Direct & Inverse Proportion
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Higher June 2025 Paper 1 Q18
18 \(y\) is inversely proportional to \(x^3\)
\(y = 4\) when \(x = 3\)
Work out the value of \(x\) when \(y = 864\)
(4)
Mark scheme| Scheme | Marks |
|---|
\(y = \dfrac{k}{x^3}\)
or\(hy = \dfrac{1}{x^3}\) | M1 |
\(4 = \dfrac{k}{3^3}\) or \(k = 4 \times 3^3\) (= 108)
or\(h \times 4 = \dfrac{1}{3^3}\) or \(h = \dfrac{1}{4 \times 3^3}\left(= \dfrac{1}{108}\right)\) | M1 |
| eg \(\left(x^3 =\right)\dfrac{4 \times 3^3}{864}\left(= \dfrac{1}{8}\right)\) or \(\left(x^3 =\right)\dfrac{\text{``}{108}\text{''}}{864}\left(= \dfrac{1}{8}\right)\) oe | M1 |
Correct answer scores full marks (unless from obvious incorrect working)
Answer: 0.5 | A1 |
| (4) |
| (4 marks) |
Notes
M1: do not award for \(y = \dfrac{1}{x^3}\)
Constant of proportionality must be a symbol such as \(k\)
Condone use of \(\propto\) for method marks
M1: for substitution of \(x\) and \(y\) into a correct formula
Condone use of \(\propto\) for method marks
M1: for method to find \(x^3\)
Condone use of \(\propto\) for method marks
A1: oe
Higher June 2025 Paper 1R Q18
18 \(F\) is inversely proportional to the cube of \(r\)
\(F = 6\) when \(r = 2\)
(a) Find a formula for \(F\) in terms of \(r\) (3)
(b) Find the value of \(r\) when \(F = 3072\) (2)
Mark scheme (a)| Scheme | Marks |
|---|
| \(F = \dfrac{k}{r^3}\) or \(Fr^3 = k\) or \(kF = \dfrac{1}{r^3}\) | M1 |
\(6 = \dfrac{k}{2^3}\) oe or \(k = 48\) or \(6k = \dfrac{1}{2^3}\) oe or \(k = \dfrac{1}{48}\) | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(F = \dfrac{48}{r^3}\) | A1 |
| (3) |
Notes
M1: oe \(k\) can be any letter (must be a letter and not 1)
M1: For substitution of \(F\) and \(r\) into a correct formula, implies the first M1 if you see this stage
Condone use of \(\propto\) for method marks
A1: oe with \(F\) the subject eg \(F = 48 \times \dfrac{1}{r^3}\) or
\(F = 48 \times r^{-3}\)
Award 3 marks if answer is \(F = \dfrac{k}{r^3}\) and
\(k = 48\) clearly given in the body of the script
M2A0 for \(Fr^3 = 48\) or \(r = \sqrt[3]{\dfrac{48}{F}}\) or \(r^3 = \dfrac{48}{F}\)
Mark scheme (b)| Scheme | Marks |
|---|
| \((r^3 =)\dfrac{\text{``}{48}\text{''}}{3072}\) oe eg \(\dfrac{1}{64}\) or (0.01(5625)) rounded or truncated | M1ft |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(\dfrac{1}{4}\) | A1 |
| (2) |
| (5 marks) |
Notes
M1ft: allow use of their “48” as long as M2 gained in (a)
A1: oe
Higher November 2024 Paper 1 Q14
14 \(B\) is inversely proportional to the square of \(d\)
\(B = 0.25\) when \(d = 12\)
Find a formula for \(B\) in terms of \(d\)
(3)
Mark scheme| Scheme | Marks |
|---|
\(B = \dfrac{k}{d^2}\) oe
or\(Bk = \dfrac{1}{d^2}\) oe | M1 |
\(0.25 = \dfrac{k}{12^2}\) oe or \(0.25 = \dfrac{k}{144}\) oe or \(k = 36\)
or\(0.25 \times k = \dfrac{1}{12^2}\) oe or \(0.25 \times k = \dfrac{1}{144}\) oe or \(k = \dfrac{1}{36}\) | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(B = \dfrac{36}{d^2}\) | A1 |
| (3) |
| (3 marks) |
Notes
M1: (NB Not for \(B = \dfrac{1}{d^2}\))
Constant of proportionality must be a symbol such as \(k\)
Allow \(D\) or \(b\)
M1: for substitution of \(B\) and \(d\) into a correct formula
A1: e.g \(B = 36 \times \dfrac{1}{d^2}\) or \(B = 36 \times d^{-2}\)
Award 3 marks if answer is \(B = \dfrac{k}{d^2}\) on the answer line and \(k = 36\) clearly given in the body of working of the script
M2A0 for \(Bd^2 = 36\) or \(d = \dfrac{6}{\sqrt{B}}\) or \(d^2 = \dfrac{36}{B}\)
M2 for
\(0.25 = \dfrac{k}{12^2}\) oe or
\(0.25 \times k = \dfrac{1}{12^2}\) oe or
No questions match these filters.
