New Zealand Scholarship · Calculus and Advanced Mathematics

Conic Sections, Vectors, Mathematical Proof and Sequences

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

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All 39 questions

  1. What are the coordinates of the foci of the ellipse x^2 / 25 + y^2 / 16 = 1?
  2. What are the equations of the asymptotes of the hyperbola x^2 / 9 - y^2 / 16 = 1?
  3. What is the scalar (dot) product of vectors u = 3i - 4j and v = 2i + 5j, and what is their angular relationship?
  4. What is the sum to infinity of the geometric series 12 + 4 + 4/3 + 4/9 + ...?
  5. In mathematical proof by induction, what are the two required procedural steps?
  6. Prove that sqrt(2) is irrational by contradiction. What initial assumption is made and where does the contradiction arise?
  7. Find the parametric equations of the tangent line to the ellipse x = 3cos(t), y = 2sin(t) at parameter t = pi/4.
  8. Using de Moivre's theorem, evaluate (cos(pi/6) + i*sin(pi/6))^6.
  9. What geometric locus in the complex Argand plane is represented by the equation |z - 3| = |z + 1|?
  10. What geometric shape in the complex plane is described by |z - (2 + 3i)| = 5?
  11. What is the sum of all n distinct nth roots of unity for any integer n >= 2?
  12. If a polynomial with real coefficients has a non-real root z = 2 - 5i, what other root must it possess?
  13. What is the vector projection of vector u = (4, 2) onto vector v = (3, 0)?
  14. What is the cross product of vectors u = (1, 0, 0) and v = (0, 1, 0) in R^3?
  15. What is the acute angle between the two planes with normal vectors n1 = (1, 1, 0) and n2 = (0, 1, 1)?
  16. What is the perpendicular distance from the point (1, 2, 3) to the plane 2x - 2y + z - 10 = 0?
  17. What is the determinant of the 2x2 matrix A = [[3, 2], [1, 4]]?
  18. What is the inverse of the matrix A = [[2, 1], [5, 3]] (det = 23 - 15 = 1)?
  19. What are the eigenvalues of the matrix A = [[4, 1], [2, 3]]?
  20. What matrix represents an anticlockwise rotation by 90 degrees (pi/2 radians) about the origin in R^2?
  21. What matrix represents a reflection across the line y = x in R^2?
  22. What is the eccentricity of the ellipse x^2 / 25 + y^2 / 16 = 1?
  23. What is the eccentricity e of any hyperbola in conic section geometry?
  24. What are the coordinates of the focus of the parabola y^2 = 12x?
  25. Convert the Cartesian point (x, y) = (-2, 2sqrt(3)) to polar coordinates (r, theta) with r > 0 and 0 <= theta < 2pi.
  26. What geometric curve is represented by the polar equation r = 4*cos(theta)?
  27. What is the closed-form formula for the sum of the first n positive integers, 1 + 2 + 3 + ... + n?
  28. In the binomial expansion of (x + 2)^5, what is the coefficient of the x^3 term?
  29. What does Pascal's Identity state regarding binomial combinations C(n, k)?
  30. What does the Cauchy-Schwarz inequality state for any real vectors u and v?
  31. What is the shortest distance between two parallel lines in R^2: 3x + 4y - 5 = 0 and 3x + 4y + 15 = 0?
  32. What is the Cartesian equation of the hyperbola with parametric equations x = 3sec(t) and y = 4tan(t)?
  33. What is the area of the triangle in R^3 with vertices at the origin O(0, 0, 0), A(2, 0, 0), and B(0, 3, 0)?
  34. What is the modulus and principal argument of the complex number z = -1 - i*sqrt(3)?
  35. For an invertible n x n matrix A, what is the determinant of its inverse A^(-1)?
  36. Find the sum to infinity of the geometric series 8 + 4 + 2 + 1 + ...
  37. The third term of an arithmetic sequence is 11 and the seventh term is 23. What is the first term?
  38. Find the scalar (dot) product of the vectors (1, 2, 3) and (4, -5, 6).
  39. What is the equation of the directrix of the parabola x^2 = 8y?

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