Question
Five pipes labelled, “6 metres in length”, were delivered to a building site. The contractor measured each pipe to check its length (in metres) and recorded the following;
5.96, 5.95, 6.02, 5.95, 5.99.
(i) Find the mean of the contractor’s measurements.
(ii) Calculate the percentage error between the mean and the stated, approximate length of 6 metres.[3]
Calculate \(\sqrt {{{3.87}^5} – {{8.73}^{ – 0.5}}} \), giving your answer
(i) correct to the nearest integer,
(ii) in the form \(a \times 10^k\), where 1 ≤ a < 10, \(k \in {\mathbb{Z}}\) .[3]
Answer/Explanation
Markscheme
(i) Mean = (5.96 + 5.95 + 6.02 + 5.95 + 5.99) / 5 = 5.974 (5.97) (A1)
(ii) \({\text{% error}} = \frac{{error}}{{actualvalue}} \times 100\% \)
\( = \frac{{6 – 5.974}}{{5.974}} \times 100\% = 0.435\% \) (M1)(A1)(ft)
(M1) for correctly substituted formula.
Allow 0.503% as follow through from 5.97
Note: An answer of 0.433% is incorrect. (C3)[3 marks]
number is 29.45728613
(i) Nearest integer = 29 (A1)
(ii) Standard form = 2.95 × 101 (accept 2.9 × 101) (A1)(ft)(A1)
Award (A1) for each correct term
Award (A1)(A0) for 2.95 × 10 (C3)[3 marks]
Question
Write down the following numbers in increasing order.
\(3.5\), \(1.6 \times 10^{−19}\), \(60730\), \(6.073 \times 10^{5}\), \(0.006073 \times 10^6\), \(\pi\), \(9.8 \times 10^{−18}\).[3]
Write down the median of the numbers in part (a).[1]
State which of the numbers in part (a) is irrational.[1]
Answer/Explanation
Markscheme
\(1.6 \times 10^{−19}\), \(9.8 \times 10^{−18}\), \(\pi\), \(3.5\), \(0.006073 \times 10^6\), \(60730\), \(6.073 \times 10^{5}\) (A4)
Award (A1) for \(\pi\) before 3.5
Award (A1) for \(1.6 \times 10^{−19}\) before \(9.8 \times 10^{−18}\)
Award (A1) for the three numbers containing 6073 in the correct order.
Award (A1) for the pair with negative indices placed before 3.5 and \(\pi\) and the remaining three numbers placed after (independently of the other three marks).
Award (A3) for numbers given in correct decreasing order.
Award (A2) for decreasing order with at most 1 error (C4)[3 marks]
The median is 3.5. (A1)(ft)
Follow through from candidate’s list. (C1)[1 mark]
\(\pi\) is irrational. (A1) (C1)[1 mark]
Question
Calculate \(\frac{{77.2 \times {3^3}}}{{3.60 \times {2^2}}}\).[1]
Express your answer to part (a) in the form \(a \times 10^k\), where \(1 \leqslant a < 10\) and \(k \in {\mathbb{Z}}\).[2]
Juan estimates the length of a carpet to be 12 metres and the width to be 8 metres. He then estimates the area of the carpet.
(i) Write down his estimated area of the carpet.
When the carpet is accurately measured it is found to have an area of 90 square metres.
(ii) Calculate the percentage error made by Juan.[3]
Answer/Explanation
Markscheme
\(144.75\left( { = \frac{{579}}{4}} \right)\) (A1)
accept 145 (C1)[1 mark]
\(1.4475 \times 10^2\) (A1)(ft)(A1)(ft)
accept \(1.45 \times 10^2\) (C2)[2 marks]
Unit penalty (UP) is applicable in question part (c)(i) only.
(UP) (i) Area = 96 m2 (A1)
(ii) \(\% {\text{ error}} = \frac{{(96 – 90)}}{{90}} \times 100\) (M1)
\( = \frac{{6 \times 100}}{{90}}\)
\(\frac{{20}}{3}\% \) or 6.67 % (A1)(ft) (C3)[3 marks]
Question
Calculate exactly \(\frac{{{{(3 \times 2.1)}^3}}}{{7 \times 1.2}}\).[1]
Write the answer to part (a) correct to 2 significant figures.[1]
Calculate the percentage error when the answer to part (a) is written correct to 2 significant figures.[2]
Write your answer to part (c) in the form \(a \times {10^k}\) where \(1 \leqslant a < 10{\text{ and }}k \in \mathbb{Z}\).[2]
Answer/Explanation
Markscheme
\(29.7675\) (A1) (C1)
Note: Accept exact answer only.[1 mark]
\(30\) (A1)(ft) (C1)[1 mark]
\(\frac{{30 – 29.7675}}{{29.7675}} \times 100\% \) (M1)
For correct formula with correct substitution.
\( = 0.781\% \) accept \(0.78\% \) only if formula seen with \(29.7675\) as denominator (A1)(ft) (C2)[2 marks]
\(7.81 \times {10^{ – 1}}\% \) (\(7.81 \times {10^{ – 3}}\) with no percentage sign) (A1)(ft)(A1)(ft) (C2)[2 marks]
Question
Given that \(h = \sqrt {{\ell ^2} – \frac{{{d^2}}}{4}} \) ,
Calculate the exact value of \(h\) when \(\ell = 0.03625\) and \(d = 0.05\) .[2]
Write down the answer to part (a) correct to three decimal places.[1]
Write down the answer to part (a) correct to three significant figures.[1]
Write down the answer to part (a) in the form \(a \times {10^k}\) , where \(1 \leqslant a < 10{\text{, }}k \in \mathbb{Z}\).[2]
Answer/Explanation
Markscheme
\(h = \sqrt {{{0.03625}^2} – \frac{{{{0.05}^2}}}{4}} \) (M1)
\( = 0.02625\) (A1) (C2)
Note: Award (A1) only for \(0.0263\) seen without working[2 marks]
\(0.026\) (A1)(ft) (C1)[1 mark]
\(0.0263\) (A1)(ft) (C1)[1 mark]
\(2.625 \times {10^{ – 2}}\)
for \(2.625\) (ft) from unrounded (a) only (A1)(ft)
for \( \times {10^{ – 2}}\) (A1)(ft) (C2)[2 marks]