Qatar Senior Secondary Certificate (Thanaweya) · Advanced Mathematics

Calculus and Analytical Geometry

74 practice questions, each with the answer and an explanation.

Practise Calculus and Analytical Geometry now

Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 74 questions

  1. Evaluate the limit: lim_(x -> 0) [sin(5x) / (2x)].
  2. Find the derivative dy/dx of the implicit equation x^3 + y^3 = 6xy at the point (3, 3) on the Folium of Descartes.
  3. What is the equation of the normal line to the curve y = x^2 - 4x + 5 at the point (1, 2)?
  4. Evaluate the definite integral: integral from 0 to 1 of [x / (x^2 + 1)] dx.
  5. Find the absolute maximum value of the function f(x) = 2x^3 - 3x^2 - 12x + 1 on the closed interval [-2, 3].
  6. What is the total area of the finite region enclosed between the parabola y = 4 - x^2 and the line y = 0?
  7. Evaluate the indefinite integral: integral of x * e^(2x) dx using integration by parts.
  8. A cylindrical water tank with a radius of 2 meters is being filled at a constant rate of 0.8 pi cubic meters per minute. At what rate is the water level rising in the tank?
  9. What are the coordinates of the inflection point of the cubic curve f(x) = x^3 - 6x^2 + 9x + 2?
  10. What is the volume of the solid generated by revolving the region bounded by y = sqrt(x), y = 0, and x = 4 about the x-axis?
  11. Find the slope of the tangent line to the polar curve r = 2 + 2 cos(theta) at theta = pi / 2.
  12. Evaluate the limit using L'Hôpital's Rule: lim_(x -> 0) [(e^(3x) - 1 - 3x) / x^2].
  13. What is the center and length of the semi-major axis of the ellipse given by the equation: 4x^2 + 9y^2 - 16x + 54y + 61 = 0?
  14. What is the length of the arc of the curve y = (2/3) x^(3/2) from x = 0 to x = 3?
  15. Find the general solution of the separable differential equation: dy/dx = 2xy.
  16. Find the radius of convergence R of the power series: sum from n=1 to infinity of [(-1)^n (x - 2)^n] / [n 3^n].
  17. Evaluate the limit: lim_(x -> inf) [(2x^2 + 5x) / (3x^2 - 4)].
  18. At what values of x does the graph of f(x) = x^4 - 4x^3 have points of inflection?
  19. Find the derivative of the inverse trigonometric function: f(x) = arctan(3x).
  20. What is the average value of the function f(x) = sin(x) on the interval [0, pi]?
  21. Find the vector equation of the line passing through point A(1, 2, 3) with direction vector d = (2, -1, 4).
  22. A curve is defined by parametric equations x = t^2 + 1, y = t^3 - 3t. At what values of parameter t is the tangent line horizontal?
  23. What is the maximum area of a rectangle that can be inscribed in a semicircle of radius R with one side lying along the diameter?
  24. Evaluate the limit: lim_(x -> 0) [(1 - cos(x)) / x^2].
  25. Find the equation of the plane perpendicular to the line with direction vector n = (3, -2, 5) and passing through point P(1, -1, 2).
  26. Find the curvature kappa of the circle x^2 + y^2 = 25 at any point on its circumference.
  27. The region between y = x^2 and y = 2x is rotated about the x-axis. What is the volume of the solid formed?
  28. The curve x^2 y + y^3 = 10 passes through the point (1, 2). What is dy/dx at this point?
