Edexcel FP3 (Further Pure Mathematics 3)

An ellipse has equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\).

1. Sketch the ellipse. [1]
2. Find the value of the eccentricity \(e\). [2]
3. State the coordinates of the foci of the ellipse. [2]

Solve the equation $$10 \cosh x + 2 \sinh x = 11.$$ Give each answer in the form \(\ln a\) where \(a\) is a rational number. [7]

$$I_n = \int_0^{\frac{\pi}{2}} x^n \cos x \, dx, \quad n \geq 0.$$

1. Prove that \(I_n = \left(\frac{\pi}{2}\right)^n - n(n-1)I_{n-2}\), \(n \geq 2\). [5]
2. Find an exact expression for \(I_6\). [4]

1. Given that \(y = \arctan 3x\), and assuming the derivative of \(\tan x\), prove that $$\frac{dy}{dx} = \frac{3}{1 + 9x^2}.$$ [4]
2. Show that $$\int_0^{\frac{\sqrt{3}}{3}} 6x \arctan 3x \, dx = \frac{1}{3}(4\pi - 3\sqrt{3}).$$ [6]

\includegraphics{figure_6} The curve \(C\) shown in Fig. 1 has equation \(y^2 = 4x\), \(0 \leq x \leq 1\). The part of the curve in the first quadrant is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Show that the surface area of the solid generated is given by $$4\pi \int_0^1 \sqrt{1+x} \, dx.$$ [4]
2. Find the exact value of this surface area. [3]
3. Show also that the length of the curve \(C\), between the points \((1, -2)\) and \((1, 2)\), is given by $$2 \int_0^1 \sqrt{\frac{x+1}{x}} \, dx.$$ [3]
4. Use the substitution \(x = \sinh^2 \theta\) to show that the exact value of this length is $$2[\sqrt{2} + \ln(1 + \sqrt{2})].$$ [6]

Prove that \(\sinh(i\pi - \theta) = \sinh \theta\). [4]

$$\mathbf{A} = \begin{pmatrix} 1 & 0 & 4 \\ 0 & 5 & 4 \\ 4 & 4 & 3 \end{pmatrix}.$$

1. Verify that \(\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\) and find the corresponding eigenvalue. [3]
2. Show that \(9\) is another eigenvalue of \(\mathbf{A}\) and find the corresponding eigenvector. [5]
3. Given that the third eigenvector of \(\mathbf{A}\) is \(\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\), write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{D}.$$ [5]

The plane \(\Pi\) passes through the points $$A(-1, -1, 1), B(4, 2, 1) \text{ and } C(2, 1, 0).$$

1. Find a vector equation of the line perpendicular to \(\Pi\) which passes through the point \(D(1, 2, 3)\). [3]
2. Find the volume of the tetrahedron \(ABCD\). [3]
3. Obtain the equation of \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [3]

The perpendicular from \(D\) to the plane \(\Pi\) meets \(\Pi\) at the point \(E\).

1. Find the coordinates of \(E\). [4]
2. Show that \(DE = \frac{11\sqrt{35}}{35}\). [2]

The point \(D'\) is the reflection of \(D\) in \(\Pi\).

1. Find the coordinates of \(D'\). [3]

Find the values of \(x\) for which $$4 \cosh x + \sinh x = 8,$$ giving your answer as natural logarithms. [6]

1. Prove that the derivative of \(\operatorname{artanh} x\), \(-1 < x < 1\), is \(\frac{1}{1-x^2}\). [3]
2. Find \(\int \operatorname{artanh} x \, dx\). [4]

\includegraphics{figure_12} Figure 1 shows the cross-section \(R\) of an artificial ski slope. The slope is modelled by the curve with equation $$y = \frac{10}{\sqrt{4x^2 + 9}}, \quad 0 \leq x \leq 5.$$ Given that 1 unit on each axis represents 10 metres, use integration to calculate the area \(R\). Show your method clearly and give your answer to 2 significant figures. [7]

\includegraphics{figure_13} A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation $$y = a \cosh\left(\frac{x}{a}\right), \quad -ka \leq x \leq ka,$$ where \(a\) and \(k\) are positive constants.

1. Prove that the length of the rope is \(2a \sinh k\). [5]

Given that the length of the rope is \(8a\),

1. find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate. [4]

The curve \(C\) has equation $$y = \operatorname{arcsec} e^x, \quad x > 0, \quad 0 < y < \frac{1}{2}\pi.$$

1. Prove that \(\frac{dy}{dx} = \frac{1}{\sqrt{e^{2x} - 1}}\). [5]
2. Sketch the graph of \(C\). [2]

The point \(A\) on \(C\) has \(x\)-coordinate \(\ln 2\). The tangent to \(C\) at \(A\) intersects the \(y\)-axis at the point \(B\).

