Questions — Edexcel (10514 questions)

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Edexcel F3 2021 June Q7
8 marks Challenging +1.2
Using calculus, find the exact values of
  1. \(\int_1^2 \frac{1}{x^2 - 4x + 5} \, dx\) [3]
  2. \(\int_{\sqrt{3}}^3 \frac{\sqrt{x^2 - 3}}{x^2} \, dx\) [5]
Edexcel F3 2021 June Q8
14 marks Challenging +1.8
The hyperbola \(H\) has equation $$4x^2 - y^2 = 4$$
  1. Write down the equations of the asymptotes of \(H\). [1]
  2. Find the coordinates of the foci of \(H\). [2]
The point \(P(\sec \theta, 2 \tan \theta)\) lies on \(H\).
  1. Using calculus, show that the equation of the tangent to \(H\) at the point \(P\) is $$y \tan \theta = 2x \sec \theta - 2$$ [4]
The point \(V(-1, 0)\) and the point \(W(1, 0)\) both lie on \(H\). The point \(Q(\sec \theta, -2 \tan \theta)\) also lies on \(H\). Given that \(P\), \(Q\), \(V\) and \(W\) are distinct points on \(H\) and that the lines \(VP\) and \(WQ\) intersect at the point \(S\),
  1. show that, as \(\theta\) varies, \(S\) lies on an ellipse with equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where \(a\) and \(b\) are integers to be found. [7]
Edexcel F3 2018 Specimen Q1
6 marks Standard +0.3
The curve \(C\) has equation $$y = 9 \cosh x + 3 \sinh x + 7x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm. [6]
Edexcel F3 2018 Specimen Q2
11 marks Challenging +1.2
An ellipse has equation $$\frac{x^2}{25} + \frac{y^2}{4} = 1$$ The point \(P\) lies on the ellipse and has coordinates \((5\cos \theta, 2\sin \theta)\), \(0 < \theta < \frac{\pi}{2}\) The line \(L\) is a normal to the ellipse at the point \(P\).
  1. Show that an equation for \(L\) is $$5x \sin \theta - 2y \cos \theta = 21 \sin \theta \cos \theta$$ [5]
Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(PQ\),
  1. find the exact area of triangle \(OPM\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2\theta\) [6]
Edexcel F3 2018 Specimen Q3
12 marks Challenging +1.2
Without using a calculator, find
  1. \(\int_{-2}^{1} \frac{1}{x^2 + 4x + 13} \, dx\), giving your answer as a multiple of \(\pi\), [5]
  2. \(\int_{-1}^{4} \frac{1}{\sqrt{4x^2 - 12x + 34}} \, dx\), giving your answer in the form \(p \ln\left(q + r\sqrt{2}\right)\), [7]
where \(p\), \(q\) and \(r\) are rational numbers to be found.
Edexcel F3 2018 Specimen Q4
9 marks Standard +0.3
$$\mathbf{M} = \begin{pmatrix} 1 & k & 0 \\ -1 & 1 & 1 \\ 1 & k & 3 \end{pmatrix}, \text{ where } k \text{ is a constant}$$
  1. Find \(\mathbf{M}^{-1}\) in terms of \(k\). [5]
Hence, given that \(k = 0\)
  1. find the matrix \(\mathbf{N}\) such that $$\mathbf{MN} = \begin{pmatrix} 3 & 5 & 6 \\ 4 & -1 & 1 \\ 3 & 2 & -3 \end{pmatrix}$$ [4]
Edexcel F3 2018 Specimen Q5
7 marks Challenging +1.3
Given that \(y = \text{artanh}(\cos x)\)
  1. show that $$\frac{dy}{dx} = -\text{cosec } x$$ [2]
  2. Hence find the exact value of $$\int_{0}^{\frac{\pi}{4}} \cos x \, \text{artanh}(\cos x) \, dx$$ giving your answer in the form \(a \ln\left(b + c\sqrt{3}\right) + d\pi\), where \(a\), \(b\), \(c\) and \(d\) are rational numbers to be found. [5]
Edexcel F3 2018 Specimen Q6
9 marks Challenging +1.2
The coordinates of the points \(A\), \(B\) and \(C\) relative to a fixed origin \(O\) are \((1, 2, 3)\), \((-1, 3, 4)\) and \((2, 1, 6)\) respectively. The plane \(\Pi\) contains the points \(A\), \(B\) and \(C\).
