Total marks: 45 · Duration: 90 minutes
Instructions to candidates
- Answer ALL questions.
- Each question is worth 1 mark.
- A 2 × 2 matrix is written row by row with a semicolon between the rows: (2 3 ; 1 4) has first row 2, 3 and second row 1, 4.
- Write your answers on the Multiple Choice Answer Sheet provided.
- You may use a silent, non-programmable scientific calculator.
- If you need to change an answer, erase your first answer completely.
- Work as quickly and as carefully as you can.
- The use of electronic devices other than silent, non-programmable calculators is prohibited.
Paper
Section A — Multiple Choice (45 marks)
1. If f(x) = 3x² - 5x + 2, what is the value of f(-2)?
A) 0
B) 12
C) 24
D) 32
2. The gradient of the line perpendicular to the line 2y + 3x = 6 is
A) 2/3
B) 3/2
C) -3/2
D) -2/3
3. Given that log₂ 8 = 3 and log₂ 4 = 2, the value of log₂ 32 is
A) 5
B) 6
C) 12
D) 8
4. The roots of the equation x² - 7x + 12 = 0 are
A) x = 1 and x = 12
B) x = -3 and x = -4
C) x = 3 and x = 4
D) x = 2 and x = 6
5. If vectors a = 3i + 4j and b = -2i + 5j, then a + b equals
A) 5i + 9j
B) i - j
C) -i + j
D) i + 9j
6. The function y = x³ - 3x² + 2 has a stationary point when
A) x = 0 only
B) x = 0 and x = 2
C) x = 2 only
D) x = -2 and x = 2
7. The curve y = 2x² + 5 is translated 3 units to the right. The equation of the new curve is
A) y = 2(x + 3)² + 5
B) y = 2x² + 2
C) y = 2x² + 8
D) y = 2(x - 3)² + 5
8. A cone has base radius 5 cm and height 12 cm. The volume of the cone, in cm³, is
A) 100π
B) 200π
C) 300π
D) 600π
9. The remainder when the polynomial P(x) = 2x³ + 5x² - 4x + 1 is divided by (x - 1) is
A) 8
B) 14
C) 4
D) 1
10. A geometric progression has first term 8 and common ratio 1/2. The sum to infinity of this progression is
A) 8
B) 24
C) 16
D) 12
11. If sin θ = 3/5 and θ is acute, then cos θ equals
A) 3/4
B) 5/4
C) 4/3
D) 4/5
12. The solution set for the inequality 2x - 5 < 3x + 1 is
A) x > -6
B) x < 6
C) x < -6
D) x > 6
13. A particle moves in a straight line such that its displacement, s metres, from a fixed point O at time t seconds is given by s = 2t³ - 9t² + 12t. The velocity of the particle when t = 2 is
A) 8 m/s
B) 0 m/s
C) 12 m/s
D) 4 m/s
14. The matrix M = (2 3 ; 1 4) has determinant equal to
A) 11
B) 14
C) 8
D) 5
15. If 3^(2x+1) = 27, then x equals
A) 1
B) 2
C) 3/2
D) 1/2
16. The angle between the vectors p = 2i + j and q = i + 2j can be found using the formula
A) tan θ = (p · q) / (|p||q|)
B) cos θ = |p| / |q|
C) cos θ = (p · q) / (|p||q|)
D) sin θ = (p · q) / (|p||q|)
17. A company in Kingston, Jamaica, models its profit P (in thousands of dollars) over t years by the function P(t) = -2t² + 16t + 10. The maximum profit occurs after
A) 10 years
B) 4 years
C) 8 years
D) 2 years
18. The expression (x + 2)² - (x - 3)² simplifies to
A) 5x + 10
B) 4x - 5
C) 10x - 5
D) 10x + 5
19. Given that y = ln(2x + 1), then dy/dx equals
A) 1/(2x + 1)
B) 2/(2x + 1)
C) ln 2
D) 2x + 1
20. The roots of the quadratic equation ax² + bx + c = 0 are equal when
A) b² < 4ac
B) b² > 4ac
C) b² + 4ac = 0
D) b² = 4ac
21. ∫(6x² - 4x + 3) dx equals
A) 2x³ - 2x² + 3x + c
B) 6x³ - 4x² + 3x + c
C) 12x - 4 + c
D) 2x³ - 4x² + 3 + c
22. The nth term of the arithmetic sequence 5, 9, 13, 17, ... is
