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CAIE P3 2018 November Q1
4 marks Standard +0.8
Find the set of values of \(x\) satisfying the inequality \(2|2x - a| < |x + 3a|\), where \(a\) is a positive constant. [4]
CAIE P3 2018 November Q2
4 marks Moderate -0.3
Showing all necessary working, solve the equation \(\frac{2e^x + e^{-x}}{e^x - e^{-x}} = 4\), giving your answer correct to 2 decimal places. [4]
CAIE P3 2018 November Q3
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation \(x^3 = 3 - x\) has exactly one real root. [2]
  2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{2x_n^3 + 3}{3x_n^2 + 1}$$ converges, then it converges to the root of the equation in part (i). [2]
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
CAIE P3 2018 November Q4
7 marks Standard +0.3
The parametric equations of a curve are $$x = 2\sin\theta + \sin 2\theta, \quad y = 2\cos\theta + \cos 2\theta,$$ where \(0 < \theta < \pi\).
  1. Obtain an expression for \(\frac{dy}{dx}\) in terms of \(\theta\). [3]
  2. Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the \(y\)-axis. [4]
CAIE P3 2018 November Q5
7 marks Standard +0.3
The coordinates \((x, y)\) of a general point on a curve satisfy the differential equation $$x\frac{dy}{dx} = (2 - x^2)y.$$ The curve passes through the point \((1, 1)\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\). [7]
CAIE P3 2018 November Q6
8 marks Standard +0.8
  1. Show that the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\) can be expressed in the form \(R\sin(x - \alpha) = \sqrt{2}\), where \(R > 0\) and \(0° < \alpha < 90°\). [4]
  2. Hence solve the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\), for \(0° < x < 180°\). [4]
CAIE P3 2018 November Q7
9 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve \(y = 5\sin^2 x \cos^3 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places. [5]
  2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]
CAIE P3 2018 November Q8
9 marks Standard +0.3
  1. Showing all necessary working, express the complex number \(\frac{2 + 3i}{1 - 2i}\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures. [5]
  2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(|z - 3 + 2i| = 1\). Find the least value of \(|z|\) for points on this locus, giving your answer in an exact form. [4]
CAIE P3 2018 November Q9
10 marks Standard +0.3
Let \(f(x) = \frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence, showing all necessary working, show that \(\int_{-1}^0 f(x) dx = 1 + \frac{1}{2}\ln\left(\frac{4}{3}\right)\). [5]
CAIE P3 2018 November Q10
10 marks Standard +0.3
The planes \(m\) and \(n\) have equations \(3x + y - 2z = 10\) and \(x - 2y + 2z = 5\) respectively. The line \(l\) has equation \(\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})\).
  1. Show that \(l\) is parallel to \(m\). [3]
  2. Calculate the acute angle between the planes \(m\) and \(n\). [3]
  3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2. Find the position vectors of the two possible positions of \(P\). [4]
CAIE Further Paper 1 2024 November Q1
10 marks Standard +0.3
The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k\) (\(k \neq 0\)), followed by a shear, \(x\)-axis fixed, with \((0, 1)\) mapped to \((k, 1)\).
  1. Show that \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
  2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]
The unit square \(S\) in the \(x\)-\(y\) plane is transformed by \(\mathbf{M}\) onto the parallelogram \(P\).
  1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
  2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]
CAIE Further Paper 1 2024 November Q2
6 marks Challenging +1.2
Prove by mathematical induction that, for all positive integers \(n\), $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\tan^{-1}x\right) = P_n(x)\left(1+x^2\right)^{-n},$$ where \(P_n(x)\) is a polynomial of degree \(n-1\). [6]
CAIE Further Paper 1 2024 November Q3
10 marks Challenging +1.8
The quartic equation \(x^4 + 2x^3 - 1 = 0\) has roots \(\alpha, \beta, \gamma, \delta\).
  1. Find a quartic equation whose roots are \(\alpha^4, \beta^4, \gamma^4, \delta^4\) and state the value of \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\). [5]
  2. Find the value of \(\alpha^5 + \beta^5 + \gamma^5 + \delta^5\). [3]
  3. Find the value of \(\alpha^8 + \beta^8 + \gamma^8 + \delta^8\). [2]
CAIE Further Paper 1 2024 November Q4
8 marks Challenging +1.2
  1. Use the method of differences to find \(\sum_{r=1}^{n} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. [4]
It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).
