Bagrut (Secondary School Matriculation) · Mathematics

Differential and Integral Calculus and Complex Numbers

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

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All 26 questions

  1. Solve the trigonometric equation 2sin(x)^2 - 3sin(x) + 1 = 0 on the interval [0, 2*pi].
  2. Which expression is identical to cos(2*x) / (cos(x) + sin(x)) where defined?
  3. Find all real solutions to sin(3*x) + sin(x) = cos(x) on the interval [0, pi].
  4. If sin(x) = 3/5 and x is in the second quadrant (pi/2 < x < pi), what is the exact value of tan(2*x)?
  5. What is the fundamental period of the function f(x) = 3tan(2x - pi/4) + 1?
  6. Determine all intersection points of y = 2cos(x) and y = 1/cos(x) on the interval [0, 2pi].
  7. Evaluate the trigonometric limit: lim_{x -> 0} (1 - cos(4x)) / (x sin(2*x)).
  8. What is the principal value of arcsin(sin(5*pi/6))?
  9. In an arithmetic progression, the 3rd term is 11 and the 7th term is 27. What is the 15th term?
  10. Find the sum of the infinite geometric series: 12 - 4 + 4/3 - 4/9 + ...
  11. A sequence is defined by a_1 = 3 and a_{n+1} = sqrt(6 + a_n). Assuming the sequence converges, find its limit L.
  12. An arithmetic sequence a_n has a_1 = 2 and common difference d = 3. A sequence b_n is formed by b_n = 2^(a_n). Find the product P = b_1 b_2 ... * b_10.
  13. Evaluate the telescoping sum: sum_{k=1}^n (1 / (k*(k + 1))).
  14. A sequence satisfies a_1 = 1/2 and a_{n+1} = 2*a_n / (1 + a_n). Find the explicit expression for a_n and its limit as n approaches infinity.
  15. For which value of k do the expressions 2k + 1, 4k - 2, and 7*k - 9 form consecutive terms of an arithmetic progression?
  16. The sum of an infinite geometric series is 18, and the sum of the squares of its terms is 108. What is the common ratio r?
  17. What is the vector equation of the line passing through points A(1, 2, -1) and B(3, 0, 4)?
  18. Find the cosine of the angle between vectors u = (2, -1, 2) and v = (3, 4, 0).
  19. Find the area of the triangle with vertices at A(1, 0, 0), B(0, 2, 0), and C(0, 0, 3).
  20. Find the equation of the plane passing through points P(1, 1, 1), Q(2, 3, 1), and R(3, 1, 4).
  21. What is the perpendicular distance from point P(2, -3, 4) to the plane 2x - y + 2z - 6 = 0?
  22. Find the point of intersection between the line r = (1, -1, 2) + t(2, 1, -1) and the plane 3x + 2*y - z = 8.
  23. An ellipse centered at the origin has foci at (4, 0) and (-4, 0), and a major axis of length 10. What is its standard equation?
  24. What is the equation of the tangent line to the parabola y^2 = 8*x at the point (2, 4)?
  25. Find the locus of points P(x, y, z) that are equidistant from the two parallel planes x + 2y - 2z = 1 and x + 2y - 2z = 7.
  26. Find the length of the chord cut by the line 3x + 4y - 15 = 0 on the circle x^2 + y^2 = 25.

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