(a) Work out the height, \(h\) cm, of the smaller jar. Give your answer correct to 1 decimal place. (4)
The surface area of the smaller jar is \(A\) cm2 The surface area of the larger jar is \(pA\) cm2
(b) Find the exact value of \(p\). (1)
Mark scheme (a)
Answer
Mark
Mark scheme
4.9
P1
for beginning to work with scale factors eg \(\left(\dfrac{h}{12}\right)^3\) or \(\left(\dfrac{12}{h}\right)^3\) or \(\dfrac{90}{1350}\ \left(= \dfrac{1}{15}\right)\) or \(\dfrac{1350}{90}\ (= 15)\) or 90 : 1350 oe
P1
for equating scale factors eg \(\left(\dfrac{h}{12}\right)^3 = \dfrac{90}{1350}\) or \(\left(\dfrac{12}{h}\right)^3 = \text{``}15\text{''}\) or for finding the linear scale factor, eg \(\sqrt[3]{\dfrac{90}{1350}}\) or \(\sqrt[3]{\text{``}15\text{''}}\ (= 2.46(6\ldots))\)
P1
for a complete method eg \(\sqrt[3]{12^3 \times \dfrac{90}{1350}}\) or \(12 \times \sqrt[3]{\dfrac{90}{1350}}\) or \(\dfrac{12}{\text{``}\left(\sqrt[3]{\text{``}15\text{''}}\right)\text{''}}\)
A1
for answers in the range 4.8 to 4.9
Mark scheme (b)
Answer
Mark
Mark scheme
\((15)^{\frac{2}{3}}\)
B1
for \(15^{\frac{2}{3}}\) or \(\left(\sqrt[3]{15}\right)^2\) or \(\sqrt[3]{15^2}\) or \(225^{\frac{1}{3}}\)
Additional guidance
If given in an acceptable form, then given as a decimal, isw and award
\(AEC\) and \(ADB\) are straight lines. \(ED\) is parallel to \(CB\).
(a) Prove that triangle \(ABC\) is similar to triangle \(ADE\). (2)
\(ED = 20\) cm \(CB = 30\) cm \(AC = 18\) cm
(b) Work out the length of \(EC\). (3)
Mark scheme (a)
Answer
Mark
Mark scheme
Proof
M1
for identifying angle \(A\) as common or for angle \(AED\) = angle \(ACB\) or angle \(ADE\) = angle \(ABC\) with appropriate reason(s) eg corresponding angles are equal or co-interior angles add up to 180 and angles on a straight line add up to 180
C1
for completing the proof by identifying a second pair of equal angles with appropriate reason(s) and a statement that the angles in each triangle are the same
Additional guidance
Statement can be implied by identifying a third pair of equal angles (no reason needed)
Mark scheme (b)
Answer
Mark
Mark scheme
6
P1
for a scale factor of 1.5 or \(\dfrac{2}{3}\) oe or for \(\dfrac{AE}{20} = \dfrac{18}{30}\) oe
P1
for a process to find the length of \(AE\), eg \(18 \div 1.5\ (= 12)\) oe or \((AE =)\ \dfrac{20 \times 18}{30}\) oe or for a complete process to find the length of \(EC\), eg \(18 \times \left(1 - \dfrac{2}{3}\right)\) oe
13 \(ABC\) and \(AED\) are straight lines. \(BE\) and \(CD\) are parallel.
\(BE = 4.2\) cm \(CD = 6.3\) cm \(AC = 10.8\) cm
Work out the area of trapezium \(BCDE\). (3)
Mark scheme
Answer
Mark
Mark scheme
18.9
P1
for using length scale factor to find \(AB\), eg \((AB =)\ 10.8 \div \left(\dfrac{3}{2}\right)\) or \((AB =)\ 10.8 \times \left(\dfrac{2}{3}\right)\ (= 7.2)\) or for using length scale factor to find \(BC\), eg \((BC =)\ 10.8 \div \dfrac{6.3}{6.3 - 4.2}\) or \((BC =)\ 10.8 \times \dfrac{6.3 - 4.2}{6.3}\ (= 3.6)\) or finds area scale factor, eg \(\left(\dfrac{3}{2}\right)^2\) or \(\left(\dfrac{2}{3}\right)^2\)
P1
for a complete process to find the area of trapezium, eg \(\dfrac{6.3 + 4.2}{2} \times (10.8 - \text{``}7.2\text{''})\) or \(\dfrac{(6.3 - 4.2) \times \text{``}3.6\text{''}}{2} + \text{``}3.6\text{''} \times 4.2\) or \(\dfrac{10.8 \times 6.3}{2} - \dfrac{7.2 \times 4.2}{2}\) or \(\dfrac{10.8 \times 6.3}{2} - \dfrac{10.8 \times 6.3}{2} \div \left(\text{``}\dfrac{3}{2}\text{''}\right)^2\) or \(\dfrac{10.8 \times 6.3}{2} - \dfrac{10.8 \times 6.3}{2} \times \left(\text{``}\dfrac{2}{3}\text{''}\right)^2\)
A1
accept trailing zeros eg 18.90
Additional guidance
Can use a combination of skills but must have a complete process to find \(AB\) or \(BC\) to score this mark
Sphere A has a volume of 64 cm3 Sphere B has a volume of 125 cm3
The radius of sphere C is 50% of the radius of sphere B.
Work out the ratio of the surface area of sphere A to the surface area of sphere C. Give your answer in the form \(a : b\) where \(a\) and \(b\) are integers. (4)
Mark scheme
Answer
Mark
Mark scheme
64 : 25
P1
for start of process to find ratio of lengths of A to B, eg \(\sqrt[3]{64}\ (= 4)\) or \(\sqrt[3]{125}\ (= 5)\) or 4 : 5
P1
for \(\sqrt[3]{125} \div 2\ (= 2.5)\) oe or \(\left(\text{``}\sqrt[3]{64}\text{''}\right)^2\ (= 16)\)
P1
for process to find ratio of areas of A to C, eg \(\text{``}4\text{''}^2 : \text{``}2.5\text{''}^2\ (= 16 : 6.25)\)
A1
for 64 : 25 oe in form \(a : b\) where \(a\) and \(b\) are integers
\(15 \div 6\) or 2.5 or \(6 \div 3\) or 2 or \(6 \div 15\) or 0.4 or \(3 \div 6\) or 0.5 or \(\dfrac{x}{15} = \dfrac{3}{6}\) or \(\dfrac{x}{15} = 0.5\)
M1
oe may be seen on the diagram
7.5 or \(7\dfrac{1}{2}\) or \(\dfrac{15}{2}\)
A1
Additional guidance
Accept correct answer unambiguously placed as \(x\) on the diagram provided it is not contradicted in the working space.
\(6 \div 3\) or 2 or \(9 \div 2\) or \(3 \div 6\) or 0.5 or \(9 \times 0.5\) or \(9 \div 6\) or 1.5 or \(3 \times 1.5\) or \(6 \div 9\) or \(\dfrac{2}{3}\) or \(3 \div \dfrac{2}{3}\)
M1
oe
4.5
A1
oe
Additional guidance
Accept embedded answer \(4.5 \times 2 = 9\)
M1A1
Ignore further working in attempt to round after answer 4.5 eg \(9 \div 2 = 4.5\) with answer 5