  1. Find a cartesian equation of the plane \(\Pi\). [5]
The point \(D\) has coordinates \((k, 4, 14)\) where \(k\) is a positive constant. Given that the volume of the tetrahedron \(ABCD\) is 6 cubic units,
  1. find the value of \(k\). [4]
Edexcel F3 2018 Specimen Q7
11 marks Challenging +1.2
The curve \(C\) has parametric equations $$x = 3t^4, \quad y = 4t^3, \quad 0 \leq t \leq 1$$ The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).
  1. Show that $$S = k\pi \int_{0}^{1} t^2(t^2 + 1)^{\frac{1}{2}} dt$$ where \(k\) is a constant to be found. [4]
  2. Use the substitution \(u^2 = t^2 + 1\) to find the value of \(S\), giving your answer in the form \(p\pi\left(11\sqrt{2} - 4\right)\) where \(p\) is a rational number to be found. [7]
Edexcel F3 2018 Specimen Q8
10 marks Challenging +1.8
$$I_n = \int_{0}^{\ln 2} \tanh^{2n} x \, dx, \quad n \geq 0$$
  1. Show that, for \(n \geq 1\) $$I_n = I_{n-1} - \frac{1}{2n-1}\left(\frac{3}{5}\right)^{2n-1}$$ [5]
  2. Hence show that $$\int_{0}^{\ln 2} \tanh^{-1} x \, dx = p + \ln 2$$ where \(p\) is a rational number to be found. [5]
Edexcel FP3 2011 June Q1
5 marks Challenging +1.2
The curve \(C\) has equation \(y = 2x^3\), \(0 \leq x \leq 2\). The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, find the area of the surface generated, giving your answer to 3 significant figures. [5]
Edexcel FP3 2011 June Q2
8 marks Standard +0.8
  1. Given that \(y = x \arcsin x\), \(0 \leq x \leq 1\), find
    1. an expression for \(\frac{dy}{dx}\),
    2. the exact value of \(\frac{dy}{dx}\) when \(x = \frac{1}{2}\).
    [3]
  2. Given that \(y = \arctan(3e^{2x})\), show that $$\frac{dy}{dx} = \frac{3}{5\cosh 2x + 4\sinh 2x}.$$ [5]
Edexcel FP3 2011 June Q3
9 marks Challenging +1.2
Show that
  1. \(\int_5^8 \frac{1}{x^2 - 10x + 34} dx = k\pi\), giving the value of the fraction \(k\), [5]
  2. \(\int_5^8 \frac{1}{\sqrt{x^2 - 10x + 34}} dx = \ln(A + \sqrt{n})\), giving the values of the integers \(A\) and \(n\). [4]
Edexcel FP3 2011 June Q4
8 marks Challenging +1.2
$$I_n = \int_1^e x^2 (\ln x)^n dx, \quad n \geq 0$$
  1. Prove that, for \(n \geq 1\), $$I_n = \frac{e^3}{3} - \frac{n}{3} I_{n-1}$$ [4]
  2. Find the exact value of \(I_3\). [4]
Edexcel FP3 2011 June Q5
9 marks Standard +0.8
The curve \(C_1\) has equation \(y = 3\sinh 2x\), and the curve \(C_2\) has equation \(y = 13 - 3e^{2x}\).
  1. Sketch the graph of the curves \(C_1\) and \(C_2\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes. [4]
  2. Solve the equation \(3\sinh 2x = 13 - 3e^{2x}\), giving your answer in the form \(\frac{1}{2}\ln k\), where \(k\) is an integer. [5]
Edexcel FP3 2011 June Q6
10 marks Standard +0.3
The plane \(P\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix}$$
  1. Find a vector perpendicular to the plane \(P\). [2] The line \(l\) passes through the point \(A(1, 3, 3)\) and meets \(P\) at \((3, 1, 2)\). The acute angle between the plane \(P\) and the line \(l\) is \(\alpha\).
