Calculate the exact value of \(T\).

Give your answer to \(T\) correct to

(i)     two significant figures;

(ii)     three decimal places.

Pyotr estimates the value of \(T\) to be \(0.002\).

Calculate the percentage error in Pyotr’s estimate.

\(\frac{{\left( {\tan (2 \times 30) + 1} \right)\left( {2\cos (30) - 1} \right)}}{{{{41}^2} - {9^2}}}\)     (M1)

Note: Award (M1) for correct substitution into formula.

\( = 0.00125\;\;\;\left( {\frac{1}{{800}}} \right)\)     (A1)     (C2)

Note: Using radians the answer is \( - 0.000570502\), award at most (M1)(A0).

(i)     \(0.0013\)    (A1)(ft)

Note: Follow through from part (a).

(ii)     \(0.001\)     (A1)(ft)     (C2)

Note: Follow through from part (a).

\(\left| {\frac{{0.002 - 0.00125}}{{0.00125}}} \right| \times 100\)     (M1)

Notes: Award (M1) for their correct substitution into the percentage error formula. Absolute value signs are not required.

Their unrounded answer from part (a) must be used.

Do not accept use of answers from part (b).

\( = 60{\text{ (%)}}\)     (A1)(ft)     (C2)

Notes: The \({\text{%}}\) sign is not required.

The answer from radians is \(450.568{\text{%}}\), award (M1)(A1)(ft).

Follow through from part (a).

Despite a significant number of candidates scoring well on this question, many candidates failed to use their calculator correctly. Common errors identified were: the use of radians; incorrect use of the double parentheses, calculating tan(61) instead of tan(60) + 1 ; or premature rounding. Such candidates earned, at most, method in part (a).

Despite errors in part (a), part (b)(i) tended to be often a correct follow through answer but some candidates struggled to give a 2 sf answer correctly, using truncation instead of rounding or dropping the leading zeros. Part (b)(ii) was more often answered correctly.

In part (c) many candidates used the percentage error formula incorrectly, reversing the estimated and the exact value, or using one of the rounded answers from part (b) as the exact value.