Questions (33218 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
SPS SPS FM 2024 October Q4
3 marks Moderate -0.8
The curve \(y = \sqrt{2x - 1}\) is stretched by scale factor \(\frac{1}{4}\) parallel to the \(x\)-axis and by scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt{ax - b}\) where \(a\) and \(b\) are rational numbers. [3]
SPS SPS FM 2024 October Q5
9 marks Standard +0.3
In this question you must show detailed reasoning. The polynomial \(f(x)\) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of \(f(x)\). [1]
    2. Hence find the exact roots of the equation \(f(x) = 0\). [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form \(f(x) = 0\). [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS FM 2024 October Q6
7 marks Standard +0.8
The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,
  1. Find the value of \(k\), giving a reason for your answer. [4]
  2. Find the value of \(\sum_{r=2}^{\infty} u_r\). [3]
SPS SPS FM 2024 October Q7
9 marks Standard +0.8
The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}
  1. Determine the coordinates of \(B\). [6]
  2. Hence determine the exact value of \(\tan\alpha\). [3]
SPS SPS FM 2024 October Q8
5 marks Standard +0.3
Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by \(3\), for all integers \(n > 0\). [5]
SPS SPS FM 2024 October Q9
9 marks Standard +0.8
A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.
  1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
    1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
    2. Give a geometrical interpretation of the case in part (b)(i). [1]
  3. Give a geometrical interpretation of the case where \(a = 5\). [1]
SPS SPS SM 2024 October Q1
3 marks Easy -1.2
A is inversely proportional to B. B is inversely proportional to the square of C. When A is 2, C is 8. Find C when A is 12. [3]
SPS SPS SM 2024 October Q2
5 marks Moderate -0.8
  1. Write \(3x^2 + 24x + 5\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]
The finite region R is enclosed by the curve \(y = 3x^2 + 24x + 5\) and the \(x\)-axis.
  1. State the inequalities that define R, including its boundaries. [2]
SPS SPS SM 2024 October Q3
5 marks Moderate -0.8
The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]
SPS SPS SM 2024 October Q4
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS SM 2024 October Q5
8 marks Moderate -0.8
\includegraphics{figure_5} Figure 4 The line \(l_1\) has equation \(y = \frac{3}{5}x + 6\) The line \(l_2\) is perpendicular to \(l_1\) and passes through the point \(B(8, 0)\), as shown in the sketch in Figure 4.
  1. Show that an equation for line \(l_2\) is $$5x + 3y = 40$$ [3]
Given that
  • lines \(l_1\) and \(l_2\) intersect at the point C
  • line \(l_1\) crosses the \(x\)-axis at the point A
  1. find the exact area of triangle \(ABC\), giving your answer as a fully simplified fraction in the form \(\frac{p}{q}\) [5]
SPS SPS SM 2024 October Q6
8 marks Moderate -0.8
In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30e^{-0.1t}.$$
  1. Find the initial mass of the chemical. What is the mass of chemical in the long term? [3]
  2. Find the time when the mass is 30 grams. [3]
  3. Sketch the graph of \(m\) against \(t\). [2]
SPS SPS SM 2024 October Q7
4 marks Moderate -0.8
Express \(\frac{a^{\frac{1}{2}} - a^{\frac{2}{3}}}{a^{\frac{1}{3}} - a}\) in the form \(a^m + \sqrt{a^n}\), where \(m\) and \(n\) are integers and \(a \neq 0\) or 1. [4]
SPS SPS SM 2024 October Q8
5 marks Standard +0.3
A circle, C, has equation \(x^2 - 6x + y^2 = 16\). A second circle, D, has the following properties:
  • The line through the centres of circle C and circle D has gradient 1.
  • Circle D touches circle C at exactly one point.
  • The centre of circle D lies in the first quadrant.
  • Circle D has the same radius as circle C.
Find the coordinates of the centre of circle D. [5]
SPS SPS SM 2024 October Q9
9 marks Moderate -0.3
In this question you must show detailed reasoning. The polynomial f(x) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of f(x). [1]
    2. Hence find the exact roots of the equation f(x) = 0. [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form f(x) = 0. [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS SM 2024 October Q10
7 marks Standard +0.3
The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,
  1. Find the value of k, giving a reason for your answer. [4]
  2. Find the value of \(\sum_{r=2}^{\infty} u_r\) [3]
SPS SPS SM 2024 October Q1
3 marks Moderate -0.8
The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(v\)ms\(^{-1}\). When \(v = 3.6\), \(P = 50\). Determine the wind speed that will give a power output of 225 watts. [3]
SPS SPS SM 2024 October Q2
7 marks Easy -1.2
Solve the inequalities
  1. \(3 - 8x > 4\), [2]
  2. \((2x - 4)(x - 3) < 12\). [5]
SPS SPS SM 2024 October Q3
6 marks Standard +0.3
The first three terms of an arithmetic series are \(9p\), \(8p - 3\), \(5p\) respectively, where \(p\) is a constant. Given that the sum of the first \(n\) terms of this series is \(-1512\), find the value of \(n\). [6]
SPS SPS SM 2024 October Q4
7 marks Standard +0.3
The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]
SPS SPS SM 2024 October Q5
11 marks Moderate -0.3
A line has equation \(y = 2x\) and a circle has equation \(x^2 + y^2 + 2x - 16y + 56 = 0\).
  1. Show that the line does not meet the circle. [3]
    1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2x\). [4]
    2. Hence find the shortest distance between the line \(y = 2x\) and the circle, giving your answer in an exact form. [4]
SPS SPS SM 2024 October Q6
6 marks Moderate -0.8
The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).
\(t\)051025
\(m\)200160
  1. Find the values missing from the table. [2]
  2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams. [4]
SPS SPS SM 2024 October Q7
6 marks Moderate -0.3
A student was asked to solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\). The student's attempt is written out below. \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\) \(4\log_3 x - 3 \log_3 x - 2 = 0\) \(\log_3 x - 2 = 0\) \(\log_3 x = 2\) \(x = 8\)
  1. Identify the two mistakes that the student has made. [2]
  2. Solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\), giving your answers in an exact form. [4]
SPS SPS SM 2024 October Q8
8 marks Standard +0.3
In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac{5}{3x-4} + \left(\frac{5}{3x-4}\right)^2 + \left(\frac{5}{3x-4}\right)^3 + \ldots$$ is convergent.
  1. Find the set of possible values of \(x\), giving your answer in set notation. [5]
  2. Given that the sum to infinity of the series is \(\frac{2}{3}\), find the value of \(x\). [3]
SPS SPS FM Pure 2025 January Q1
4 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the curve \(y = 6x - x^2\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning. [4]