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.
A circular inspection window has a radius recorded as correct to the nearest
For this question, percentage error is
Write down the lower and upper bounds for the radius.
Calculate the estimated area of the window, giving your answer to three significant figures.
Determine the greatest possible percentage error in this estimated area. Use unrounded values in your calculations and give your answer to three significant figures.
A comet lander collects dust during a -minute sampling pulse. Figure 1 shows the rate at which particles enter the collection chamber. Assume that varies linearly between consecutive readings. The mean mass of a particle is
Determine the total mass of dust collected during the pulse. Give your answer in scientific notation to 2 significant figures.
A return canister has a maximum dust payload of Is it sufficient for the dust collected during the full pulse? Justify your answer with a numerical comparison.
A remote-sensing archive receives data from two instruments. Instrument A records observations, each requiring bytes. Instrument B produces bytes of data. The archive has bytes of free space. Each additional storage module can hold at most bytes.
Show that the data from Instrument A require bytes. Give your answer in scientific notation.
Determine the least number of additional storage modules required to store all the data from both instruments. Justify that this number is least.
A neighbourhood repair fund plans to increase its annual contribution by the same amount each year. Figure 1 shows the planned contributions for the first three years.
The planned contributions form an arithmetic sequence.
Write down the first term and the common difference of the sequence.
Find the planned contribution for year 8.
Calculate the total planned contributions for the first eight years.
Two accounts each receive a deposit of $10 000. The principal is returned in full by either account, so only interest and fees are compared.
Account A pays simple interest at per year and has no fees.
The annual interest payments observed for account B are shown below.
| Year | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Interest payment ($) | 246 | 267 | 284 | 306 |
For a projection, an adviser models account B’s annual interest payments as an arithmetic sequence. The first term is the payment in year 1, and the common difference is the mean of the three consecutive changes shown in the table. Assume this model continues for years. Account B has a one-time opening fee of $200.
State the first term and common difference of the arithmetic model for account B.
Let be the total interest paid by account B during the first years. Show that
Hence determine the first whole number of years for which account B’s cumulative interest, after its opening fee, exceeds account A’s cumulative interest. Justify that this is the first such year.
A laboratory compares the hydrogen ion concentrations of two aqueous solutions. Figure 1 shows the measured concentrations. For these solutions, the one with the greater hydrogen ion concentration is more acidic.
For the pH calculations, use
where is the numerical value of the hydrogen ion concentration in
Write down the hydrogen ion concentration of solution B in standard form, including its unit.
Calculate the pH of solution A, giving your answer to 2 decimal places.
Determine the ratio of the hydrogen ion concentration of solution A to that of solution B, giving your answer in standard form.
Hence state which solution is more acidic.
A theatre records the total revenue from three group ticket sales. Let and be the prices, in dollars, of one adult, student and child ticket, respectively. The records are shown in Figure 1.
For a later performance, the regular adult ticket price is dollars. If the discount is dollars, the estimated number of discounted adult tickets sold is The revenue from these tickets is modelled by
where
Use your GDC to determine the values of and
Using your answer to (a), use your GDC to find all values of for which Give your answers to the nearest cent.
During an acoustic test, sound level in decibels (dB), is defined by
where is the sound intensity and is a fixed reference intensity. When machines operate together, their sound intensities add.
Machine A alone produces a sound level of dB and machine B alone produces a sound level of dB.
Show that if the sound levels of A and B are and respectively, then
Determine the sound level when A and B operate together. Give your answer to the nearest dB.
Identical copies of machine A are used instead. Determine the least whole number of copies needed for their combined sound level to be at least the level found in part (b).
A point on an Argand diagram is represented by
Write down the real part and the imaginary part of
Write down the complex conjugate of
Determine the distance of from the origin, in units.
Use your GDC to calculate giving your answer in Cartesian form.
A field station sends four-letter locker identifiers using a linear code. The letters A to Z are represented by the integers 1 to 26, respectively. Each successive pair of letters is written as a column vector and is encoded as
The transmitted pairs, in order, are
The same encoding matrix was used for both pairs. It was one of
Show that
Determine which matrix was used and hence decode the locker identifier. Justify your answer.
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