1. (3 marks) Evaluate:
(a) $\displaystyle 16^{\frac{1}{4}} + 2^{-2} - \left(\frac{16}{25}\right)^{\frac{-1}{2}}$
(b) $\sqrt{12} - \sqrt{27} + \sqrt{3}$
2. (3 marks) Find the relation between p and q if px² + qx + 1 = 0 has equal roots.
3. (3 marks) A rectangle has area (6 − √3) cm² and one side is (2 + √3) cm. Find the other side in the form a + b√c.
4. (4 marks) Solve the system of equations:
$5x = 2 - y$
$x = 13 + 4y$
5. (4 marks) Aaron stays at a hotel: $99.95/night + 8% tax on room rate + one-time $5 fee (untaxed).
(a) Write a formula for total charge for x nights.
(b) How many nights can he stay for $1520?
6. (3 marks) The toll for cars = 35 kn, trucks = 81 kn. In 2 hours, 165 vehicles crossed. Total tolls = 7845 kn. Find the number of cars and trucks.
7. (4 marks) One corner of a cube (side 8 cm) is removed by cutting through midpoints of 3 adjacent edges.
(a) Volume of piece removed?
(b) Surface area of remaining solid?
8. (3 marks) The sides of a triangle are in ratio 2:3:4. Find the cosine of the largest angle.
9. (3 marks) A golden rectangle: when a square (side = width) is removed, the remaining rectangle has the same proportions as the original. Find the exact value of φ (the ratio length/width).
10. (4 marks) Sketch these graphs on separate diagrams:
(a) $y = |1 - x|$
(b) $y = 1 - |x|$
(c) $\displaystyle y = \frac{1}{x}$
11. (4 marks) Find exactly the values of sin α and tan α given that cos α = −1/4 and α is an obtuse angle.
12. (4 marks) In how many ways can 7 students be arranged:
(a) in a row?
(b) in a circle?
(c) in a circle if the tallest girl and tallest boy must be adjacent?
13. (4 marks) Six lines are drawn, no two parallel, at most two through any point. How many triangles are formed?
14. (3 marks) Kate is 5 years older than her brother. 6 years ago, her brother was half her age. Find their current ages.
15. (4 marks) How many numbers smaller than 20000, divisible by 5, can be formed using the digits 0, 1, 3, 5, 7 and 9? (Repetition of digits is allowed.)
16. (4 marks) Find the domain and range of:
(a) $\displaystyle y = \frac{-x}{|x|}$
(b) $\displaystyle y = \frac{1}{2x - 7}$
17. (4 marks) A small business has two printers. Machine A has a 6% chance of malfunctioning on any day, Machine B has a 4% chance. Determine the probability that on any given day:
(a) both machines will work effectively
(b) at least one of the machines will malfunction
18. (3 marks) Find the perimeter of the triangle with vertices A(0, 0), B(6, 0), C(2, 8).
19. (4 marks) Anna collects 6 litres of strawberries in 27 minutes. Mika collects 7 litres in 36 minutes. How long does it take to collect 5 litres when they work together?
20. (4 marks) Solve the inequality: $(x + 1)^{2} \leq 6x^{2} + x + 1$
TOTAL: approximately 70 marks
1.
(a) 1
(b) 0
2. $q^{2} = 4p$
3. $15 - 8\sqrt{3}$
4. $x = 1, y = -3$
5.
(a) $T = 107.946x + 5$
(b) $x \approx 14 nights$
6. 120 cars, 45 trucks
7.
(a) $\displaystyle \frac{32}{3} \approx 10.67 cm^{3}$
(b) $384 - 24 + 8\sqrt{3} \approx 373.9 cm^{2}$
8. $\displaystyle \frac{-1}{4}$
9. $\displaystyle \frac{1+\sqrt{5}}{2}$
10.
(a) V-shape, vertex at (1, 0), opening upward
(b) Inverted V-shape, vertex at (0, 1), crossing x-axis at x = ±1
(c) Hyperbola in quadrants 1 and 3
11. $\displaystyle \sin \alpha = \frac{\sqrt{15}}{4}, \tan \alpha = -\sqrt{15}$
12.
(a) 5040
(b) 720
(c) 240
13. 20 triangles
14. Kate = 16, brother = 11
15. 863
16.
(a) Domain: $x \neq 0. Range: {-1, 1}$
(b) Domain: $x \neq 7/2. Range: y \neq 0$
17.
(a) 0.9024
(b) 0.0976
18. 23.2
19. 12 minutes
20. x ≤ 0 or x ≥ 1/5