Thirty thousand = 30 000, and fifty = 50.
Combine: \(30\,000 + 50 = 30\,050\).
Answer: 30 050The units digit is 6 (\(\ge 5\)), so round the tens up: the 2 tens become 3 tens and the units become 0.
Answer: 5930Measure the full length of ST, halve it, and mark the point at that distance from S (equivalently from T).
Answer: midpoint of ST marked (the centre point of the segment)The grid is a \(6\times 6 = 36\)-square array.
\[\frac{2}{9}\times 36 = \frac{72}{9} = 8 \text{ squares}\]
(a) Shade 8 squares\[\frac{2}{9} = 2 \div 9 = 0.2222\ldots = 22.22\ldots\%\]
(b) 22.2% (22.22…%)Split the interval at midnight:
\[2\text{ h }10\text{ min} + 5\text{ h }18\text{ min} = 7\text{ h }28\text{ min}\]
Answer: 7 h 28 min\(55 = 5\times 11\) and \(121 = 11\times 11\); the others are not multiples of 11.
(a) 55 and 121Alternately subtract and add the digits 9, 1, 8, 2, 7, 1, 9, 3, 7:
\[9 - 1 + 8 - 2 + 7 - 1 + 9 - 3 + 7 = 33\]
Since \(33 = 3\times 11\) is a multiple of 11, the number is a multiple of 11.
(b) 33 = 3 × 11 ⇒ multiple of 11For the seven known numbers: maximum = 42, minimum = 16. Let the eighth number be \(x\).
Case 1 — \(x\) is the new maximum: \(\;x - 16 = 31 \Rightarrow x = 47\) (47 > 42 ✓)
Case 2 — \(x\) is the new minimum: \(\;42 - x = 31 \Rightarrow x = 11\) (11 < 16 ✓)
Answer: 11 or 47\[\sqrt{5.76} = 2.4\]
\[2.8^{3} = 2.8\times 2.8\times 2.8 = 21.952\]
\[2.4 + 21.952 = 24.352\]
Answer: 24.352Collect like terms: \(4m - m = 3m\) and \(7k + 3k = 10k\).
Answer: 3m + 10kTwo large negatives give a large positive: \((-9)\times(-7) = 63\). Compare \(6\times 8 = 48\).
(a) 63Most negative result: largest negative × two largest positives: \((-9)\times 8\times 6 = -432\). (e.g. \((-9)(-7)(-3) = -189\) is larger.)
(b) −4322 | 8 3 | 4 | 5 6 9 5 | 0 4 8 6 | 4 5 8 7 | 0 1 2 7
Ordered: 28, 45, 46, 49, 50, 54, 58, 64, 65, 68, 70, 71, 72, 77.
With 14 values the median is the mean of the 7th and 8th: \[\frac{58 + 64}{2} = \frac{122}{2} = 61\]
(b) 61\[2(lw + lh + wh) = 2(10\times4 + 10\times5 + 4\times5) = 2(40+50+20) = 220\ \text{cm}^2\]
(a) 220 cm²\[10\times 4\times 5 = 200\ \text{cm}^3\]
(b) 200 cm³\[\frac{3}{20}\times 360^\circ = \frac{1080}{20} = 54^\circ\]
(a) 54°\[P(\text{not blue}) = \frac{20-3}{20} = \frac{17}{20}\]
(b) 17/20Common factor \(x\): \(\;3x^{3} - 7xy = x(3x^{2} - 7y)\).
Answer: x(3x² − 7y)\[\vec{AB} = \binom{x_B - x_A}{y_B - y_A} = \binom{-3-7}{4-1} = \binom{-10}{3}\]
Answer: (−10, 3)ᵀRupees → dollars: \[17850 \times 0.013 = 232.05 \text{ dollars}\]
Dollars → euros: \[x = \frac{232.05}{1.05} = 221\]
Answer: 221Set \(y = 3\): \(\;3 = 2x - 5 \Rightarrow 2x = 8 \Rightarrow x = 4\).
Answer: P = (4, 3)A (upper-left) maps onto B (lower-middle) by a quarter turn about the origin.
(a) Rotation, 90° anticlockwise, centre (0, 0)Under reflection in \(x = -1\), each point \((x, y)\mapsto(-2 - x,\ y)\). Apply to every vertex of A and draw the mirror image on the right of the line.
(b) Image of A reflected in x = −1 drawn correctly\[16 \times \pi \times 7^{2} = \pi R^{2}\]
\[R^{2} = 16\times 49 = 784 \Rightarrow R = \sqrt{784} = 28\]
Answer: R = 28\(n=1: 1-3=-2;\quad n=2: 4-3=1;\quad n=3: 9-3=6\).
