IBDP Maths AI: Topic : SL 1.6: Approximation: decimal places, significant figures: IB style Questions SL Paper 1

Question 1 [Maximum mark: 4]

Katya approximates p, correct to four decimal places, by using the following expression.

\( 3+ \frac{1}{6+\frac{13}{16}}\)

(a) Calculate Katya’s approximation of p, correct to four decimal places. [2]

(b) Calculate the percentage error in using Katya’s four decimal place approximation of p, compared to the exact value of p in your calculator. [2]

Answer/Explanation

(a) \(3+ \frac{1}{6+\frac{13}{16}}\) 

\(=3.14678.. \((\frac{343}{109},3 \frac{16}{109})\)

\(= 3.1465\)

(b) \(|\frac{3.1468-\pi }{\pi }|\times 100\) = 0.166(%) (0.165754…)

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]
a.

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]
b.
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]

a.

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]

b.

Question

1 Brazilian Real (BRL) = 2.607 South African Rand (ZAR). Giving answers correct to two decimal places,

(i) convert 300 BRL to ZAR,

(ii) find how many Real it costs to purchase 300 Rand.

Answer/Explanation

Markscheme

Financial accuracy penalty (FP) is applicable where indicated in the left hand column.

1 BRL = 2.607 ZAR

(FP) (i) \(300 \times 2.607 = 782.10 {\text{ ZAR}}\)     (A1)

Note: 782.1 is (A0)(FP)

(FP) (ii) \(300 \times \frac{1}{{2.607}} = 115.07{\text{ BRL}}\)     (A1)(ft)

Note: Follow through only if processes are reversed.     (C2)

[2 marks]

Question

Ben inherits $6500. Ben invests his money in a bank that pays compound interest at a rate of 4.5% per annum.

Calculate the value of Ben’s investment at the end of 6 years. Give your answer correct to 2 decimal places.

Answer/Explanation

Markscheme

\({\text{Ben Amount}} = 6500{\left( {1 + \frac{{4.5}}{{100}}} \right)^6}\)     (M1)(A1)

\( = $8464.69\)     (A1)

(M1)(A1)(A0) if interest only found (=$1964.69)     (C3)

[3 marks]

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