1. Find the exact value of the \(y\)-coordinate of \(B\). [4]

$$I_n = \int_0^1 x^n e^x \, dx \text{ and } J_n = \int_0^1 x^n e^{-x} \, dx, \quad n \geq 0.$$

1. Show that, for \(n \geq 1\), $$I_n = e - nI_{n-1}.$$ [2]
2. Find a similar reduction formula for \(J_n\). [3]
3. Show that \(J_2 = 2 - \frac{5}{e}\). [3]
4. Show that \(\int_0^1 x^n \cosh x \, dx = \frac{1}{2}(I_n + J_n)\). [1]
5. Hence, or otherwise, evaluate \(\int_0^1 x^2 \cosh x \, dx\), giving your answer in terms of \(e\). [4]

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 the point \(P(a \sec t, b \tan t)\) is $$ax \sin t + by = (a^2 + b^2) \tan t.$$ [6]

The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and \(S\) is a focus of \(C\). Given that the eccentricity of \(C\) is \(\frac{3}{2}\), and that \(OA = 3OS\), where \(O\) is the origin,

1. determine the possible values of \(t\), for \(0 \leq t < 2\pi\). [8]

Referred to a fixed origin \(O\), the position vectors of three non-collinear points \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. By considering \(\overrightarrow{AB} \times \overrightarrow{AC}\), prove that the area of \(\triangle ABC\) can be expressed in the form \(\frac{1}{2}|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|\). [5]

$$\mathbf{M} = \begin{pmatrix} 4 & -5 \\ 6 & -9 \end{pmatrix}$$

1. Find the eigenvalues of \(\mathbf{M}\). [4]

A transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{M}\). There is a line through the origin for which every point on the line is mapped onto itself under \(T\).

1. Find a cartesian equation of this line. [3]

$$\mathbf{A} = \begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & u \end{pmatrix}, \quad u \neq 1.$$

1. Show that \(\det \mathbf{A} = 2(u - 1)\). [2]
2. Find the inverse of \(\mathbf{A}\). [6]

The image of the vector \(\begin{pmatrix} a \\ b \\ c \end{pmatrix}\) when transformed by the matrix \(\begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & 6 \end{pmatrix}\) is \(\begin{pmatrix} 3 \\ 1 \\ 6 \end{pmatrix}\).

1. Find the values of \(a\), \(b\) and \(c\). [3]

The plane \(\Pi_1\) passes through the \(P\), with position vector \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), and is perpendicular to the line \(L\) with equation $$\mathbf{r} = 3\mathbf{i} - 2\mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}).$$

1. Show that the Cartesian equation of \(\Pi_1\) is \(x - 5y - 3z = -6\). [4]

The plane \(\Pi_2\) contains the line \(L\) and passes through the point \(Q\), with position vector \(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\).

1. Find the perpendicular distance of \(Q\) from \(\Pi_1\). [4]
2. Find the equation of \(\Pi_2\) in the form \(\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}\). [4]

Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of exponentials,

1. prove that \(\cosh^2 x - \sinh^2 x = 1\), [3]
2. solve \(\operatorname{cosech} x - 2 \coth x = 2\), giving your answer in the form \(k \ln a\), where \(k\) and \(a\) are integers. [4]

$$4x^2 + 4x + 17 \equiv (ax + b)^2 + c, \quad a > 0.$$

1. Find the values of \(a\), \(b\) and \(c\). [3]
2. Find the exact value of $$\int_{-0.5}^{1.5} \frac{1}{4x^2 + 4x + 17} \, dx.$$ [4]

An ellipse, with equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), has foci \(S\) and \(S'\).

1. Find the coordinates of the foci of the ellipse. [4]
2. Using the focus-directrix property of the ellipse, show that, for any point \(P\) on the ellipse, $$SP + S'P = 6.$$ [3]

Given that \(y = \sinh^{n-1} x \cosh x\),

1. show that \(\frac{dy}{dx} = (n-1) \sinh^{n-2} x + n \sinh^n x\). [3]

The integral \(I_n\) is defined by \(I_n = \int_0^{\operatorname{arsinh} 1} \sinh^n x \, dx\), \(n \geq 0\).

1. Using the result in part (a), or otherwise, show that $$nI_n = \sqrt{2} - (n-1)I_{n-2}, \quad n \geq 2$$ [2]
2. Hence find the value of \(I_4\). [4]

\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]

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]

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]

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]

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)

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)

\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]

$$I_n = \int_0^1 x^n e^{2x} \, dx, \quad n \geq 0.$$

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]

\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)

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)

1. 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]

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]

$$\mathbf{A} = \begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & k \end{pmatrix}.$$

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]

Evaluate \(\int_1^4 \frac{1}{\sqrt{x^2 - 2x + 17}} \, dx\), giving your answer as an exact logarithm. [5]

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]