Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2015 June

Determine the volume of tetrahedron \(OABC\), where \(O\) is the origin and \(A\), \(B\) and \(C\) are, respectively, the points \((2, 3, -2)\), \((2, 0, 4)\) and \((6, 1, 7)\). [3]

The Taylor series expansion, about \(x = 1\), of the function \(y\) is $$y = 1 + \sum_{n=1}^{\infty} \frac{(-2)^{n-1}(x-1)^n}{1 \times 3 \times 5 \times \ldots \times (2n-1)}.$$ Find the value of \(\frac{\text{d}^4 y}{\text{d}x^4}\) when \(x = 1\). [3]

\(\mathbf{M}\) is the matrix \(\begin{pmatrix} 1 & -2 & 2 \\ 2 & -1 & 2 \\ 2 & -2 & 3 \end{pmatrix}\). Use induction to prove that, for all positive integers \(n\), $$\mathbf{M}^n \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2n + 1 \\ 2n^2 + 2n \\ 2n^2 + 2n + 1 \end{pmatrix}.$$ [6]

A curve has polar equation \(r = \sin \frac{1}{4}\theta\) for \(0 \leqslant \theta < 2\pi\).

1. Sketch the curve. [3]
2. Determine the area of the region enclosed by the curve. [4]

A curve has equation \(y = \frac{2x^2 + 5x - 25}{x - 3}\).

1. Determine the equations of the asymptotes. [3]
2. Find the coordinates of the turning points. [5]
3. Sketch the curve. [3]

1. Given the complex number \(z = \cos \theta + \text{i} \sin \theta\), show that \(z^n + \frac{1}{z^n} = 2 \cos n\theta\). [1]
2. Deduce the identity \(16 \cos^5 \theta \equiv \cos 5\theta + 5 \cos 3\theta + 10 \cos \theta\). [4]
3. For \(0 < \theta < 2\pi\), solve the equation \(\cos 5\theta + 5 \cos 3\theta + 9 \cos \theta = 0\). [4]

1. On an Argand diagram, shade the region whose points satisfy $$|z - 20 + 15\text{i}| \leqslant 7.$$ [3]
2. The complex number \(z_1\) represents that value of \(z\) in the region described in part (i) for which \(\arg(z)\) is least. Mark \(z_1\) on your Argand diagram and determine \(\arg(z_1)\) correct to 3 decimal places. [4]

The group \(G\), of order 8, consists of the elements \(\{e, a, b, c, ab, bc, ca, abc\}\), together with a multiplicative binary operation, where \(e\) is the identity and $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb \quad \text{and} \quad ca = ac.$$

1. Construct the group table of \(G\). [You are not required to show how individual elements of the table are determined.] [4]
2. List all the proper subgroups of \(G\). [5]

The differential equation \((\star)\) is $$\frac{\text{d}^2 u}{\text{d}x^2} + 4u = 8x + 1.$$

1. Find the general solution of \((\star)\). [5]
2. The differential equation \((\star \star)\) is $$x \frac{\text{d}^2 v}{\text{d}x^2} + 2 \frac{\text{d}v}{\text{d}x} + 4xv = 8x + 1.$$ By using the substitution \(u = xv\), show that \((\star)\) becomes \((\star \star)\) and deduce the general solution of \((\star \star)\). [4]

1. Find a vector equation for the line of intersection of the planes with cartesian equations $$x + 7y - 6z = -10 \quad \text{and} \quad 3x - 5y + 8z = 48.$$ [5]
2. Determine the value of \(k\) for which the system of equations \begin{align} x + 7y - 6z &= -10
   3x - 5y + 8z &= 48
   kx + 2y + 3z &= 16 \end{align} does not have a unique solution and show that, for this value of \(k\), the system of equations is inconsistent. [6]

1. The cubic equation \(x^3 + 2x^2 + 3x - 4 = 0\) has roots \(p\), \(q\) and \(r\). A second cubic equation has roots \(qr\), \(rp\) and \(pq\). Show how the substitution \(y = \frac{4}{x}\) can be used to determine this second equation. Hence, or otherwise, find this equation in the form \(y^3 + ay^2 + by + c = 0\). [6]
2. The cubic equation \(x^3 - 4x^2 + 5x - 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha\) is real and positive, and that \(\beta\) and \(\gamma\) are complex.
   1. Describe the relationship between \(\beta\) and \(\gamma\). [1]
   2. Explain why \(|\beta| = \frac{2}{\sqrt{\alpha}}\). [2]
   3. Verify that \(\alpha = 2.70\) correct to 3 significant figures, and deduce that \(\text{Re}(\beta) = 0.65\) correct to 2 significant figures. [4]

Let \(I_n = \int_0^2 x^n \sqrt{1 + 2x^2} \, \text{d}x\) for \(n = 0, 1, 2, 3, \ldots\).

1. 1. Evaluate \(I_1\). [3]
   2. Prove that, for \(n \geqslant 2\), $$(2n + 4)I_n = 27 \times 2^{n-1} - (n - 1)I_{n-2}.$$ [6]
   3. Using a suitable substitution, or otherwise, show that $$I_0 = 3 + \frac{1}{\sqrt{2}} \ln(1 + \sqrt{2}).$$ [8]
2. The curve \(y = \frac{1}{\sqrt{2}} x^2\), between \(x = 0\) and \(x = 2\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface with area \(S\). Find the exact value of \(S\). [5]

1. By sketching a suitable triangle, show that \(\tan^{-1} a + \tan^{-1} \left(\frac{1}{a}\right) = \frac{1}{4}\pi\), for \(a > 0\). [1]
2. Given that \(a\) and \(b\) are positive and less than 1, express \(\tan(\tan^{-1} a \pm \tan^{-1} b)\) in terms of \(a\) and \(b\). [2]
3. By letting \(a = \frac{1}{n-1}\) and \(b = \frac{1}{n+1}\), use the method of differences to prove that $$\sum_{n=1}^{\infty} \tan^{-1} \left(\frac{2}{n^2}\right) = \frac{3}{4}\pi.$$ [7]