Questions — Edexcel (10514 questions)

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Edexcel FP3 Q25
12 marks Challenging +1.8
\includegraphics{figure_25} Figure 1 shows the curve with parametric equations $$x = a \cos^3 \theta, \quad y = a \sin^3 \theta, \quad 0 \leq \theta < 2\pi.$$
  1. Find the total length of this curve. [7]
The curve is rotated through \(\pi\) radians about the \(x\)-axis.
  1. Find the area of the surface generated. [5]
Edexcel FP3 Q26
10 marks Standard +0.3
The points \(A\), \(B\) and \(C\) lie on the plane \(\Pi\) and, relative to a fixed origin \(O\), they have position vectors $$\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 4\mathbf{k}, \quad \mathbf{b} = -\mathbf{i} + 2\mathbf{j}, \quad \mathbf{c} = 5\mathbf{i} - 3\mathbf{j} + 7\mathbf{k}$$ respectively.
  1. Find \(\overrightarrow{AB} \times \overrightarrow{AC}\). [4]
  2. Find an equation of \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
The point \(D\) has position vector \(5\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\).
  1. Calculate the volume of the tetrahedron \(ABCD\). [4]
Edexcel FP3 Q27
12 marks Standard +0.8
The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} 1 & 4 & -1 \\ 3 & 0 & p \\ a & b & c \end{pmatrix},$$ where \(p\), \(a\), \(b\) and \(c\) are constants and \(a > 0\). Given that \(\mathbf{M}\mathbf{M}^T = k\mathbf{I}\) for some constant \(k\), find
  1. the value of \(p\), [2]
  2. the value of \(k\), [2]
  3. the values of \(a\), \(b\) and \(c\), [6]
  4. \(|\det \mathbf{M}|\). [2]
Edexcel FP3 Q28
14 marks Standard +0.3
The transformation \(R\) is represented by the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}.$$
  1. Find the eigenvectors of \(\mathbf{A}\). [5]
  2. Find an orthogonal matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}.$$ [5]
  3. Hence describe the transformation \(R\) as a combination of geometrical transformations, stating clearly their order. [4]
Edexcel FP3 Q29
7 marks Standard +0.8
  1. Find \(\int \frac{1+x}{\sqrt{1-4x^2}} \, dx\). [5]
  2. Find, to 3 decimal places, the value of $$\int_0^{0.3} \frac{1+x}{\sqrt{1-4x^2}} \, dx.$$ [2]
(Total 7 marks)
Edexcel FP3 Q30
7 marks Standard +0.3
  1. Show that, for \(x = \ln k\), where \(k\) is a positive constant, $$\cosh 2x = \frac{k^4 + 1}{2k^2}.$$ [3]
Given that \(f(x) = px - \tanh 2x\), where \(p\) is a constant,
  1. find the value of \(p\) for which \(f(x)\) has a stationary value at \(x = \ln 2\), giving your answer as an exact fraction. [4]
(Total 7 marks)
Edexcel FP3 Q31
8 marks Challenging +1.8
\includegraphics{figure_31} Figure 1 shows a sketch of the curve with parametric equations $$x = a \cos^3 t, \quad y = a \sin^3 t, \quad 0 \leq t \leq \frac{\pi}{2},$$ where \(a\) is a positive constant. The curve is rotated through \(2\pi\) radians about the \(x\)-axis. Find the exact value of the area of the curved surface generated. [8]
Edexcel FP3 Q32
8 marks Challenging +1.2
$$I_n = \int_0^1 x^n e^{2x} \, dx, \quad n \geq 0.$$
  1. Prove that, for \(n \geq 1\), $$I_n = \frac{1}{2}(x^n e^{2x} - nI_{n-1}).$$ [3]
  2. Find, in terms of \(e\), the exact value of $$\int_0^1 x^2 e^{2x} \, dx.$$ [5]
Edexcel FP3 Q33
Challenging +1.8
