MAT1501 Supp 2022 exam paper — questions

Free sample

  1. Question 1(a) · Sets, intervals and number lines · 4 marks

    Consider four sets given by A = {-1, 1, 3, 4, 7, 10}, B = {0, 2, 3, 4}, C = (-3, 9) (an open interval), and D = (0, 7] (a half-open interval). For the sets A = {-1, 1, 3, 4, 7, 10}, B = {0, 2, 3, 4}, C = (-3, 9), and D = (0, 7], draw a separate number line for each of the four sets and sketch the set on it.Show the full question

  2. Question 1(b) · Sets, intervals and number lines · 4 marks

    Consider four sets given by A = {-1, 1, 3, 4, 7, 10}, B = {0, 2, 3, 4}, C = (-3, 9) (an open interval), and D = (0, 7] (a half-open interval). Using A = {-1, 1, 3, 4, 7, 10} and B = {0, 2, 3, 4}, work out the intersection A ∩ B and the union A ∪ B.Show the full question

  3. Question 1(c) · Sets, intervals and number lines · 2 marks

    Consider four sets given by A = {-1, 1, 3, 4, 7, 10}, B = {0, 2, 3, 4}, C = (-3, 9) (an open interval), and D = (0, 7] (a half-open interval). Using D = (0, 7] and C = (-3, 9), determine D ∩ C and express your answer using interval notation.Show the full question

  4. Question 1(d) · Sets, intervals and number lines · 2 marks

    Consider four sets given by A = {-1, 1, 3, 4, 7, 10}, B = {0, 2, 3, 4}, C = (-3, 9) (an open interval), and D = (0, 7] (a half-open interval). Using C = (-3, 9) and D = (0, 7], determine C ∪ D and express your answer using set builder notation.Show the full question

  5. Question 2.1 · Sequences and series · 2 marks

    Express the repeating decimal number 2.7̇18̇ (that is, 2.718718718...) as an improper fraction.Show the full question

  6. Question 2.2 · Sequences and series · 2 marks

    Express the repeating decimal number 10.2𝟏̄3̄ (that is, 10.21313131...) as an improper fraction.Show the full question

  7. Question 3.1 · Sequences and series · 5 marks

    Calculate the sum of the first 200 natural numbers.Show the full question

  8. Question 3.2 · Sequences and series · 5 marks

    A golf ball is released from a height of 30 feet above the pavement and falls straight down. Each time it bounces, it rebounds to three fourths of the distance it just fell. Determine the total distance (counting both the downward falls and the upward rebounds) that the ball has travelled by the time it strikes the pavement for the sixth time.Show the full question

  9. Question 4.1 · Word problems and models · 6 marks

    The Tour de France is scheduled to take place on 26 June 2021, running from Brest to Mür de Bretagne, a route of approximately 131 km. Mark starts cycling from Brest towards Mür de Bretagne while Jay starts cycling from the opposite end towards Brest, with both cyclists setting off towards each other at exactly the same time. Mark's cycling speed is twice that of Jay's. If the two cyclists meet each other one and a half hours after starting, determine the average speed at which each of them was travelling.Show the full question

  10. Question 4.2 · Word problems and models · 8 marks

    A body initially at temperature T1 is placed into surroundings held at a different temperature T0. Over time t (measured in minutes) the body's temperature T(t) changes according to the formula T(t) = T0 + (T1 - T0)e^(-kt). A cup of coffee starts at a temperature of 40.56°C and is placed into a freezer kept at -17.78°C. After 5 minutes, the coffee's temperature has dropped to 21.11°C. Using this information, determine what the coffee's temperature will be after 10 minutes.Show the full question

  11. Question 5(a) · Limits and continuity · 2 marks

    Work out the value of the limit as x approaches 2 of the product (x^2 + x)(3x^3 + 6).Show the full question

  12. Question 5(b) · Limits and continuity · 3 marks

    Work out the value of the limit as u approaches -2 of the square root of (u^2 + 3u + 6).Show the full question

  13. Question 5(c)(i) · Limits and continuity · 3 marks

    Given the function f(x) = (2x + 12) / |x + 6|, determine the limit of f(x) as x approaches 6 from the right (x \u2192 6+).Show the full question

  14. Question 5(c)(ii) · Limits and continuity · 2 marks

    Using the same function f(x) = (2x + 12) / |x + 6|, determine the limit of f(x) as x approaches 6 from the left (x \u2192 6-).Show the full question

