a^2}.}

{\tan \theta = \frac{2}{\sqrt{4a+a^2}}.}

(c)

The optimal point and optimal angle may not correspond to the highest chance of scoring.

(c)

{\angle KDA \approx \frac{\pi}{2} \textrm{ rad}.}

(e)

{0.0801 \leq \theta < \frac{\pi}{2}}

Solutions

Full solutions (with working)