OCR Further Pure Core 2 (Further Pure Core 2) 2022 June

1

1. Find a vector which is perpendicular to both \(3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\). The equations of two lines are \(\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } + 11 \mathbf { j } - 4 \mathbf { k } + \mu ( - \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\).
2. Show that the lines intersect, stating the point of intersection.

2 Two polar curves, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by \(C _ { 1 } : r = 2 \theta\) and \(C _ { 2 } : r = \theta + 1\) where \(0 \leqslant \theta \leqslant 2 \pi\). \(C _ { 1 }\) intersects the initial line at two points, the pole and the point \(A\).

1. Write down the polar coordinates of \(A\).
2. Determine the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). The diagram below shows a sketch of \(C _ { 1 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
3. On the copy of this sketch in the Printed Answer Booklet, sketch \(C _ { 2 }\).

3 In this question you must show detailed reasoning. The roots of the equation \(4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

4 In this question you must show detailed reasoning.
Determine the smallest value of \(n\) for which \(\frac { 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } } { 1 + 2 + \ldots + n } > 341\).

5

1. By using the exponential definitions of \(\sinh x\) and \(\cosh x\), prove the identity \(\cosh 2 x \equiv \cosh ^ { 2 } x + \sinh ^ { 2 } x\).
2. Hence find an expression for \(\cosh 2 x\) in terms of \(\cosh x\).
3. Determine the solutions of the equation \(5 \cosh 2 x = 16 \cosh x + 21\), giving your answers in exact logarithmic form.

6 A particle, \(P\), positioned at the origin, \(O\), is projected with a certain velocity along the \(x\)-axis. \(P\) is then acted on by a single force which varies in such a way that \(P\) moves backwards and forwards along the \(x\)-axis. When the time after projection is \(t\) seconds, the displacement of \(P\) from the origin is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~ms} ^ { - 1 }\). The motion of \(P\) is modelled using the differential equation \(\ddot { x } + \omega ^ { 2 } x = 0\), where \(\omega\) rads \(^ { - 1 }\) is a positive constant.

1. Write down the general solution of this differential equation.
   \(D\) is the point where \(x = d\) for some positive constant, \(d\). When \(P\) reaches \(D\) it comes to instantaneous rest.
2. Using the answer to part (a), determine expressions, in terms of \(\omega\), \(d\) and \(t\) only, for the following quantities
   • \(X\)
   • \(v\)
   • Hence show that, according to the model, \(v ^ { 2 } = \omega ^ { 2 } \left( d ^ { 2 } - x ^ { 2 } \right)\).
   The quantity \(z\) is defined by \(z = \frac { 1 } { v }\).
3. Using part (c), determine an expression for \(\mathrm { Z } _ { \mathrm { m } }\), the mean value of z with respect to the displacement, as \(P\) moves directly from \(O\) to \(D\). One measure of the validity of the model is consideration of the value of \(\mathrm { z } _ { \mathrm { m } }\). If \(\mathrm { z } _ { \mathrm { m } }\) exceeds 8 then the model is considered to be valid. The value of \(d\) is measured as 0.25 to 2 significant figures. The value of \(\omega\) is measured as \(0.75 \pm 0.02\).
4. Determine what can be inferred about the validity of the model from the given information.
5. Find, according to the model, the least possible value of the velocity with which \(P\) was initially projected. Give your answer to \(\mathbf { 2 }\) significant figures.

7 You are given that \(a\) is a parameter which can take only real values.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c r } 2 & 4 & - 6
- 3 & 10 - 4 a & 9
7 & 4 & 4 \end{array} \right)\).

1. Find an expression for the determinant of \(\mathbf { A }\) in terms of \(a\). You are given the following system of equations in \(x , y\) and \(z\). $$\begin{array} { r r } 2 x + & 4 y - 6 z =
   - 3 x + & ( 10 - 4 a ) y + 9 z =
   7 x + & 4 y + 4 z =
   7 x + & 11 \end{array}$$ The system can be written in the form \(\mathbf { A } \left( \begin{array} { c } \mathrm { x }
   \mathrm { y }
   \mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6
   - 9
   11 \end{array} \right)\).
2. 1. In the case where \(\mathbf { A }\) is not singular, solve the given system of equations by using \(\mathbf { A } ^ { - 1 }\).
   2. In the case where \(\mathbf { A }\) is singular describe the configuration of the planes whose equations are the three equations of the system. The transformation represented by \(\mathbf { A }\) is denoted by T .
      A 3-D object of volume \(| 5 a - 20 |\) is transformed by T to a 3-D image.
3. 1. Determine the range of values of \(a\) for which the orientation of the image is the reverse of the orientation of the object.
   2. Determine the range of values of \(a\) for which the volume of the image is less than the volume of the object.

1. Show that \(\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta\), where \(a\) is an integer to be determined.
2. Hence show that \(\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }\), where \(b\) and \(c\) are integers to be determined.

9 In this question you must show detailed reasoning.

1. Show that \(\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta\), where \(a\) is an integer to be determined.
2. Hence show that \(\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }\), where \(b\) and \(c\) are integers to be determined.

10 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2 , - 1\) ) and ( \(13,10,9\) ) respectively.

• The plane \(\Pi _ { A }\) contains \(A\) and the plane \(\Pi _ { B }\) contains \(B\).
• The planes \(\Pi _ { A }\) and \(\Pi _ { B }\) are parallel.
• The \(x\) and \(y\) components of any normal to plane \(\Pi _ { A }\) are equal.
• The shortest distance between \(\Pi _ { A }\) and \(\Pi _ { B }\) is 2 .

There are two possible solution planes for \(\Pi _ { A }\) which satisfy the above conditions.
Determine the acute angle between these two possible solution planes.