(a) −2, 1, 6Common difference 7, so the term contains \(7n\): values \(7,14,21,28,35\), each 5 more than the sequence.
(b) 7n − 5Half of 0.1 m is 0.05 m: \(\;18.7 - 0.05 = 18.65\) and \(18.7 + 0.05 = 18.75\).
Answer: 18.65 ≤ l < 18.75\[n = \frac{5.46\times 10^{23}}{6.5\times 10^{19}} = \frac{5.46}{6.5}\times 10^{4} = 0.84\times 10^{4} = 8.4\times 10^{3}\]
Answer: 8.4 × 10³\[\frac{h}{97.5} = \frac{118.9}{159.9}\]
\[h = 97.5\times\frac{118.9}{159.9} = 97.5\times 0.74359\ldots = 72.5\]
Answer: h = 72.5\[1\tfrac{1}{4} = \frac{5}{4}\]
Common denominator 12: \(\;\dfrac{5}{4} = \dfrac{15}{12},\quad \dfrac{5}{6} = \dfrac{10}{12}\).
\[\frac{15}{12} - \frac{10}{12} = \frac{5}{12}\]
Answer: 5/12Write the numbers as \(6a\) and \(6b\) with \(a,b\) coprime. Then \(\text{LCM} = 6ab = 90 \Rightarrow ab = 15\).
Coprime pairs of 15: \((1,15)\) → 6 and 90 (rejected, 6 is not > 6); \((3,5)\) → 18 and 30.
Check: HCF(18, 30) = 6 ✓, LCM(18, 30) = 90 ✓.
Answer: 18 and 30\(26 = 1\times 26 = 2\times 13\).
Answer: 1, 2, 13, 26(a) AB joins two points on the circle without passing through the centre → it is a chord.
(b) A radius is a straight line from centre O to any point on the circle.
(a) Chord · (b) Radius drawn(a) \(0.25 = \dfrac{25}{100} = \dfrac{1}{4}\)
(b) \(\dfrac{1}{2} = 50\%\)
(c) \(7\% = \dfrac{7}{100} = 0.07\)
(a) 1/4 · (b) 50% · (c) 0.07(a) Cylinder. (b) Equilateral triangle.
(a) Cylinder · (b) Equilateral(a) Least likely = smallest count = blue.
(b) \(P(\text{white}) = \dfrac{2}{5}\).
(c) \(P(\text{not red}) = \dfrac{5-2}{5} = \dfrac{3}{5}\).
(a) Blue · (b) 2/5 · (c) 3/5(a) Reciprocal of 8 = \(\dfrac{1}{8} = 0.125\).
(b) \(19^{3} = 361\times 19 = 6859\).
(a) 1/8 · (b) 68591. The bars are not of equal width.
2. The vertical scale is not linear (uneven spacing of values).
Bar widths unequal; scale not linear\[6.5\times 60\times 60 = 6.5\times 3600 = 23\,400 \text{ seconds}\]
Answer: 23 400 secondsSplit into two rectangles (bottom 23 × 5, upper 8 × 7):
\[23\times 5 + 8\times 7 = 115 + 56 = 171\ \text{cm}^2\]
(Equivalently \(12\times 8 + 5\times 15 = 96 + 75 = 171\).)
Answer: 171 cm²Mode = value occurring most = 7 (twice). Range = \(21 - 3 = 18\).
(a) Mode 7 · Range 18Total after 5 = \(5\times 28 = 140\); total after 6 = \(6\times 26 = 156\). 6th score = \(156 - 140 = 16\).
(b) 16\[\frac{19}{100}\times 46.25 = 0.19\times 46.25 = 8.7875 \approx 8.79\]
Answer: $8.79\((5b + 2b) + (-8c - 3c) = 7b - 11c\).
(a) 7b − 11c\(37 = 3(4) + 5t = 12 + 5t \Rightarrow 5t = 25 \Rightarrow t = 5\).
(b) t = 5Total parts \(= 8\); one part \(= 136 \div 8 = 17\). So \(3\times 17 = 51\) and \(5\times 17 = 85\).
Answer: $51 and $85Capacity of 14 boxes: \(14\times 68 = 952\) pens. Since \(952 < 980\), not enough. (\(980\div 68 = 14.4\ldots\), so 15 boxes needed; 28 pens remain.)
