October 2021 Paper 2 Q4

OCR ACurrent spec10 marksModelling Trigonometry

4 The size, \(P\), of a population of a certain species of insect at time \(t\) months is modelled by the following formula.

\(P = 5000 - 1000\cos(30t)^\circ\)

(a) Write down the maximum size of the population. [1]

(b) Write down the difference between the largest and smallest values of \(P\). [1]

(d) Find the time taken for the population to return to its initial size for the first time. [2]

(e) Determine the time on the second occasion when \(P = 4500\). [4]

A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and \(P\) converges to 5000.

(f) Suggest a change to the model that will take account of this observation. [1]

Scheme	Marks
6000	B1
	[1]

Scheme	Marks
2000	B1f
	[1]

Scheme	Marks
Oscillates or Goes up and down. oe Fluctuates. Moves in a cycle	B1
	[1]

Scheme	Marks
\(30t = 360\)	M1
Time to return to initial size = 12 months	A1
	[2]

Mark scheme (e)

Scheme	Marks
\(4500 = 5000 - 1000\cos(30t)^\circ\)	M1
\(\cos(30t)^\circ = 0.5\)	A1
\(30t = 60\) or \(300\) (both)	M1
2nd time \(P = 4500\) is when \(t = 10\)	A1
	[4]

Notes

M1: Substitute \(P = 4500\) May be implied by next line

A1: Correct rearrangement

M1: Attempt \(30t = \cos^{-1}(\text{their } 0.5)\), giving \(\alpha\) and \(360 - \alpha\).
Condone \(30\mathrm{t} = \frac{\pi}{3}, \frac{5\pi}{3}\)

A1: or after 10 months. Allow \(t = 10\) months, or just 10
SC. (If not gained 1^st M1A1) Correct answer with no or inadequate working and/or T&I: \(t = 10\) stated: B2; \(t = 10\) embedded: B1B0

Alternative methods for 2nd M1A1

Scheme	Marks
\(30t = 60\) or \(-60\) (both) \((t = 2 \text{ or } -2)\)	M1
2nd time \(P = 4500\) is when \(t = -2 + 12 = 10\)	A1

M1: \(30t = 60\ (t = 2)\)
A1: (end of 1^st cycle at \(t = 12\)) 2^nd time \(P = 4500\) is when \(t = 12 - 2 = 10\)

Scheme	Marks
\(30t = 60\ \ (t = 2)\)	M1
\(6 - 2 = 4\); \(t = 6 + 4 = 10\)	A1

Mark scheme (f)

Scheme	Marks
eg \(P = 5000 - 1000\mathrm{e}^{-t}\cos(30t)^\circ\) \(P = 5000 - 1000\mathrm{e}^{-kt}\cos(30t)^\circ\ (k \gt 0)\) Answers in words must be equivalent to one of these	B1
	[1]

Notes

or other good answers
eg \(P = 5000 - (1000\cos(30t)^\circ)^{1/t}\)
\(P = 5000 - \dfrac{1000}{t}\cos(30t)^\circ.\ (t \gt 0)\)