Edexcel FP3 (Further Pure Mathematics 3) 2011 June

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]

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]

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]

$$I_n = \int_1^e x^2 (\ln x)^n dx, \quad n \geq 0$$

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]

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]

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]

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]

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]