University of the Philippines College Admission Test · Mathematics

Algebra, Functions and Polynomials

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

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All 35 questions

  1. What are all the real solutions to the quadratic equation 3x^2 - 11x + 6 = 0?
  2. If f(x) = (2x + 5) / (x - 3) for x != 3, what is the inverse function f^(-1)(x)?
  3. What is the remainder when the polynomial P(x) = 2x^4 - 3x^3 + x^2 - 5x + 7 is divided by x - 2?
  4. Solve for x in the logarithmic equation: log2(x - 2) + log2(x + 2) = 5.
  5. What is the solution set of the inequality |2x - 7| <= 5?
  6. Solve for x in the exponential equation 4^(x + 1) - 10 * 2^x - 24 = 0.
  7. If 3x + 2y = 16 and 2x - 5y = -21, what is the value of x + y?
  8. What is the 25th term of the arithmetic sequence 7, 11, 15, 19, ...?
  9. What is the sum of the first 6 terms of the geometric sequence 5, 15, 45, ...?
  10. What is the sum of the infinite geometric series 12 + 8 + 16/3 + ...?
  11. What is the coefficient of x^3 in the binomial expansion of (2x - 3)^5?
  12. Find all real solutions to the rational equation 3/(x - 1) - 2/(x + 1) = 1.
  13. Solve for x in the radical equation sqrt(2x + 7) - x = 2.
  14. What is the standard rectangular form of the complex quotient (3 + 4i) / (1 - 2i)?
  15. What is the vertex and maximum value of the quadratic function f(x) = -2x^2 + 8x - 3?
  16. For what values of the constant k does the equation kx^2 + 6x + 1 = 0 have two distinct real roots?
  17. If f(x) = 2x^2 - 3 and g(x) = 3x + 1, what is the composite function (f o g)(x)?
  18. What is the natural domain of the real-valued function f(x) = sqrt((x - 2) / (x + 5))?
  19. If r1 and r2 are the roots of 2x^2 - 7x + 3 = 0, what is the value of 1/r1 + 1/r2?
  20. If x - 3 is a factor of the polynomial P(x) = 2x^3 - kx^2 + 5x - 6, what is the value of k?
  21. In the partial fraction decomposition (5x - 1) / ((x - 2)(x + 1)) = A / (x - 2) + B / (x + 1), what are the values of A and B?
  22. The variable y varies directly as x^2 and inversely as z. If y = 15 when x = 3 and z = 6, what is y when x = 4 and z = 5?
  23. What is the solution set of the inequality |3x + 2| > 8?
  24. If log_b(3) = m and log_b(5) = n, express log_b(45) in terms of m and n.
  25. Simplify the algebraic expression ((27x^6 y^(-3)) / (8x^(-3) y^6))^(2/3).
  26. What are the quotient and remainder when 4x^3 - 6x^2 + 8x - 5 is divided by 2x - 1?
  27. What are the vertical and horizontal asymptotes of the rational function f(x) = (3x^2 - 5x + 2) / (x^2 - 4)?
  28. What is the range of the quadratic function f(x) = x^2 - 6x + 13 for all real numbers x?
  29. What is the determinant of the 2x2 matrix with rows [4, -3] and [2, 5]?
  30. If r1, r2, and r3 are the roots of the cubic equation x^3 - 6x^2 + 11x - 6 = 0, what is the value of r1^2 + r2^2 + r3^2?
  31. A radioactive sample decays from 80 grams to 10 grams over a period of 12 hours. What is its half-life?
  32. What is the greatest common factor of the polynomials x^2 - 9 and x^2 - 5x + 6?
  33. Find all points of intersection between the parabola y = x^2 - 4 and the straight line y = 2x - 1.
  34. A two-digit number is 4 times the sum of its digits. If 27 is added to the number, the digits reverse. What is the original number?
  35. If 2^(2x + 1) - 9 * 2^x + 4 = 0, what are the solutions for x?

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