\includegraphics{figure_33} Figure 2 shows a sketch of the curve with equation $$y = x \operatorname{arcosh} x, \quad 1 \leq x \leq 2.$$ The region \(R\), as shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line \(x = 2\). Show that the area of \(R\) is $$\frac{7}{4} \ln(2 + \sqrt{3}) - \frac{\sqrt{3}}{2}.$$ (Total 10 marks)
Edexcel FP3 Q34
13 marks Challenging +1.3
  1. Show that, for \(0 < x \leq 1\), $$\ln \left(\frac{1 - \sqrt{1-x^2}}{x}\right) = -\ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [3]
  2. Using the definition of \(\cosh x\) or \(\operatorname{sech} x\) in terms of exponentials, show that, for \(0 < x \leq 1\), $$\operatorname{arsech} x = \ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [5]
  3. Solve the equation $$3 \tanh^2 x - 4 \operatorname{sech} x + 1 = 0,$$ giving exact answers in terms of natural logarithms. [5]
(Total 13 marks)
Edexcel FP3 Q35
9 marks Challenging +1.3
    1. Explain why, for any two vectors \(\mathbf{a}\) and \(\mathbf{b}\), \(\mathbf{a} \cdot \mathbf{b} \times \mathbf{a} = 0\). [2]
    2. Given vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) such that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\), where \(\mathbf{a} \neq \mathbf{0}\) and \(\mathbf{b} \neq \mathbf{c}\), show that $$\mathbf{b} - \mathbf{c} = \lambda\mathbf{a}, \quad \text{where } \lambda \text{ is a scalar.}$$ [2]
  2. \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are \(2 \times 2\) matrices.
    1. Given that \(\mathbf{A}\mathbf{B} = \mathbf{A}\mathbf{C}\), and that \(\mathbf{A}\) is not singular, prove that \(\mathbf{B} = \mathbf{C}\). [2]
    2. Given that \(\mathbf{A}\mathbf{B} = \mathbf{A}\mathbf{C}\), where \(\mathbf{A} = \begin{pmatrix} 3 & 6 \\ 1 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix}\), find a matrix \(\mathbf{C}\) whose elements are all non-zero. [3]
Edexcel FP3 Q36
10 marks Standard +0.8
The line \(l_1\) has equation $$\mathbf{r} = \mathbf{i} + 6\mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + 3\mathbf{k})$$ and the line \(l_2\) has equation $$\mathbf{r} = 3\mathbf{i} + p\mathbf{j} + \mu(\mathbf{i} - 2\mathbf{j} + \mathbf{k}), \text{ where } p \text{ is a constant.}$$ The plane \(\Pi_1\) contains \(l_1\) and \(l_2\).
  1. Find a vector which is normal to \(\Pi_1\). [2]
  2. Show that an equation for \(\Pi_1\) is \(6x + y - 4z = 16\). [2]
  3. Find the value of \(p\). [1]
The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} + \mathbf{k}) = 2\).
  1. Find an equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form $$(\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}.$$ [5]
Edexcel FP3 Q37
14 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & k \end{pmatrix}.$$
  1. Show that \(\det \mathbf{A} = 20 - 4k\). [2]
  2. Find \(\mathbf{A}^{-1}\). [6]
Given that \(k = 3\) and that \(\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\),
  1. find the corresponding eigenvalue. [2]
Given that the only other distinct eigenvalue of \(\mathbf{A}\) is \(8\),
  1. find a corresponding eigenvector. [4]
Edexcel FP3 Q38
5 marks Challenging +1.2
Evaluate \(\int_1^4 \frac{1}{\sqrt{x^2 - 2x + 17}} \, dx\), giving your answer as an exact logarithm. [5]
Edexcel FP3 Q39
7 marks Standard +0.3
The hyperbola \(H\) has equation \(\frac{x^2}{16} - \frac{y^2}{4} = 1\). Find
  1. the value of the eccentricity of \(H\), [2]
  2. the distance between the foci of \(H\). [2]
The ellipse \(E\) has equation \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).
