Mathematics paper 1- Practice Questions

Mathematics paper 1- Practice Questions

Below are past exam questions focused on paper 1. Attempt to simplify without using calculators or tables, as per exam requirements.

Surds Questions

  1. 2012-5. Without using a calculator or four-figure tables, simplify (8√2) / (√98 - 3√2), giving your answer in its simplest form. (4 marks)
  2. 2011-3. Without using a calculator or four-figure tables, simplify 1/√2 - √2/3 leaving your answer with a rational denominator. (4 marks)
  3. 2010-5. Without using a calculator or four-figure tables, simplify (√54 + 3√3) / √3 in its simplest form. (4 marks)
  4. 2009-14. Without using a calculator or four-figure tables, simplify 11 / (5 - √3), leaving your answer with a rational denominator. (4 marks)
  5. 2008-2. Without using a calculator or four-figure tables, simplify √27 × √32, leaving your answer in surd form. (3 marks)
  6. 2007-1. Express √2 / √3 with a rational denominator. (3 marks)
  7. 2006-23. Simplify (x² - 2) / (x - √2). (3 marks)
  8. 2005-2. Without using a calculator or four-figure tables, simplify √125 + √5 - √45, leaving your answer in surd form. (3 marks)
  9. 2004-14. Simplify √2 / (√2 - 1). (4 marks)
  10. 2003-3. Express 3 / √2 as a fraction with a rational denominator. (3 marks)

Exponential and Logarithmic Functions

Exponential Equations

  1. 2009-7. Solve the equation 3y = 9y / 81. (5 marks)

Logarithmic Equations

  1. 2011-4. Given that logx 6¼ = 2, solve for x. (4 marks)
  2. 2009-11. Given that ½ log3 x = log3 (6 - x)½, find the value of x. (5 marks)
  3. 2008-6. Given that logx (½) + logx 16 = 3, find the value of x. (4 marks)
  4. 2004-19. Solve the equation log10 (2m + 6) = log10 (m - 1). (5 marks)

Evaluation of Logarithms

  1. 2007-11. Given that loga 2 = 0.668 and loga 3 = 0.884, evaluate loga 12. (5 marks)
  2. 2003-8. Given loga 2 = 0.6110 and loga 3 = 0.7039, calculate loga 6. (4 marks)

Rules of Logarithms

  1. 2012-11. Given that log p - log q = 2 log r, find p in terms of q and r. (5 marks)
  2. 2010-8. Given that log5 x + log5 y = 3 log5 q, show that x = q³ / y. (3 marks)
  3. 2006-20. Given that log10 n - log10 m = 2 log10 h, show that n = m h². (4 marks)
  4. 2005-8. Given that logm 27 = 3, find m. (3 marks)

Variations - Joint Variation

  1. Given that x ∝ y / Z² and x = 12 when y = 2 and Z = 1. Find y when x = 15 and Z = 2. (6 marks)
  2. p varies directly as r and inversely as the square root of q. Given that p = 4 when q = 9 and r = 1, calculate q when r = 2 and p = 6. (6 marks)
  3. Given that V ∝ r d and V = 54 when r = 2 and d = 3. Find V when r = ½ and d = 6. (5 marks)
  4. A quantity T is proportional to m and the square of v. When v = 3 and m = 5, T = 90. Calculate the value of T when m = 2 and v = 10. (5 marks)
  5. Given that q ∝ √p and p = 4 when q = 3, find the value of p when q = 15. (5 marks)
  6. P varies directly as x³ and inversely as y. When x = 2 and y = 4, P = 3. Find the value of x when P = 12 and y = 4. (6 marks)
  7. Given that x ∝ y / z. When x = 10, y = 2 and z = 4. Find value of x when y = 1 and z = 5. (5 marks)
  8. Given that P varies as the product of q and r², and that P = 50 when q = 1 and r = 5, find P when q = 3 and r = 8. (5 marks)
  9. A quantity b varies jointly with r and t, and b = 108 when r = 3 and t = 6. Find an equation which expresses b in terms of r and t. (4 marks)
  10. Given that x varies jointly as y and inversely as the square of z, calculate the missing value in Table 1.
    x y z
    3 1 2
    1 3 ?
    (6 marks)

