Diagnostic colle

diagnostic
Published

October 8, 2026

45 minutes. Pen and paper. No calculator. Skip a problem after 3 minutes and mark it “skipped”. A skip is useful data.

Arithmetic

  1. \frac{3}{4} - \frac{2}{3} + \frac{1}{6}
  2. \frac{2}{5} \div \frac{4}{15}
  3. \dfrac{2^{-3} \cdot 4^{2}}{8^{1/3}}

Algebra

  1. Expand (2x - 3)^2.
  2. Factor x^2 - 5x + 6.
  3. Solve 3(x - 2) = 2x + 5.
  4. Solve x^2 + 2x - 15 = 0.
  5. Solve the system 2x + y = 7, x - y = 2.
  6. Solve \lvert x - 1 \rvert < 3.
  7. Simplify \dfrac{x^2 - 9}{x^2 + 3x}. State for which x it is valid.

Exponentials and logarithms

  1. \log_2 32
  2. Solve 3^{x+1} = 27.
  3. \ln(e^3) - \ln 1

Functions

  1. f(x) = 2x + 1, g(x) = x^2. Find f(g(3)) and g(f(3)).
  2. Find the inverse of f(x) = \frac{x - 1}{3}.
  3. Find the line through (1, 2) and (3, 8).

Trigonometry and geometry

  1. \sin\frac{\pi}{6}, \cos\frac{\pi}{3}, \tan\frac{\pi}{4}
  2. Convert 135^\circ to radians.
  3. A right triangle has legs 5 and 12. Find the hypotenuse.
  4. \sin\theta = \frac{3}{5} and \frac{\pi}{2} < \theta < \pi. Find \cos\theta.
  5. A circle has radius 3. Find its area and circumference.
  6. Find the sum of the interior angles of a hexagon.

Calculus

  1. \frac{d}{dx}\left(x^3 - 4x\right)
  2. \frac{d}{dx}\left(x \sin x\right)
  3. \int_0^2 3x^2 \, dx

Proof

  1. Prove: the sum of two odd integers is even. Bring it to supervision. It has no answer below.
  1. \frac{1}{4}
  2. \frac{3}{2}
  3. 1
  4. 4x^2 - 12x + 9
  5. (x - 2)(x - 3)
  6. x = 11
  7. x = 3 or x = -5
  8. x = 3, y = 1
  9. -2 < x < 4
  10. \frac{x - 3}{x}, for x \neq 0 and x \neq -3
  11. 5
  12. x = 2
  13. 3
  14. f(g(3)) = 19, g(f(3)) = 49
  15. f^{-1}(x) = 3x + 1
  16. y = 3x - 1
  17. \frac{1}{2}, \frac{1}{2}, 1
  18. \frac{3\pi}{4}
  19. 13
  20. -\frac{4}{5}
  21. Area 9\pi, circumference 6\pi
  22. 720^\circ
  23. 3x^2 - 4
  24. \sin x + x \cos x
  25. 8