  29. A 10 m ladder leans against a wall. Its foot slides away from the wall at 2 m/s. How fast is the top sliding down when the foot is 6 m from the wall?
  30. An open box is made from a 12 cm by 12 cm square sheet by cutting equal squares from the corners and folding up the sides. What is the largest possible volume?
  31. What is the value of the limit as x approaches 0 of (cos x - 1 + x^2/2) / x^4?
  32. Find f'(2) for f(x) = x^3+2x^2-x.
  33. Find f'(1) for f(x) = 2x^3-3x^2+x.
  34. Find f'(3) for f(x) = x^3+5x.
  35. Find f'(2) for f(x) = 3x^3+x^2-2x.
  36. Find f'(4) for f(x) = x^3-6x^2+9x.
  37. Evaluate lim_(x -> 2) (x^2 - 4) / (x - 2).
  38. Evaluate lim_(x -> 3) (x^2 - 9) / (x - 3).
  39. Evaluate lim_(x -> 4) (x^2 - 16) / (x - 4).
  40. Evaluate lim_(x -> 5) (x^2 - 25) / (x - 5).
  41. Evaluate lim_(x -> 6) (x^2 - 36) / (x - 6).
  42. Evaluate lim_(x -> 7) (x^2 - 49) / (x - 7).
  43. Evaluate the integral of (3x^2+2x) dx from x = 0 to x = 2.
  44. Evaluate the integral of (6x^2+4x) dx from x = 0 to x = 1.
  45. Evaluate the integral of (2x^2+6x) dx from x = 0 to x = 3.
  46. Evaluate the integral of (3x^2+3x) dx from x = 0 to x = 2.
  47. Evaluate the integral of (6x^2+2x) dx from x = 0 to x = 2.
  48. Evaluate the integral of (3x^2+6x) dx from x = 0 to x = 1.
  49. Find the local maximum value of f(x) = x^3-3x^2-9x.
  50. Find the local maximum value of f(x) = x^3-12x.
  51. Find the local maximum value of f(x) = x^3-3x.
  52. A rectangle has a fixed perimeter of 20 cm. What is the largest possible area?
  53. A rectangle has a fixed perimeter of 28 cm. What is the largest possible area?
  54. A rectangle has a fixed perimeter of 36 cm. What is the largest possible area?
  55. A rectangle has a fixed perimeter of 40 cm. What is the largest possible area?
  56. A rectangle has a fixed perimeter of 60 cm. What is the largest possible area?
  57. A rectangle has a fixed perimeter of 100 cm. What is the largest possible area?
  58. Differentiate y = (2x + 1)^3 and evaluate dy/dx at x = 1.
  59. Differentiate y = (3x - 1)^2 and evaluate dy/dx at x = 1.
  60. Differentiate y = (1x + 2)^4 and evaluate dy/dx at x = 0.
  61. Differentiate y = (2x + 3)^3 and evaluate dy/dx at x = -1.
  62. Differentiate y = (4x + 1)^2 and evaluate dy/dx at x = 1.
  63. Differentiate y = (3x + 2)^3 and evaluate dy/dx at x = 0.
  64. Find the distance from the point (1, 2) to the line 3x+4y-5 = 0.
  65. Find the distance from the point (2, 1) to the line 3x+4y+7 = 0.
  66. Find the distance from the point (0, 0) to the line 5x+12y+26 = 0.
  67. Find the distance from the point (3, -1) to the line 4x+3y+6 = 0.
  68. Find the distance from the point (1, 1) to the line 5x+12y+3 = 0.
  69. Find the distance from the point (2, 3) to the line 6x+8y-5 = 0.
  70. Find the radius of the circle x^2+y^2-2x-4y-4 = 0.
  71. Find the radius of the circle x^2+y^2+4x-6y-3 = 0.
  72. Find the radius of the circle x^2+y^2-6x+2y+6 = 0.
  73. Find the radius of the circle x^2+y^2-4x-8y-5 = 0.
  74. Find the radius of the circle x^2+y^2+2x+4y-31 = 0.

Keep practising

More Qatar Senior Secondary Certificate (Thanaweya) topics

All Qatar Senior Secondary Certificate (Thanaweya) practice →

Ready for the full Qatar Senior Secondary Certificate (Thanaweya) test?

Take a mock testQatar Senior Secondary Certificate (Thanaweya) hub