\(w\) is directly proportional to the cube root of \(d\).
When \(w = 6\), \(T = 20\) When \(w = 1\), \(d = 8\)
Find the value of \(d\) when \(T = 48\) (5)
Mark scheme
Answer
Mark
Mark scheme
125
P1
for setting up an equation with a constant term, eg \(T = \dfrac{k}{w}\) or \(w = K\sqrt[3]{d}\)
P1
for a process to substitute values in one equation, eg \(20 = \dfrac{k}{6}\) or \(k = 120\) or \(1 = K\sqrt[3]{8}\) or \(K = \dfrac{1}{2}\) oe
P1
(dep P2) for combining the two equations ft their values of \(k\) and \(K\), eg \(T = \dfrac{\text{``}120\text{''}}{\text{``}\frac{1}{2}\text{''}\sqrt[3]{d}}\) oe OR for a correct process to find the value of \(w\) when \(T = 48\), eg \(w = \dfrac{\text{``}120\text{''}}{48}\) (= 2.5 oe)
P1
for substitution into their formula, eg \(48 = \dfrac{\text{``}120\text{''}}{\text{``}\frac{1}{2}\text{''}\sqrt[3]{d}}\) OR for substitution of found value of \(w\) in the equation for \(d\), eg \(\text{``}2.5\text{''} = \text{``}\dfrac{1}{2}\text{''}\sqrt[3]{d}\)
A1
cao
Additional guidance
Condone the use of ‘\(\alpha\)’ instead of ‘=’ for the first two P marks Equation can be implied by correct substitution
\(T = \dfrac{120}{w} \qquad w = \dfrac{1}{2}\sqrt[3]{d}\)
Nadia says that \(y\) is directly proportional to \(x\) because the value of \(y\) increases by 4 as the value of \(x\) increases by 2
(a) Is Nadia correct? You must give a reason for your answer. (1)
\(w\) is directly proportional to the square root of \(t\).
\(w = 140\) when \(t = 64\)
(b)
(i) Calculate the value of \(w\) when \(t = 7.84\) (3)
(ii) On the axes below, sketch a graph to show the relationship between \(w\) and \(t\).(1)
Mark scheme (a)
Answer
Mark
Mark scheme
No with reason
C1
for No with reason,
Acceptable examples No, \(4 \div 4 \neq 8 \div 6\) No, \(4 \div 4 \neq 8 \div 6\) so they don’t have the same constant No because \(y = kx\) doesn’t work No, it wont pass through (0, 0)
Not acceptable examples Yes, …. No, they don’t increase by the same number No, they don’t have the same constant
Mark scheme (b)
Answer
Mark
Mark scheme
(i) 49
M1
for stating a correct relationship, eg \(w = k \times \sqrt{t}\) or \(140 = k \times \sqrt{64}\) oe
M1
for method to find constant of proportionality, eg \(140 \div \sqrt{64}\ (= 17.5\) oe\()\)
A1
cao
(ii) sketch
C1
for sketch
Additional guidance
(i) Condone the use of ‘\(\alpha\)’ instead of ‘=’ for both M marks
16 Water flows through each of the pipes that fill a lake at the same rate.
It takes 4 of the pipes 12 hours to fill the lake.
Work out how many hours it would take 6 pipes to fill \(\dfrac{1}{4}\) of the lake. (3)
Mark scheme
Answer
Mark
Mark scheme
2
P1
for a calculation from within the list \(4 \times 12 \div 4 \div 6\) eg \(4 \times 12\ (= 48)\) or \(12 \div 4\ (= 3)\) or \(6 \div 4\ (=1.5)\) or \(4 \div 6\ (= 0.66..)\)
P1
for a complete process, eg \((\text{``}48\text{''} \div 6) \div 4\) or for \(\text{``}0.\dot{6}\text{''} \times 12 \div 4\)