Polynomials

  1. Given that (x + 1) and (x - 3) are two factors of the polynomial a x³ + b x - 6, calculate the values of a and b. (7 marks)
  2. Show that k + 3 is a factor of k³ + 3k² - 4k - 12. (4 marks)
  3. Given that (x + 3)(x + 1)² ≡ A x³ + B x² + C x + D, find the value of C. (4 marks)
  4. When the polynomial x³ + 5x² + K x + 3 is divided by (x + 2), it gives a remainder of 1. Find the value of K. (5 marks)
  5. Given that (4x² - 9)(B x + C) is identical to 16 x³ + 24 x² - 36 x - 54, calculate the values of B and C, given that they are all positive. (5 marks)
  6. Use the remainder theorem to prove that (x - y) is a factor of the polynomial x² (y - 2) + y² (2 - x). (5 marks)
  7. Find the remainder when 2 x³ - 13 x² - 8 x + 12 is divided by 2 x - 1. (4 marks)

Coordinate Geometry

Gradient of a Line & Equation of a Line

  1. Find the gradient of a straight line whose equation is y - 1 = m x and passes through (-2, -5). (3 marks)
  2. A straight line passing through (3t, 7) and (t, -5) has gradient 3. Find the equation of the line. (6 marks)
  3. Given that tan θ = 2/3, find the equation of the line passing through point P(3, 4) in the form y = mx + c. (4 marks)
  4. Find the gradient of the line whose equation is (y + 2x) / 4 = x / 3. (4 marks)
  5. Given points P(-1, 4), Q(3, 0) and R(2, a) are collinear, find the value of a. (5 marks)
  6. The gradient of a line passing through point P(-2, 5) is -½. Find the equation of the line in the form y = mx + c. (4 marks)
  7. Lines G and H intersect at point P. G passes through (-4, 0) and (0, 6). Given that H has equation y = 4x - 4, find the coordinates of P. (3 marks)
  8. A line passes through (1, 6) and cuts the y-axis at 4. Calculate its gradient. (4 marks)
  9. The line joining A(3, q) and B(5 - q, 8) has gradient ½. Calculate the value of q. (4 marks)

Parallel Lines

  1. Given y = 2x - 3 and y = (b - 1) x + 5 are graphs of two parallel lines, calculate the value of b. (3 marks)
  2. Find the equation of a straight line passing through (0, 7) and parallel to y = 2x + 5. (4 marks)

Triangles

  1. Triangle ABC has vertices A(-1, 2), B(3, 3), and C(2, -1). Prove that ∠BAC = ∠ACB. (5 marks)

Similarity - Area Factor

  1. The areas of two similar triangles are 75 cm² and 243 cm². If the base of the smaller triangle is 30 cm and that of the bigger triangle is b cm, calculate the value of b. (5 marks)
  2. The ratio of two circles is 4:9. Given that the radius of the bigger circle is 18 cm, find the radius of the smaller circle. (4 marks)
  3. Figure 3 shows two similar triangles TQU and RSU. TU = 3 cm, UR = 6 cm and RS is parallel to QT. Calculate the ratio of the area of triangle TQU to the area of triangle RSU, leaving your answer in its simplest form. (4 marks)
  4. The areas of two similar triangles ABC and XYZ are in the ratio 1:16. If the height of the smaller triangle is 2 cm, calculate the height of the bigger triangle. (4 marks)
  5. In Figure 7, triangle ABC is similar to triangle DBA. The area of triangle DBA is 24 cm², AB = 8 cm and DB = 4 cm. Calculate the area of triangle ABC. (4 marks)
  6. A trapezium has a height of 3 cm and its area is 6 cm². Calculate the area of a similar trapezium with a height of 12 cm. (4 marks)
  7. The areas of two similar triangles ABC and HKL are 100 cm² and 256 cm² respectively. If the length of AB is 5 cm, calculate the length of HK. (5 marks)