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 SM Pure 2023 September Q6
8 marks Moderate -0.8
Liquid is kept in containers, which due to evaporation and ongoing chemical reactions, at the end of each month the volume of the liquid in these containers reduces by 10% compared with the volume at the start of the same month. One such container is filled up with 250 litres of liquid.
  1. Show that the volume of the liquid in the container at the end of the second month is 202.5 litres. [1]
  2. Find the volume of the liquid in the container at the end of the twelfth month. [2]
At the start of each month a new container is filled up with 250 litres of liquid, so that at the end of twelve months there are 12 containers with liquid.
  1. Use an algebraic method to calculate the total amount of liquid in the 12 containers at the end of 12 months. [5]
SPS SPS SM Pure 2023 September Q7
5 marks Moderate -0.3
\includegraphics{figure_7} The figure above shows a circular sector \(OAB\) whose centre is at \(O\). The radius of the sector is 60 cm. The points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively, so that \(|OC| = |OD| = 24\) cm. Given that the length of the arc \(AB\) is 48 cm, find the area of the shaded region \(ABDC\), correct to the nearest cm\(^2\). [5 marks]
SPS SPS SM Pure 2023 September Q8
9 marks Moderate -0.3
A cubic curve \(C\) has equation $$y = (3-x)(4+x)^2.$$
  1. Sketch the graph of \(C\). [3] The sketch must include any points where the graph meets the coordinate axes.
  2. Sketch in separate diagrams the graph of \(\ldots\)
    1. \(\ldots y = (3-2x)(4+2x)^2\). [2]
    2. \(\ldots y = (3+x)(4-x)^2\). [2]
    3. \(\ldots y = (2-x)(5+x)^2\). [2]
    Each of the sketches must include any points where the graph meets the coordinate axes.
SPS SPS SM Pure 2023 September Q9
6 marks Challenging +1.2
Solve the following trigonometric equation in the range given. $$4\tan^2\theta\cos\theta = 15, \quad 0 \leq \theta < 360°.$$ [6 marks]
SPS SPS SM Pure 2023 September Q10
12 marks Standard +0.3
\includegraphics{figure_10} The figure above shows solid right prism of height \(h\) cm. The cross section of the prism is a circular sector of radius \(r\) cm, subtending an angle of 2 radians at the centre.
  1. Given that the volume of the prism is 1000 cm\(^3\), show clearly that $$S = 2r^2 + \frac{4000}{r},$$ where \(S\) cm\(^2\) is the total surface area of the prism. [5]
  2. Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer. [7]
SPS SPS SM Pure 2023 September Q11
4 marks Moderate -0.3
It is given that $$f(x) = x^2 - kx + (k+3),$$ where \(k\) is a constant. If the equation \(f(x) = 0\) has real roots find the range of the possible values of \(k\). [4]
SPS SPS SM Pure 2023 September Q12
8 marks Standard +0.3
\includegraphics{figure_12} The figure above shows the curve \(C\) with equation $$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$
  1. Determine the coordinates of the minimum point of \(C\), labelled as \(M\). [5]
The point \(N\) lies on the \(x\) axis so that \(MN\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(MN\) and the straight line with equation \(x = 1\).
