Questions F3 (138 questions)

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Edexcel F3 2021 June Q4
8 marks Standard +0.8
  1. \(f(x) = x \arccos x \quad -1 \leq x \leq 1\) Find the exact value of \(f'(0.5)\). [3]
  2. \(g(x) = \arctan(e^{2x})\) Show that $$g''(x) = k \operatorname{sech}(2x) \tanh(2x)$$ where \(k\) is a constant to be found. [5]
Edexcel F3 2021 June Q5
10 marks Challenging +1.8
$$I_n = \int \sec^n x \, dx \quad n \geq 0$$
  1. Prove that for \(n \geq 2\) $$(n-1)I_n = \tan x \sec^{n-2} x + (n-2)I_{n-2}$$ [6]
  2. Hence, showing each step of your working, find the exact value of $$\int_0^{\pi/4} \sec^6 x \, dx$$ [4]
Edexcel F3 2021 June Q6
13 marks Standard +0.8
The line \(l_1\) has equation $$\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{k})$$ and the line \(l_2\) has equation $$\mathbf{r} = 2\mathbf{i} + s\mathbf{j} + \mu(\mathbf{i} - 2\mathbf{j} + \mathbf{k})$$ where \(s\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters. Given that \(l_1\) and \(l_2\) both lie in a common plane \(\Pi_1\)
  1. show that an equation for \(\Pi_1\) is \(3x + y - z = 3\) [4]
  2. Find the value of \(s\). [1]
The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 3\)
  1. Find an equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\) [4]
  2. Find the acute angle between \(\Pi_1\) and \(\Pi_2\) giving your answer in degrees to 3 significant figures. [4]
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]