  15. Question 5(c)(iii) · Limits and continuity · 3 marks

    Using your results from parts (i) and (ii) for f(x) = (2x + 12) / |x + 6|, deduce whether the limit of f(x) exists as x approaches 6, and state its value if it does.Show the full question

  16. Question 5(d)(i) · Limits and continuity · 2 marks

    Consider the piecewise function f(x) defined as cos(x) for x < 0, equal to 0 when x = 0, and equal to 1 - x^2 for x > 0. Sketch the graph of this function f(x).Show the full question

  17. Question 5(d)(ii) · Limits and continuity · 3 marks

    For the same piecewise function f(x) (equal to cos(x) for x < 0, equal to 0 at x = 0, and equal to 1 - x^2 for x > 0), determine whether f(x) is continuous at x = 0, providing reason(s) to support your answer.Show the full question

  18. Question 6(a) · Differentiation · 6 marks

    Apply the first principles definition of the derivative to find f'(x) for the function f(x) = 6/x.Show the full question

  19. Question 6(b)(i) · Differentiation · 5 marks

    Using a suitable differentiation technique, find the derivative of f(x) = cos(√(sin(tan(πx)))), simplifying your result as much as possible.Show the full question

  20. Question 6(b)(ii) · Differentiation · 5 marks

    Using a suitable differentiation technique, determine the derivative of p(t) = cos(t) / (1 − sin(t)), simplifying your answer as far as possible.Show the full question

  21. Question 6(b)(iii) · Differentiation · 5 marks

    Using a suitable differentiation technique, determine the derivative of g(x) = ln( e^x / (1 + e^x) ), simplifying your answer as far as possible.Show the full question

  22. Question 7(a) · Tangents and normals · 5 marks

    Given the function f(x) = x / √(x² + 1), For the function f(x) = x divided by the square root of (x² + 1), determine the equation of the tangent line to the curve at x = 1.Show the full question

  23. Question 7(b) · Tangents and normals · 5 marks

    Given the function f(x) = x / √(x² + 1), Using the same function f(x) = x divided by the square root of (x² + 1), determine the equation of the normal line to the tangent at x = 1.Show the full question

  24. Question 8(a) · Applications of differentiation · 3 marks

    A glass tank measuring 25 cm long, 20 cm wide and 30 cm high contains water, with the water surface sitting 5 cm below the top of the tank. A solid spherical metal ball B1 is then carefully lowered into the tank, after which the water surface rises to a level 3 cm below the top of the tank. Draw a sketch diagram that illustrates the tank described above, showing the tank's dimensions (25 cm long, 20 cm wide, 30 cm high), the initial water level 5 cm below the top, and the new water level of 3 cm below the top after ball B1 has been placed in the tank.Show the full question

  25. Question 8(b) · Applications of differentiation · 2 marks

    A glass tank measuring 25 cm long, 20 cm wide and 30 cm high contains water, with the water surface sitting 5 cm below the top of the tank. A solid spherical metal ball B1 is then carefully lowered into the tank, after which the water surface rises to a level 3 cm below the top of the tank. Using the change in water level caused by placing ball B1 into the tank (from 5 cm below the top to 3 cm below the top), calculate the volume of the solid spherical metal ball B1.Show the full question

  26. Question 8(c) · Applications of differentiation · 2 marks

    A glass tank measuring 25 cm long, 20 cm wide and 30 cm high contains water, with the water surface sitting 5 cm below the top of the tank. A solid spherical metal ball B1 is then carefully lowered into the tank, after which the water surface rises to a level 3 cm below the top of the tank. Using the volume of ball B1 found above, determine the radius R1 of the spherical ball B1.Show the full question

  27. Question 8(d)(i) · Applications of differentiation · 4 marks

    A glass tank measuring 25 cm long, 20 cm wide and 30 cm high contains water, with the water surface sitting 5 cm below the top of the tank. A solid spherical metal ball B1 is then carefully lowered into the tank, after which the water surface rises to a level 3 cm below the top of the tank. Suppose a second solid metal ball B2 has a radius R2 equal to half the radius R1 of ball B1. Assume that instead of placing B1 into the tank of water, ball B2 is placed into the tank instead. Determine, with reasons, whether the surface of the water would then be 4 cm below the top of the tank.Show the full question