  1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [3]
SPS SPS SM Pure 2023 September Q13
6 marks Standard +0.8
Prove or disprove each of the following statements:
  1. If \(n\) is an integer, then \(3n^2 - 11n + 13\) is a prime number. [2]
  2. If \(x\) is a real number, then \(x^2 - 8x + 17\) is positive. [2]
  3. If \(p\) and \(q\) are irrational numbers, then \(pq\) is irrational. [2]
SPS SPS SM Pure 2023 September Q14
8 marks Standard +0.3
\includegraphics{figure_14} The diagram above shows the curve with equation $$y = (x-4)^2, \quad x \in \mathbb{R},$$ intersected by the straight line with equation \(y = 4\), at the points \(A\) and \(B\). The curve meets the \(y\) axis at the point \(C\). Calculate the exact area of the shaded region, bounded by the curve and the straight line segments \(AB\) and \(BC\). [8]
SPS SPS FM Pure 2023 February Q1
4 marks Moderate -0.8
Find \(\sum_{r=1}^{n}(2r^2 - 1)\), expressing your answer in fully factorised form. [4]
SPS SPS FM Pure 2023 February Q2
4 marks Moderate -0.8
Solve the equation \(2z - 5iz^* = 12\). [4]
SPS SPS FM Pure 2023 February Q3
3 marks Standard +0.8
In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac{1}{2}x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{2}\pi\). \includegraphics{figure_4} This region is rotated through \(2\pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated. [3]
SPS SPS FM Pure 2023 February Q4
8 marks Standard +0.3
The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) is perpendicular to \(\Pi\). [2]
  2. Hence find a Cartesian equation of \(\Pi\). [2]
The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)
  1. determine the possible coordinates of \(A\). [4]
SPS SPS FM Pure 2023 February Q5
6 marks Standard +0.3
Prove by induction that for all positive integers \(n\) $$f(n) = 3^{2n+4} - 2^{2n}$$ is divisible by 5 [6]
SPS SPS FM Pure 2023 February Q6
4 marks Standard +0.8
In this question you must show detailed reasoning. Find \(\int_{2}^{\infty} \frac{1}{4+x^2} \, dx\). [4]
SPS SPS FM Pure 2023 February Q7
10 marks Challenging +1.3
  1. Prove that $$\tanh^{-1}(x) = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad -k < x < k$$ stating the value of the constant \(k\). [5]
  2. Hence, or otherwise, solve the equation $$2x = \tanh\left(\ln \sqrt{2-3x}\right)$$ [5]
SPS SPS FM Pure 2023 February Q8
6 marks Challenging +1.8
The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\) The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\) Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\). [6]
SPS SPS FM Pure 2023 February Q9
7 marks Standard +0.8
  1. Sketch, on the Argand diagram below, the locus of points satisfying the equation \(|z - 3| = 2\) [1]
\includegraphics{figure_9}
  1. There is a unique complex number \(w\) that satisfies both \(|w - 3| = 2\) and \(\arg(w + 1) = \alpha\) where \(\alpha\) is a constant such that \(0 < \alpha < \pi\).
    1. Find the value of \(\alpha\). [2]
    2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4]
SPS SPS FM Pure 2023 February Q10
10 marks Challenging +1.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + \frac{2y}{x} = \frac{x+3}{x(x-1)(x^2+3)} \quad (x > 1)$$ [8]
  2. Find the particular solution for which \(y = 0\) when \(x = 3\). Give your answer in the form \(y = f(x)\). [2]
SPS SPS FM Pure 2023 February Q11
9 marks Challenging +1.2
In an Argand diagram, the points \(A\), \(B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2i\).
  1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and exact. [6]
The points \(D\), \(E\) and \(F\) are the midpoints of the sides of triangle \(ABC\).
  1. Find the exact area of triangle \(DEF\). [3]
SPS SPS FM Pure 2023 February Q12
11 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$
  1. Find the values of \(k\) for which the matrix \(\mathbf{M}\) has an inverse. [2]
  2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect \begin{align} 2x - y + z &= p
    3x - 6y + 4z &= 1
    3x + 2y - z &= 0 \end{align} [5]
    1. Find the value of \(q\) for which the set of simultaneous equations \begin{align} 2x - y + z &= 1
      3x - 5y + 4z &= q
      3x + 2y - z &= 0 \end{align} can be solved.
    2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically. [4]
SPS SPS FM Pure 2023 February Q13
11 marks Challenging +1.8
In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin \theta} e^{\frac{1}{2}\cos \theta}\) for \(0 \leqslant \theta \leqslant \pi\). \includegraphics{figure_13}
  1. Find the exact area enclosed by the curve. [4]
  2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}} e^{\frac{1}{6}}\). [7]
SPS SPS FM Pure 2023 February Q14
7 marks Challenging +1.3
  1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln\left(\frac{1}{2} + \cos x\right)\). [4]
  2. By considering the root of the equation \(\ln\left(\frac{1}{2} + \cos x\right) = 0\) deduce that \(\pi \approx 3\sqrt{3 \ln\left(\frac{3}{2}\right)}\). [3]
SPS SPS FM 2024 October Q1
8 marks Moderate -0.8
    1. Show that \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}}\) can be written in the form \(\frac{a}{b+cx}\), where \(a\), \(b\) and \(c\) are constants to be determined. [2]
    2. Hence solve the equation \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}} = 2\). [2]
  2. In this question you must show detailed reasoning. Solve the equation \(2^{2x} - 7 \times 2^x - 8 = 0\). [4]
SPS SPS FM 2024 October Q2
5 marks Easy -1.2
  1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
  2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]