June 2024 Paper 1 Q12

Notes

Note: Candidates working in radians are able to score all the M and B marks in this question.
Condone the absence of the degrees symbol throughout the whole question.

B1: Correct value for \(K\). Condone \(R = 500\)

M1: Award for \(\tan\alpha = \pm\dfrac{480}{140} \Rightarrow \alpha = \ldots\), \(\tan\alpha = \pm\dfrac{140}{480} \Rightarrow \alpha = \ldots\), \(\sin\alpha = \pm\dfrac{480}{\text{``}500\text{''}} \Rightarrow \alpha = \ldots\) or \(\cos\alpha = \pm\dfrac{140}{\text{``}500\text{''}} \Rightarrow \alpha = \ldots\)
Note \(\alpha =\) awrt 1.3 (rad) implies this mark.

A1: \(\alpha =\) awrt \(73.74\{^\circ\}\) or correct expression \(500\cos(\theta + 73.74)\{^\circ\}\)

Notes

(b)(i) Note: mark parts (b)(i) and (b)(ii) together.

B1ft: Correct equation of the model in either form including the \(R\) = following through on their numerical \(K\) (\(0 \lt K \leqslant 750\)) and their numerical \(\alpha\).
Allow for e.g. \(R = 1500 - \text{``}500\text{''} + \text{``}500\text{''}\cos(30t + \text{``}73.74\text{''})\{^\circ\}\) or for \(R = 1500 - \text{``}500\text{''} + 140\cos(30t)^\circ - 480\sin(30t)^\circ\) but not e.g. \(R = 1500 - K + K\cos(30t + \alpha)^\circ\)
\(R = 1000 + 140\cos 30t - 480\sin 30t\) (without the brackets) is correct.
Allow this mark if they have truncated or rounded an otherwise correct \(\alpha\) (to 3s.f.)

(b)(ii)

B1ft: 500 or follow through on (their \(A\) – their \(K\)) or (1500 – 2 × their \(K\)) provided it is non-negative and less than 1500. It must be clear this is their answer to (b)(ii) so expect to see e.g. (b) or \(R_{\min} =\) or an indication it is the minimum.

Notes

Note: if \(\theta\) is used in place of \(30t\) then they must revert back to \(30t\) correctly to access the marks.

M1: Substitutes \(t = 3.5\) into their model for the number of rabbits (you may need to check if no method is shown)
or substitutes \(t = 3.5\) into their \(\cos(30t + \alpha)^\circ\)
Condone substitution of a value of \(t\) in the range \(3 \leqslant t \leqslant 4.5\) for this mark.

A1: \(R =\) awrt 500.1… or 500 (not awrt) following substitution of \(t = 3.5\), suggesting that the model is valid/reliable/appropriate/good.
or \(\cos(30(3.5) + 73.74)\{^\circ\} \approx -1\) suggesting that the model is valid/reliable/appropriate/good.
Allow this mark if they have truncated or rounded an otherwise correct \(\alpha\) (to 3s.f.)

Alt:

M1: Minimum occurs when \(A + K\cos(30t + \alpha)^\circ = R_{\min} \Rightarrow \cos(30t + \alpha)^\circ = \lambda\) with \(|\lambda| \leqslant 1\) leading to \(t = \ldots\)
May just see \(\cos(30t + 73.74)^\circ = -1 \Rightarrow t = \ldots\) (or their \(\cos(30t + \alpha)^\circ = -1 \Rightarrow t = \ldots\))
\(30t + \alpha = 180 \Rightarrow t = \ldots\) implies this mark. Condone \(30t + \alpha = \pi \Rightarrow t = \ldots\) for this mark.

A1: \(t = 3.54\ldots\) (i.e. the middle of April) so the model is valid/reliable/appropriate/good.
Do not condone incorrect statements, e.g., \(t = 3.54\ldots\) i.e. the middle of March so close to middle of April. If using \(A + K\cos(30t + \alpha)^\circ = R_{\min}\) then \(R_{\min}\) must be = their \(A -\) their \(K\)
Allow this mark if they have truncated or rounded an otherwise correct \(\alpha\) (to 3s.f.)

Alt 2 using differentiation (Condoned)

M1: Condone finding the minimum using \(\ldots\sin(30t + 73.74)^\circ = 0 \Rightarrow t = \ldots\) (or their \(\sin(30t + \alpha)^\circ = 0 \Rightarrow t = \ldots\))
\(30t + \alpha = 180 \Rightarrow t = \ldots\) implies this mark. Condone \(30t + \alpha = \pi \Rightarrow t = \ldots\) for this mark.

A1: \(t = 3.54\ldots\) (i.e. the middle of April) suggesting that the model is valid/reliable/appropriate.
Do not condone incorrect statements, e.g., \(t = 3.54\ldots\) i.e. the middle of March so close to middle of April.
The complete derivative for \(\dfrac{\mathrm{d}R}{\mathrm{d}t}\) does not need to be seen.
Allow this mark if they have truncated or rounded an otherwise correct \(\alpha\) (to 3s.f.)

Notes

Note: if \(\theta\) is used in place of \(30t\) then they must revert back to \(30t\) correctly to access the marks.

M1: Realises that \(\sin(30t + 70)^\circ = -1\), reaches \(30t + 70 = 270\) or \(-90\) and attempts to find \(t\) (or \(30t\))
Condone attempts using differentiation. The minimum occurs when \(\cos(30t + 70)^\circ = 0 \Rightarrow 30t + 70 = 270 \Rightarrow 30t = \ldots\) (or \(t = \ldots\)). They must use 270 or \(-90\) and not 90 to achieve the minimum. Condone \(30t + \alpha = \dfrac{3\pi}{2} \Rightarrow t = \ldots\) for this mark but not \(30t + \alpha = \dfrac{\pi}{2} \Rightarrow t = \ldots\).

A1: Correct value for \(30t\) (or \(t\)) Accept rounded or truncated values to at least 3s.f. e.g. 6.66 or 6.67

dM1: Substitutes their value of \(t \gt 0\) (or \(30t \gt 0\)) coming from \(30t + 70 = \mathbf{270}\) into their model for \(R\)

A1: Correct number of rabbits. Allow 1032 or 1033 but must be whole numbers and not just 1030.
Allow this mark if they have truncated or rounded an otherwise correct \(\alpha\) (to 3s.f.)