IB Maths AI SL AI 1.1 Number and approximation questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

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Question 1

Paper 1Medium6 marks

A laboratory sample contains $3.6 \times 10^{13}$ identical nanoparticles. Each nanoparticle has a mass of $4.7 \times 10^{-19}$ kg.

(a)

Calculate the total mass of the nanoparticles in the sample, in kg. Give your answer in scientific notation.

[2]

(b)

A filter removes nanoparticles with a combined mass of $5.64 \times 10^{-6}$ kg. Using your answer to part (a), determine the number of nanoparticles remaining. Give your answer in scientific notation.

[4]

Question 2

Paper 1Hard16 marks

A workshop makes a circular cover. Its radius is measured as $6.4\text{ cm}$ correct to the nearest $0.1\text{ cm}$ The workshop estimates the area using this displayed radius and rounds the estimate to three significant figures for its label.

For all parts, use unrounded calculator values in subsequent calculations unless a specified rounding is requested. Percentage error is defined as

\frac{|\text{estimate}-\text{true value}|}{\text{true value}}\times 100\%.

(a)

Calculate the estimated area of the cover, giving your answer to three significant figures.

[2]

(b)

Write down the lower and upper bounds for the radius.

[2]

(c)

Let $A\text{ cm}^2$ be the true area of the cover. Show that a guaranteed interval for $A$ is

126.67\leq A<130.70.

[3]

(d)

Calculate the maximum percentage error in the area estimate.

[3]

(e)

The workshop prints $129\text{ cm}^2$ on the cover’s label. Determine whether this label is guaranteed to have a percentage error of less than $2\%$ Justify your answer.

[2]

(f)

A new gauge displays the radius as $6.42\text{ cm}$ correct to the nearest $q\text{ cm}$ Determine the greatest possible value of $q$ for which the estimated area is guaranteed to have a percentage error of at most $1\%$ State whether rounding to the nearest $0.1\text{ cm}$ and to the nearest $0.01\text{ cm}$ meet this requirement.

[4]

Question 3

Paper 1Hard15 marks

A workshop is producing identical circular badges for a display. Their common radius is measured as $3.4$ cm, correct to the nearest $0.1$ cm. The combined face area of a batch is the sum of the areas of its badges. The display requires a combined face area of at least $680\text{ cm}^2$

(a)

State the lower and upper bounds for the actual radius $r$

[2]

(b)

Let $A\text{ cm}^2$ be the area of one badge. Show that

11.2225\pi \leq A < 11.9025\pi.

[3]

(c)

Calculate the estimated area of one badge, based on the reported radius, giving your answer to 3 significant figures. Find a conservative upper bound for the percentage error in this estimate, using the estimated area as the denominator. Give the percentage bound to 3 significant figures, rounding upwards.

[4]

(d)

Using the estimated area from part (c), determine the minimum number of badges indicated by the estimate to meet the display’s area requirement.

[2]

(e)

The workshop orders the number of badges found in part (d). Assess the supervisor’s claim that this order is guaranteed to meet the display’s area requirement, and determine the smallest order that would guarantee it.

[4]

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