IB Maths AI HL AI 1.3 Exponents, logarithms and equations 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 1Paper 2
Difficulty
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Question 1

Paper 1Easy6 marks

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

pH=−log⁡10(C),\mathrm{pH}=-\log_{10}(C),

where CC is the numerical value of the hydrogen ion concentration in mol dm−3\mathrm{mol\,dm^{-3}}

Solution ASolution BHydrogen ionconcentration =0.000032mol dm-3Hydrogen ionconcentration =0.00000080mol dm-3
(a)

Write down the hydrogen ion concentration of solution B in standard form, including its unit.

[1]
(b)

Calculate the pH of solution A, giving your answer to 2 decimal places.

[2]
(c)

Determine the ratio of the hydrogen ion concentration of solution A to that of solution B, giving your answer in standard form.

[2]
(d)

Hence state which solution is more acidic.

[1]

Question 2

Paper 1Hard6 marks

During an acoustic test, sound level LL in decibels (dB), is defined by

L=10log⁡10(II0),L=10\log_{10}\left(\frac{I}{I_0}\right),

where II is the sound intensity and I0I_0 is a fixed reference intensity. When machines operate together, their sound intensities add.

Machine A alone produces a sound level of 67.467.4 dB and machine B alone produces a sound level of 72.172.1 dB.

(a)

Show that if the sound levels of A and B are LAL_A and LBL_B respectively, then

IBIA=10(LB−LA)/10.\frac{I_B}{I_A}=10^{(L_B-L_A)/10}.
[2]
(b)

Determine the sound level when A and B operate together. Give your answer to the nearest 0.10.1 dB.

[2]
(c)

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).

[2]

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