Bagrut (Secondary School Matriculation) · Mathematics

Vectors, Analytic Geometry and Probability

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

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

All 26 questions

  1. What is the modulus and principal argument of the complex number z = -1 + i*sqrt(3)?
  2. Compute the value of (1 + i)^10 using De Moivre's theorem.
  3. What is the sum of all complex solutions to the equation z^5 - 1 = 0?
  4. Find all four roots of the equation z^4 + 4 = 0 in the complex plane.
  5. Describe the geometric locus of points in the complex plane satisfying |z - (3 - 2*i)| = 5.
  6. Solve the complex equation z^2 + |z|^2 = 2 - 4i, where z = x + yi.
  7. Express the complex quotient (3 + 4i) / (1 - 2i) in standard form a + b*i.
  8. What is the monic quadratic equation with real coefficients having 2 + 3*i as one of its roots?
  9. If z_1 = 6(cos(5pi/6) + isin(5pi/6)) and z_2 = 2(cos(pi/3) + isin(pi/3)), what is z_1 / z_2?
  10. The six solutions to z^6 = 64 form the vertices of a regular hexagon in the complex plane. What is the area of this hexagon?
  11. What is the derivative of f(x) = (x^2 + 1) / (x - 2) for x != 2?
  12. Find the derivative of the composite exponential function f(x) = e^(sin(2*x)).
  13. What is the derivative of f(x) = ln((x + 1) / (x - 1)) on its domain (|x| > 1)?
  14. Find the slope of the tangent line to the curve y = x^3 - 3*x^2 + 4 at its inflection point.
  15. Find the coordinates of the local maximum of the function f(x) = x e^(-2x).
  16. A right circular cylinder is inscribed in a right circular cone of base radius R and height H. What height h of the cylinder maximizes its volume?
  17. How many inflection points does the function f(x) = x^4 - 4x^3 + 6x^2 have?
  18. Find all asymptotes of the rational function f(x) = (2x^3 - 5x + 1) / (x^2 - 4).
  19. Evaluate the limit using L'Hopital's rule: lim_{x -> 0} (e^x - 1 - x) / x^2.
  20. A 5-meter ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at 2 m/s, how fast is the top sliding down when the bottom is 3 meters from the wall?
  21. Evaluate the definite integral: int_0^1 (x / (x^2 + 1)) dx.
  22. Evaluate the indefinite integral: int x * cos(x) dx.
  23. Find the total area enclosed between the parabola y = x^2 and the line y = 2*x.
  24. Find the positive constant k such that the area bounded between the parabola y = x^2 and the line y = k*x is equal to 36.
  25. Find the volume of the solid generated by rotating the region bounded by y = sqrt(x), y = 0, and x = 4 around the y-axis.
  26. If g(x) = int_1^(x^2) sqrt(t^3 + 1) dt, find g'(x).

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