  2. Find \(\alpha\) to the nearest degree. [4]
  3. Find the perpendicular distance from \(A\) to the plane \(P\). [4]
Edexcel FP3 2011 June Q7
12 marks Challenging +1.2
The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} k & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}, \quad k \neq 1$$
  1. Show that \(\det \mathbf{M} = 2 - 2k\). [2]
  2. Find \(\mathbf{M}^{-1}\), in terms of \(k\). [5] The straight line \(l_1\) is mapped onto the straight line \(l_2\) by the transformation represented by the matrix \(\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}\). The equation of \(l_2\) is \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\), where \(\mathbf{a} = 4\mathbf{i} + \mathbf{j} + 7\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
  3. Find a vector equation for the line \(l_1\). [5]
Edexcel FP3 2011 June Q8
14 marks Challenging +1.3
The hyperbola \(H\) has equation $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
  1. Use calculus to show that the equation of the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\) may be written in the form $$xb\cosh\theta - ya\sinh\theta = ab$$ [4] The line \(l_1\) is the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\), \(\theta \neq 0\). Given that \(l_1\) meets the \(x\)-axis at the point \(P\),
  2. find, in terms of \(a\) and \(\theta\), the coordinates of \(P\). [2] The line \(l_2\) is the tangent to \(H\) at the point \((a, 0)\). Given that \(l_1\) and \(l_2\) meet at the point \(Q\),
  3. find, in terms of \(a\), \(b\) and \(\theta\), the coordinates of \(Q\). [2]
  4. Show that, as \(\theta\) varies, the locus of the mid-point of \(PQ\) has equation $$x(4y^2 + b^2) = ab^2$$ [6]
Edexcel FP3 2014 June Q1
8 marks Standard +0.3
The line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is $$\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3$$ Find
  1. a vector equation of the line \(l\), [2]
  2. the position vector of the point where \(l\) meets \(\Pi\). [4]
  3. Hence find the perpendicular distance of \(P\) from \(\Pi\). [2]
Edexcel FP3 2014 June Q2
13 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$
  1. Show that matrix \(\mathbf{M}\) is not orthogonal. [2]
  2. Using algebra, show that \(1\) is an eigenvalue of \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). [5]
  3. Find an eigenvector of \(\mathbf{M}\) which corresponds to the eigenvalue \(1\) [2]
The transformation \(M : \mathbb{R}^3 \to \mathbb{R}^3\) is represented by the matrix \(\mathbf{M}\).
  1. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac{y}{2} = \frac{z}{-1}$$ [4]
Edexcel FP3 2014 June Q3
8 marks Standard +0.8
Using calculus, find the exact value of
  1. \(\int_1^2 \frac{1}{\sqrt{x^2 - 2x + 3}} \, dx\) [4]
  2. \(\int_0^1 e^{-x} \sinh x \, dx\) [4]
Edexcel FP3 2014 June Q4
7 marks Standard +0.3
Using the definitions of hyperbolic functions in terms of exponentials,
  1. show that $$\operatorname{sech}^2 x = 1 - \tanh^2 x$$ [3]
  2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$ [4]
Edexcel FP3 2014 June Q5
4 marks Standard +0.8
Given that \(y = \arctan \frac{x}{\sqrt{1 + x^2}}\) show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1 + x^2}}\) [4]
Edexcel FP3 2014 June Q6
10 marks Challenging +1.3
[In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.] The points \(P(3\cos \alpha, 2\sin \alpha)\) and \(Q(3\cos \beta, 2\sin \beta)\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$
  1. Show the equation of the chord \(PQ\) is $$\frac{x}{3}\cos\frac{(\alpha + \beta)}{2} + \frac{y}{2}\sin\frac{(\alpha + \beta)}{2} = \cos\frac{(\alpha - \beta)}{2}$$ [4]
  2. Write down the coordinates of the mid-point of \(PQ\). [1]
Given that the gradient, \(m\), of the chord \(PQ\) is a constant,
  1. show that the centre of the chord lies on a line $$y = -kx$$ expressing \(k\) in terms of \(m\). [5]
Edexcel FP3 2014 June Q7
9 marks Standard +0.8
A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x^2 + y^2 = r^2\) where \(r\) is a constant.
  1. Show that \(1 + \left(\frac{dy}{dx}\right)^2 = \frac{r^2}{r^2 - x^2}\) [3]
  2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4\pi r^2\). [5]
  3. Write down the length of the arc of the curve \(y = \sqrt{1 - x^2}\) from \(x = 0\) to \(x = 1\) [1]