  1. Sketch \(H\) and \(E\) on the same diagram, showing the coordinates of the points where each curve crosses the axes. [3]
Edexcel FP3 Specimen Q1
Moderate -0.5
Find the eigenvalues of the matrix \(\begin{pmatrix} 7 & 6 \\ 6 & 2 \end{pmatrix}\) (Total 4 marks)
Edexcel FP3 Specimen Q2
Standard +0.3
Find the values of \(x\) for which $$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms. (Total 6 marks)
Edexcel FP3 Specimen Q3
Challenging +1.2
\includegraphics{figure_1} The parametric equations of the curve \(C\) shown in Figure 1 are $$x = a(t - \sin t), \quad y = a(1 - \cos t), \quad 0 \leq t \leq 2\pi$$ Find, by using integration, the length of \(C\). (Total 6 marks)
Edexcel FP3 Specimen Q4
Challenging +1.8
Find \(\int \sqrt{x^2 + 4} \, dx\). (Total 7 marks)
Edexcel FP3 Specimen Q5
7 marks Standard +0.8
Given that \(y = \arcsin x\) prove that
  1. \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\) [3]
  2. \((1-x^2) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} = 0\) [4]
(Total 7 marks)
Edexcel FP3 Specimen Q6
8 marks Challenging +1.2
$$I_n = \int_0^{\pi} x^n \sin x \, dx$$
  1. Show that for \(n \geq 2\) $$I_n = n \left( \frac{\pi}{2} \right)^{n-1} - n(n-1)I_{n-2}$$ [4]
  2. Hence obtain \(I_3\), giving your answers in terms of \(\pi\). [4]
(Total 8 marks)
Edexcel FP3 Specimen Q7
12 marks Standard +0.3
$$\mathbf{A}(x) = \begin{pmatrix} 1 & x & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}, \quad x \neq \frac{5}{2}$$
  1. Calculate the inverse of \(\mathbf{A}(x)\). $$\mathbf{B} = \begin{pmatrix} 1 & 3 & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}$$ [8] The image of the vector \(\begin{pmatrix} p \\ q \\ r \end{pmatrix}\) when transformed by \(\mathbf{B}\) is \(\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}\)
  2. Find the values of \(p\), \(q\) and \(r\). [4]
(Total 14 marks)
Edexcel FP3 Specimen Q8
12 marks Standard +0.3
The points \(A\), \(B\), \(C\), and \(D\) have position vectors $$\mathbf{a} = 2\mathbf{i} + \mathbf{k}, \quad \mathbf{b} = \mathbf{i} + 3\mathbf{j}, \quad \mathbf{c} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}, \quad \mathbf{d} = 4\mathbf{j} + \mathbf{k}$$ respectively.
  1. Find \(\overrightarrow{AB} \times \overrightarrow{AC}\) and hence find the area of triangle \(ABC\). [7]
  2. Find the volume of the tetrahedron \(ABCD\). [2]
  3. Find the perpendicular distance of \(D\) from the plane containing \(A\), \(B\) and \(C\). [3]
(Total 12 marks)
Edexcel FP3 Specimen Q9
13 marks Challenging +1.8
The hyperbola \(C\) has equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
  1. Show that an equation of the normal to \(C\) at \(P(a \sec \theta, b \tan \theta)\) is $$by + ax \sin \theta = (a^2 + b^2)\tan \theta$$ [6] The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(AB\) is \(M\).
  2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies. [7]
(Total 13 marks)
Edexcel M1 2015 January Q1
7 marks Moderate -0.3
A railway truck \(A\) of mass \(m\) and a second railway truck \(B\) of mass \(4m\) are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck \(A\) is \(3u\) and the speed of truck \(B\) is \(2u\). In the collision the trucks join together. Modelling the trucks as particles, find
  1. the speed of \(A\) immediately after the collision, [3]
  2. the direction of motion of \(A\) immediately after the collision, [1]
  3. the magnitude of the impulse exerted by \(A\) on \(B\) in the collision. [3]