  1. Find the value of \(k\). [2]
  2. Hence find \(\sum_{r=1}^{n-1} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\). [2]
CAIE Further Paper 1 2024 November Q5
13 marks Challenging +1.2
  1. Show that the curve with Cartesian equation $$\left(x^2+y^2\right)^2 = 6xy$$ has polar equation \(r^2 = 3\sin 2\theta\). [2]
The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).
  1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
  2. Find the area of the region enclosed by \(C\). [2]
  3. Find the maximum distance of a point on \(C\) from the initial line. [6]
CAIE Further Paper 1 2024 November Q6
13 marks Challenging +1.2
The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).
  1. Find the equations of the asymptotes of \(C\). [2]
  2. Find the coordinates of any stationary points on \(C\). [4]
  3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
  4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]
CAIE Further Paper 1 2024 November Q7
15 marks Challenging +1.3
The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu(\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) respectively. The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).
  1. Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\). [4]
The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).
  1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [4]
The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).
  1. Find a vector equation for \(PQ\). [7]
CAIE Further Paper 1 2024 November Q4
8 marks Challenging +1.2
  1. Use the method of differences to find \(\sum_{r=1}^n \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. [4]
It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).
  1. Find the value of \(k\). [2]
  2. Hence find \(\sum_{r=7}^{n+5} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\). [2]
CAIE Further Paper 1 2024 November Q5
13 marks Challenging +1.2
  1. Show that the curve with Cartesian equation \(\left(x^2 + y^2\right)^2 = 6xy\) has polar equation \(r^2 = 3\sin 2\theta\). [2]
The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).
  1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
  2. Find the area of the region enclosed by \(C\). [2]
  3. Find the maximum distance of a point on \(C\) from the initial line. [6]
CAIE FP1 2003 November Q1
6 marks Challenging +1.2
\includegraphics{figure_1} The curve \(C\) has polar equation $$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$ where \(0 \leq \theta \leq \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]
CAIE FP1 2003 November Q2
6 marks Challenging +1.2
Given that $$u_n = \frac{1}{n^2 - n + 1} - \frac{1}{n^2 + n + 1},$$ find \(S_N = \sum_{n=N+1}^{2N} u_n\) in terms of \(N\). [3] Find a number \(M\) such that \(S_N < 10^{-20}\) for all \(N > M\). [3]
CAIE FP1 2003 November Q3
6 marks Challenging +1.2
Three \(n \times 1\) column vectors are denoted by \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\), and \(\mathbf{M}\) is an \(n \times n\) matrix. Show that if \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\) are linearly dependent then the vectors \(\mathbf{Mx}_1\), \(\mathbf{Mx}_2\), \(\mathbf{Mx}_3\) are also linearly dependent. [2] The vectors \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) and the matrix \(\mathbf{P}\) are defined as follows: $$\mathbf{y}_1 = \begin{pmatrix} 1 \\ 5 \\ 7 \end{pmatrix}, \quad \mathbf{y}_2 = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}, \quad \mathbf{y}_3 = \begin{pmatrix} 5 \\ 51 \\ 55 \end{pmatrix},$$ $$\mathbf{P} = \begin{pmatrix} 1 & -4 & 3 \\ 0 & 2 & 5 \\ 0 & 0 & -7 \end{pmatrix}$$
  1. Show that \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) are linearly dependent. [2]
  2. Find a basis for the linear space spanned by the vectors \(\mathbf{Py}_1\), \(\mathbf{Py}_2\), \(\mathbf{Py}_3\). [2]
CAIE FP1 2003 November Q4
8 marks Challenging +1.2
Given that \(y = x \sin x\), find \(\frac{d^2y}{dx^2}\) and \(\frac{d^4y}{dx^4}\), simplifying your results as far as possible, and show that $$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3] Use induction to establish an expression for \(\frac{d^{2n}y}{dx^{2n}}\), where \(n\) is a positive integer. [5]
CAIE FP1 2003 November Q5
8 marks Challenging +1.8
The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{1}{4}\pi} \sec^n x \, dx.$$ By considering \(\frac{d}{dx}(\tan x \sec^n x)\), or otherwise, show that $$(n + 1)I_{n+2} = 2^{\frac{4n}{n}} + nI_n.$$ [4] Find the value of \(I_6\). [4]
CAIE FP1 2003 November Q6
9 marks Challenging +1.2
Find the sum of the squares of the roots of the equation $$x^3 + x + 12 = 0,$$ and deduce that only one of the roots is real. [4] The real root of the equation is denoted by \(\alpha\). Prove that \(-3 < \alpha < -2\), and hence prove that the modulus of each of the other roots lies between 2 and \(\sqrt{6}\). [5]