Write down the decrease in the temperature of the coffee

(i) during the first minute (between t = 0 and t =1) ;

(ii) during the second minute;

(iii) during the third minute.

Assuming the pattern in the answers to part (a) continues, show that \(k = 19\).

Use the seven results in the table to draw a graph that shows how the temperature of the coffee changes during the first six minutes.

Use a scale of 2 cm to represent 1 minute on the horizontal axis and 1 cm to represent 10 °C on the vertical axis.

The function that models the change in temperature of the coffee is y = p (2^−t )+ q.

(i) Use the values t = 0 and y = 94 to form an equation in p and q.

(ii) Use the values t =1 and y = 54 to form a second equation in p and q.

Solve the equations found in part (d) to find the value of p and the value of q.

The graph of this function has a horizontal asymptote.

Write down the equation of this asymptote.

George decides to model the change in temperature of the coffee with a linear function using correlation and linear regression.

Use the seven results in the table to write down

(i) the correlation coefficient;

(ii) the equation of the regression line y on t.

Use the equation of the regression line to estimate the temperature of the coffee at t = 3.

Find the percentage error in this estimate of the temperature of the coffee at t = 3.

(i) 40

(ii) 20

(iii) 10     (A3)

Notes: Award (A0)(A1)(ft)(A1)(ft) for −40, −20, −10.

    Award (A1)(A0)(A1)(ft) for 40, 60, 70 seen.

    Award (A0)(A0)(A1)(ft) for −40, −60, −70 seen.

\(24 - k = 5\) or equivalent     (A1)(M1)

Note: Award (A1) for 5 seen, (M1) for difference from 24 indicated.

\(k = 19\)     (AG)

Note: If 19 is not seen award at most (A1)(M0).

     (A1)(A1)(A1)(A1)

Note: Award (A1) for scales and labelled axes (t or “time” and y or “temperature”).

   Accept the use of x on the horizontal axis only if “time” is also seen as the label.

   Award (A2) for all seven points accurately plotted, award (A1) for 5 or 6 points accurately plotted, award (A0) for 4 points or fewer accurately plotted.

   Award (A1) for smooth curve that passes through all points on domain [0, 6].

   If graph paper is not used or one or more scales is missing, award a maximum of (A0)(A0)(A0)(A1).

(i) \(94 = p + q\)     (A1)

(ii) \(54 = 0.5p + q\)     (A1)

Note: The equations need not be simplified; accept, for example \(94 = p(2^{-0}) + q\).

p = 80, q = 14     (G1)(G1)(ft)

Note: If the equations have been incorrectly simplified, follow through even if no working is shown.

y = 14     (A1)(A1)(ft)

Note: Award (A1) for y = a constant, (A1) for their 14. Follow through from part (e) only if their q lies between 0 and 15.25 inclusive.

(i) –0.878 (–0.87787...)     (G2)

Note: Award (G1) if –0.877 seen only. If negative sign omitted award a maximum of (A1)(A0).

(ii) y = –11.7t + 71.6 (y = –11.6517...t + 71.6336...)     (G1)(G1)

Note: Award (G1) for –11.7t, (G1) for 71.6.

   If y = is omitted award at most (G0)(G1).

   If the use of x in part (c) has not been penalized (the axis has been labelled “time”) then award at most (G0)(G1).

−11.6517...(3) + 71.6339...     (M1)

Note: Award (M1) for correct substitution in their part (g)(ii).

= 36.7 (36.6785...)     (A1)(ft)(G2)

Note: Follow through from part (g). Accept 36.5 for use of the 3sf answers from part (g).

\(\frac{{36.6785... - 24}}{{24}} \times 100\)     (M1)

Note: Award (M1) for their correct substitution in percentage error formula.

= 52.8% (52.82738...)     (A1)(ft)(G2)

Note: Follow through from part (h). Accept 52.1% for use of 36.5.

   Accept 52.9 % for use of 36.7. If partial working (\(\times 100\) omitted) is followed by their correct answer award (M1)(A1). If partial working is followed by an incorrect answer award (M0)(A0). The percentage sign is not required.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.

Almost all candidates were able to score on the first parts of this question; errors occurring only when insufficient care was taken in reading what the question was asking for. The graph was usually well drawn, other than for those who have no idea what centimetres are.

The majority were able to determine the simultaneous equations, if only in unsimplified form; there was less success in solving these – though this is easily done via the GDC (the preferred approach) and the equation of the asymptote proved a discriminating task. The final parts, involving correlation and regression were largely independent of the previous parts and were accessible to most. Hopefully, contrasting the large percentage error with the value of the correlation coefficient will be valuable in class discussions. Given the many scripts that gave the value of the coefficient of determination as that of \(r\) , it seems better that the former is simply not taught.