Answer: No — 14 boxes hold only 952 pens (needs 15 boxes)\[68 + x + 43 + 135 = 360 \Rightarrow x = 360 - 246 = 114\]
Answer: x = 114The height meets the base at its midpoint → right triangle with legs 5 and \(\tfrac{4}{2}=2\):
\[x = \sqrt{5^{2} + 2^{2}} = \sqrt{29} = 5.385\ldots \approx 5.39\]
(a) 5.39 (√29)Draw a 4 × 4 square (base) with one identical isosceles triangle (height 5, base 4) on each side.
(b) 4×4 square + 4 identical isosceles trianglesFirst four s.f. are 4, 6, 1, 7; next digit 9 (\(\ge 5\)) rounds the 7 up to 8.
Answer: 46.18\(8.4^{2} = 70.56;\; 70.56 + 9.3 = 79.86;\; \dfrac{79.86}{26.5} = 3.0135\ldots\)
\[\sqrt{3.0135\ldots} = 1.7359\ldots \approx 1.74\]
(a) 1.74\[4^{-3} = \frac{1}{4^{3}} = \frac{1}{64} = 0.015625\]
(b) 1/64 (0.015625)\(3(5x+2) = 15x + 6;\quad -4(x+1) = -4x - 4\).
\[15x + 6 - 4x - 4 = 11x + 2\]
Answer: 11x + 2No solution required — full marks (2) awarded automatically.
Removed — 2 marks awarded(a) Rotation, 90° clockwise, centre (0, 0).
(b) Reflection mapping points to (−4, 2), (−2, −2), (−3, −3), (−1, −3).
(a) Rotation 90° cw, centre (0,0) · (b) points as above| x | −4 | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| y | 12 | 5 | 0 | −3 | −4 | −3 | 0 | 5 | 12 |
Calculations: \((-3)^2-4=5,\; 0^2-4=-4,\; 3^2-4=5\).
Profit = \(4.64 - 3.20 = 1.44\). \[\frac{1.44}{3.20}\times 100 = 45\%\]
(a) 45%\[900 = \frac{1}{3}\pi r^{2}(8) \Rightarrow r^{2} = \frac{900\times 3}{8\pi} = 107.43\ldots\]
\[r = \sqrt{107.43\ldots} = 10.36\ldots \approx 10.4\ \text{cm}\]
(b) 10.4 cm\(m - 32 = 9p \Rightarrow p = \dfrac{m - 32}{9}\).
Answer: p = (m − 32)/9\(20 = 2^{2}\times 5,\quad 36 = 2^{2}\times 3^{2}\). Take the highest power of each prime:
\[\text{LCM} = 2^{2}\times 3^{2}\times 5 = 4\times 9\times 5 = 180\]
Answer: 180From the graph the \(y\)-intercept is 9, so \(c = 9\). The line also passes through \((-3, 0)\):
\[m = \frac{9 - 0}{0 - (-3)} = \frac{9}{3} = 3\]
Answer: y = 3x + 9No solution required — full marks (4) awarded automatically.
Removed — 4 marks awardedUsing \(\text{speed} = \dfrac{\text{distance}}{\text{time}}\) with distance 5.4 and time 36 min:
\[\frac{5.4}{36}\times 60 = 9\]
Answer: 9 (km/h)(a) Similar triangles: \(\dfrac{20.8}{x} = \dfrac{9.1}{1.4} \Rightarrow x = 3.2\).
(b) \(\cos 38° = \dfrac{y}{5.4} \Rightarrow y = 5.4\cos 38° = 4.255\ldots \approx 4.26\).
