Questions — OCR MEI (4455 questions)

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OCR MEI Further Extra Pure Specimen Q2
4 marks Challenging +1.2
A binary operation \(*\) is defined on the set \(S = \{p, q, r, s, t\}\) by the following composition table.
\(*\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(q\)\(q\)\(p\)\(s\)\(t\)\(r\)
\(r\)\(r\)\(t\)\(p\)\(q\)\(s\)
\(s\)\(s\)\(r\)\(t\)\(p\)\(q\)
\(t\)\(t\)\(s\)\(q\)\(r\)\(p\)
Determine whether \((S, *)\) is a group. [4]
OCR MEI Further Extra Pure Specimen Q3
12 marks Challenging +1.2
  1. Find the general solution of $$u_n = 8u_{n-1} - 16u_{n-2}, \quad n \geq 2. \quad (*)$$ [4]
A new sequence \(v_n\) is defined by \(v_n = \frac{u_n}{u_{n-1}}\) for \(n \geq 1\).
  1. (A) Use (*) to show that \(v_n = 8 - \frac{16}{v_{n-1}}\) for \(n \geq 2\). [2] (B) Deduce that if \(v_n\) tends to a limit then it must be 4. [2]
  2. Use your general solution in part (i) to show that \(\lim_{n \to \infty} v_n = 4\). [3]
  3. Deduce the value of \(\lim_{n \to \infty} \left(\frac{u_n}{u_{n-2}}\right)\). [1]
OCR MEI Further Extra Pure Specimen Q4
16 marks Challenging +1.8
A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).
  1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
  2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
  3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
  4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]
The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).
  1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]
OCR MEI Further Extra Pure Specimen Q5
18 marks Challenging +1.8
In this question you must show detailed reasoning. You are given that the matrix $\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix}$ represents a rotation in 3-D space.
  1. Explain why it follows that \(\mathbf{M}\) has 1 as an eigenvalue. [2]
  2. Find a vector equation for the axis of the rotation. [4]
  3. Show that the characteristic equation of \(\mathbf{M}\) can be written as $$\lambda^3 - \lambda^2 + \lambda - 1 = 0.$$ [5]
  4. Find the smallest positive integer \(n\) such that \(\mathbf{M}^n = \mathbf{I}\). [6]
  5. Find the magnitude of the angle of the rotation which \(\mathbf{M}\) represents. Give your reasoning. [1]
OCR MEI FP2 2013 June Q1
Standard +0.3
1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { l } x
y
z \end{array} \right) = \left( \begin{array} { c } p