A) 5n + 4
B) 4n - 1
C) 4n + 5
D) 4n + 1
23. If matrix A = (3 −1 ; 2 4) and B = (2 0 ; 1 3), then A + B equals
A) (5 −1 ; 3 7)
B) (5 0 ; 1 7)
C) (6 −1 ; 2 7)
D) (5 1 ; 3 7)
24. A water tank in Bridgetown, Barbados, is being filled at a rate given by dV/dt = 3t² + 2t, where V is volume in litres and t is time in minutes. The rate of filling when t = 3 minutes is
A) 27 litres/minute
B) 51 litres/minute
C) 33 litres/minute
D) 35 litres/minute
25. The equation of the tangent to the curve y = x² + 3x - 2 at the point where x = 1 is
A) y = 5x - 7
B) y = 5x + 7
C) y = 5x + 2
D) y = 5x - 3
26. If P(x) = x³ - 6x² + 11x - 6, then (x - 1) is a factor because
A) P(1) = 0
B) P(0) = 1
C) P(6) = 0
D) P(-1) = 0
27. The general solution to the equation sin 2x = 1/2 for 0° ≤ x ≤ 360° includes
A) x = 15° and x = 75°
B) x = 15° and x = 165°
C) x = 30° and x = 330°
D) x = 30° and x = 150°
28. The area bounded by the curve y = x², the x-axis, and the lines x = 0 and x = 2 is
A) 4/3 square units
B) 2/3 square units
C) 8/3 square units
D) 16/3 square units
29. A force F = 4i - 3j N acts on a particle. The magnitude of F is
A) 1 N
B) 5 N
C) 7 N
D) 25 N
30. If log₁₀ x = 2.5, then x equals
A) 100√10
B) 25
C) 1000
D) 250
31. The coefficient of x² in the expansion of (2 + x)⁴ is
A) 12
B) 6
C) 24
D) 32
32. The inverse of the function f(x) = (x - 3)/2 is
A) f⁻¹(x) = 2(x + 3)
B) f⁻¹(x) = 2x + 3
C) f⁻¹(x) = (x + 3)/2
D) f⁻¹(x) = 2x - 3
33. A particle has velocity v = 4t - 3 m/s. If the particle is at the origin when t = 0, its displacement when t = 4 seconds is
A) 20 m
B) 24 m
C) 13 m
D) 32 m
34. The solution to the equation 2^(x+1) = 8^x is
A) x = 1/5
B) x = 1/2
C) x = 2/3
D) x = 1/3
35. If cos x = -√3/2 and 90° < x < 180°, then x equals
A) 150°
B) 120°
C) 135°
D) 210°
36. The turning point of the curve y = 3(x - 2)² + 5 is
A) (2, 5)
B) (-2, 5)
C) (2, -5)
D) (-2, -5)
37. The gradient function of y = (2x - 1)³ is
A) (2x - 1)²
B) 2(2x - 1)²
C) 6(2x - 1)²
D) 3(2x - 1)²
38. In the triangle ABC, if a = 7 cm, b = 5 cm and angle C = 60°, then c² equals
A) 74
B) 39
C) 109
D) 12
39. A farmer in St. Vincent plants crops whose yield Y (in tonnes per hectare) after x weeks is modeled by Y = -0.5x² + 6x + 2. The maximum yield occurs after
A) 3 weeks
B) 12 weeks
C) 8 weeks
D) 6 weeks
40. If dy/dx = 6x² - 4 and y = 5 when x = 1, then y equals
A) 2x³ - 4x + 5
B) 6x² - 4x + 3
C) 2x³ - 4x + 7
D) 2x³ - 4x + 3
41. The sum of the first 10 terms of the arithmetic series 3 + 7 + 11 + 15 + ... is
A) 240
B) 390
C) 120
D) 210
42. The range of the function f(x) = x² + 1 for x ∈ ℝ is
A) y > 0
B) y ≥ 1
C) y ≥ 0
D) y > 1
43. The matrix M = (2 1 ; 0 3) maps the point (3, 4) onto the point
A) (10, 7)
B) (6, 12)
C) (7, 10)
D) (10, 12)
44. The equation 2 cos²x - cos x - 1 = 0 can be solved by first letting
A) y = sin x and solving 2y² - y - 1 = 0
B) y = cos x and solving 2y² - y - 1 = 0
C) y = 2 cos x and solving y² - y - 1 = 0
D) y = cos 2x and solving 2y² - y - 1 = 0
45. A particle moves along a straight line such that its acceleration a m/s² at time t seconds is given by a = 6t - 4. If the particle starts from rest, its velocity after 2 seconds is
A) 12 m/s
B) 16 m/s
C) 8 m/s
D) 4 m/s