(a) 3.2 · (b) 4.26| Q | My answer | Mark scheme | Match |
|---|---|---|---|
| 1 | 30 050 | 30 050 | ✓ |
| 2 | 5930 | 5930 | ✓ |
| 3 | Midpoint of ST marked | Midpoint of ST marked | ✓ |
| 4(a) | 8 squares shaded | 8 squares shaded | ✓ |
| 4(b) | 22.2% | 22.2 / 22.22… | ✓ |
| 5 | 7 h 28 min | 7h 28min | ✓ |
| 6(a) | 55 and 121 | 55 121 | ✓ |
| 6(b) | 33 = 3×11 | 33 = 3×11 | ✓ |
| 7 | 11 or 47 | 11, 47 | ✓ |
| 8 | 24.352 | 24.352 | ✓ |
| 9 | 3m + 10k | 3m + 10k | ✓ |
| 10(a) | 63 | 63 | ✓ |
| 10(b) | −432 | −432 | ✓ |
| 11(a) | Stem-leaf as shown | same | ✓ |
| 11(b) | 61 | 61 | ✓ |
| 12(a) | 220 cm² | 220 | ✓ |
| 12(b) | 200 cm³ | 200 | ✓ |
| 13(a) | 54° | 54 | ✓ |
| 13(b) | 17/20 | 17/20 | ✓ |
| 14 | x(3x² − 7y) | x(3x² − 7y) | ✓ |
| 15 | (−10, 3)ᵀ | column (−10, 3) | ✓ |
| 16 | 221 | 221 | ✓ |
| 17 | (4, 3) | (4, 3) | ✓ |
| 18(a) | Rotation 90° acw, (0,0) | same | ✓ |
| 18(b) | Reflection in x = −1 | Shape drawn correctly | ✓ |
| 19 | 28 | 28 | ✓ |
| 20(a) | −2, 1, 6 | −2, 1, 6 | ✓ |
| 20(b) | 7n − 5 | 7n − 5 | ✓ |
| 21 | 18.65 ≤ l < 18.75 | 18.65 … 18.75 | ✓ |
| 22 | 8.4 × 10³ | 8.4 × 10³ | ✓ |
| 23 | 72.5 | 72.5 | ✓ |
| 24 | 5/12 | 5/12 | ✓ |
| 25 | 18 and 30 | 18 30 | ✓ |
Paper 1 result: 25/25 questions match.
| Q | My answer | Mark scheme | Match |
|---|---|---|---|
| 1 | 1, 2, 13, 26 | 1 2 13 26 | ✓ |
| 2(a) | Chord | Chord | ✓ |
| 2(b) | Radius drawn | Radius drawn | ✓ |
| 3(a) | 1/4 | 1/4 | ✓ |
| 3(b) | 50% | 50 | ✓ |
| 3(c) | 0.07 | 0.07 | ✓ |
| 4(a) | Cylinder | Cylinder | ✓ |
| 4(b) | Equilateral | Equilateral | ✓ |
| 5(a) | Blue | Blue | ✓ |
| 5(b) | 2/5 | 2/5 | ✓ |
| 5(c) | 3/5 | 3/5 | ✓ |
| 6(a) | 1/8 | 1/8 / 0.125 | ✓ |
| 6(b) | 6859 | 6859 | ✓ |
| 7 | Unequal widths; non-linear scale | same | ✓ |
| 8 | 23 400 | 23 400 | ✓ |
| 9 | 171 cm² | 171 | ✓ |
| 10(a)(i) | 7 | 7 | ✓ |
| 10(a)(ii) | 18 | 18 | ✓ |
| 10(b) | 16 | 16 | ✓ |
| 11 | $8.79 | 8.79 | ✓ |
| 12(a) | 7b − 11c | 7b − 11c | ✓ |
| 12(b) | 5 | 5 | ✓ |
| 13 | $51 and $85 | 51 85 | ✓ |
| 14 | No — 952 < 980 | No, correct statement | ✓ |
| 15 | 114 | 114 | ✓ |
| 16(a) | 5.39 | 5.39 / 5.385… | ✓ |
| 16(b) | 4×4 square + 4 triangles | Correct net | ✓ |
| 17 | 46.18 | 46.18 | ✓ |
| 18(a) | 1.74 | 1.74 | ✓ |
| 18(b) | 1/64 | 0.015625 / 1/64 | ✓ |
| 19 | 11x + 2 | 11x + 2 | ✓ |
| 20 | Removed (2 marks) | Removed — award 2 | ✓ |
| 21(a) | Rotation 90° cw, (0,0) | same | ✓ |
| 21(b) | (−4,2)(−2,−2)(−3,−3)(−1,−3) | same | ✓ |
| 22(a) | 5, −4, 5 | 5 −4 5 | ✓ |
| 22(b) | Parabola vertex (0,−4) | Correct curve | ✓ |
| 23(a) | 45% | 45 | ✓ |
| 23(b) | 10.4 cm | 10.4 / 10.36… | ✓ |
| 24 | (m − 32)/9 | (m − 32)/9 | ✓ |
| 25 | 180 | 180 | ✓ |
| 26 | y = 3x + 9 | y = 3x + 9 | ✓ |
| 27 | Removed (4 marks) | Removed — award 4 | ✓ |
| 28 | 9 | 9 | ✓ |
| 29(a) | 3.2 | 3.2 | ✓ |
| 29(b) | 4.26 | 4.26 / 4.255… | ✓ |
Paper 3 result: all reproduced questions match.