1. (4 marks) Simplify:
(a) $a^{7} \times a^{4}$
(b) $w^{15} \div w^{3}$
(c) $(8x^{5}y^{3})^{2}$
(d) Make t the subject of: $c = t^{3} - 8v$
2. (3 marks) Danil, Gabriel and Hadley share money in ratio 3:5:9. The difference between Gabriel's share and Hadley's share is €196. Find Danil's share.
3. (3 marks) Triangle ABC: angle B = 90°, BC = 8.4 cm, angle C = 65°. Find AB to 3 significant figures. (tan 65° = 2.145)
4. (4 marks) Sarah makes 150 mugs. 2/5 are small (£8.50), 32% medium (£11.20), rest large (£14.20). Total cost = £1140. She sells all 150. Find percentage profit to nearest whole number.
5. (4 marks) Jenny has six cards with whole numbers in order. Smallest = 5, largest = 24, median = 14, mode = 8. A basketball team plays 6 games. Mean after 5 games = 21. Mean after 6 games = 23.
(a) Find possible values for the four unknown cards.
(b) Find points scored in game 6.
6. (4 marks)
(a) Solve the inequality: $5x - 7 \leq 2$
(b) Factorise: $y^{2} - 2y - 35$
(c) Hence solve: $y^{2} - 2y - 35 = 0$
7. (3 marks) ε = {4,5,...,15}. A∩B = {5,10,15}, B' = {7,8,9,11,12,13,14}, A' = {4,6,7,8,14}. Draw and complete the Venn diagram.
8. (2 marks) Work out (4.2 × 10⁻²⁴) × (3 × 10¹⁴⁵). Give answer in standard form.
9. (3 marks) Isosceles triangle: AB = AC = 17.5 cm, BC = 28 cm. Find the area.
10. (3 marks) Line L: $2y + 7x = 10.$
(a) Find the gradient of L.
(b) Find where L crosses the y-axis.
11. (4 marks) Himari invests 200000 yen for 3 years. Year 1 rate: 1.8%. Years 2-3 rate: x%. End value: 209754 yen. Find x to 1 d.p.
12. (4 marks) The table gives information about the times, in minutes, taken by 80 customers to do their shopping.
| Time (t minutes) | Frequency |
|---|---|
| $0 < t \leq 10$ | 7 |
| $10 < t \leq 20$ | 26 |
| $20 < t \leq 30$ | 24 |
| $30 < t \leq 40$ | 14 |
| $40 < t \leq 50$ | 7 |
| $50 < t \leq 60$ | 2 |
(a) Complete cumulative frequency table.
(b) Estimate median.
(c) Estimate P(time > 42 minutes).
13. (4 marks)
(a) Expand and simplify: $5x(x + 2)(3x - 4)$
(b) Simplify: $\displaystyle (\frac{16w^{8}}{y^{20}})^{\frac{-3}{4}}$
14. (4 marks) Tree diagram: Packet A (12 seeds, 7 sunflower), Packet B (15 seeds, 8 sunflower). One from each.
(a) Complete the tree diagram with probabilities.
(b) Find P(both sunflower).
15. (2 marks) A is inversely proportional to C². A = 40 when C = 1.5. Find C when A = 1000.
16. (3 marks) Circle, centre O. Arc ABC = 5 cm, angle AOC = 55°. Find the area of the circle (1 d.p.).
17. (3 marks) Similar vases A (height 10 cm) and B (height 15 cm). Volume difference = 1197 cm³. Find volume of A.
18. (4 marks) Circle, centre O. DOB is diameter. A, B, C, D on circle. Angle ACD = 43°. Find angle ADB. Give reasons.
19. (4 marks) Solve simultaneously: 3x² + y² − xy = 5 and y = 2x − 3.
20. (4 marks) Express 7 + 12x − 3x² in the form a + b(x + c)². Hence find the maximum point of y = 7 + 12x − 3x².
21. (5 marks) Solid: cylinder (radius x, height 3x) + hemisphere (radius x). Total SA = 81π cm². Mass = 840g. Determine the metal. (Densities: Al 2.7, Ni 8.9, Au 19.3, Ag 10.5 g/cm³)
22. (3 marks) Trapezium ABCD: AB = 6, AD = 7, DC = 11 (angles A and D = 90°).
(a) Area of trapezium. (b) Length of BC.
23. (3 marks) Arithmetic series: first term 1, common difference 4. Find sum of terms from 41st to 100th inclusive.
TOTAL: approximately 75 marks
1.
(a) $a^{11}$
(b) $w^{12}$
(c) $64x^{10}y^{6}$
(d) $\displaystyle t = \sqrt[3]{c+8v}$
2. €147
3. 18.0 cm
4. 44%
5.
(a) e.g. 5,8,8,20,22,24
(b) 33 points
6.
(a) $\displaystyle x \leq \frac{9}{5}$
(b) $(y+5)(y-7)$
(c) y = −5 or y = 7
7. A only: {9,11,12,13}; B only: {4,6}; A∩B: {5,10,15}; outside: {7,8,14}
8. $1.26 \times 10^{122}$
9. $147 cm^{2}$
10.
(a) $\displaystyle \frac{-7}{2}$
(b) (0, 5)
11. $x = 1.5$
12.
(a) 7,33,57,71,78,80
(b) $\approx 23 min$
(c) $\approx 0.095$
13.
(a) $15x^{3}+10x^{2}-40x$
(b) $\displaystyle \frac{y^{15}}{8w^{6}}$
14.
(b) $\displaystyle \frac{14}{45}$
15. $C = 0.3$
16. $\displaystyle r = 5\times \frac{360}{55\times 2\pi } \approx 5.21. Area \approx 85.2 cm^{2}$
17. $V_A = 504 cm^{3}$
18. Angle DAB = 90° (semicircle), angle ABD = 43° (same segment). ADB = 47°
19. x=1, y=−1 or x=4/5, y=−7/5
20. $19 - 3(x-2)^{2}. Max at (2, 19)$
21. Aluminium
22.
(a) $59.5 cm^{2}$
(b) $\sqrt{74} \approx 8.60 cm$
23. 16740